Numerical Measures Numerical Measures Measures of Central Tendency
![Numerical Measures Numerical Measures](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-1.jpg)
![Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-2.jpg)
![Measures of Central Tendency (Location) • Mean • Median • Mode Central Location Measures of Central Tendency (Location) • Mean • Median • Mode Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-3.jpg)
![Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-4.jpg)
![Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-5.jpg)
![Measures of Shape • Skewness • Kurtosis Measures of Shape • Skewness • Kurtosis](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-6.jpg)
![Summation Notation Summation Notation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-7.jpg)
![Summation Notation Let x 1, x 2, x 3, … xn denote a set Summation Notation Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-8.jpg)
![Example Let x 1, x 2, x 3, x 4, x 5 denote a Example Let x 1, x 2, x 3, x 4, x 5 denote a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-9.jpg)
![Then the symbol denotes the sum of these 5 numbers x 1 + x Then the symbol denotes the sum of these 5 numbers x 1 + x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-10.jpg)
![Meaning of parts of summation notation Final value for i each term of the Meaning of parts of summation notation Final value for i each term of the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-11.jpg)
![Example Again let x 1, x 2, x 3, x 4, x 5 denote Example Again let x 1, x 2, x 3, x 4, x 5 denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-12.jpg)
![Then the symbol denotes the sum of these 3 numbers = 153 + 213 Then the symbol denotes the sum of these 3 numbers = 153 + 213](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-13.jpg)
![Measures of Central Location (Mean) Measures of Central Location (Mean)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-14.jpg)
![Mean Let x 1, x 2, x 3, … xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-15.jpg)
![Example Again let x 1, x 2, x 3, x 4, x 5 denote Example Again let x 1, x 2, x 3, x 4, x 5 denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-16.jpg)
![Then the mean of the 5 numbers is: Then the mean of the 5 numbers is:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-17.jpg)
![Interpretation of the Mean Let x 1, x 2, x 3, … xn denote Interpretation of the Mean Let x 1, x 2, x 3, … xn denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-18.jpg)
![x 1 x 3 x 4 x 2 xn x 1 x 3 x 4 x 2 xn](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-19.jpg)
![In the Example 7 0 10 10 13 21 15 20 In the Example 7 0 10 10 13 21 15 20](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-20.jpg)
![The mean, , is also approximately the center of gravity of a histogram The mean, , is also approximately the center of gravity of a histogram](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-21.jpg)
![Measures of Central Location (Median) Measures of Central Location (Median)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-22.jpg)
![The Median Let x 1, x 2, x 3, … xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-23.jpg)
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-24.jpg)
![Example Again let x 1, x 2, x 3 , x 4, x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-25.jpg)
![The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-26.jpg)
![Example 2 Let x 1, x 2, x 3 , x 4, x 5 Example 2 Let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-27.jpg)
![Median = average of two “middle” observations = Median = average of two “middle” observations =](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-28.jpg)
![Example The data on N = 23 students Variables • Verbal IQ • Math Example The data on N = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-29.jpg)
![Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-30.jpg)
![Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-31.jpg)
![Summary Summary](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-32.jpg)
![Numerical Measures Numerical Measures](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-33.jpg)
![Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-34.jpg)
![Measures of Central Tendency (Location) • Mean • Median • Mode Central Location Measures of Central Tendency (Location) • Mean • Median • Mode Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-35.jpg)
![Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-36.jpg)
![Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-37.jpg)
![Measures of Shape • Skewness • Kurtosis Measures of Shape • Skewness • Kurtosis](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-38.jpg)
![Measures of Central Location Mean Median Measures of Central Location Mean Median](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-39.jpg)
![Mean Let x 1, x 2, x 3, … xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-40.jpg)
![The Median Let x 1, x 2, x 3, … xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-41.jpg)
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-42.jpg)
![Some Comments • The mean is the centre of gravity of a set of Some Comments • The mean is the centre of gravity of a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-43.jpg)
![• The median splits the area under a histogram in two parts of • The median splits the area under a histogram in two parts of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-44.jpg)
![• For symmetric distributions the mean and the median will be approximately the • For symmetric distributions the mean and the median will be approximately the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-45.jpg)
![• For Positively skewed distributions the mean exceeds the median • For Negatively • For Positively skewed distributions the mean exceeds the median • For Negatively](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-46.jpg)
![• An outlier is a “wild” observation in the data • Outliers occur • An outlier is a “wild” observation in the data • Outliers occur](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-47.jpg)
![• The mean is altered to a significant degree by the presence of • The mean is altered to a significant degree by the presence of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-48.jpg)
![Review Review](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-49.jpg)
![Summarizing Data Graphical Methods Summarizing Data Graphical Methods](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-50.jpg)
![Histogram Stem-Leaf Diagram Grouped Freq Table Histogram Stem-Leaf Diagram Grouped Freq Table](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-51.jpg)
![Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-52.jpg)
![Summation Notation Final value for i each term of the sum Quantity changing in Summation Notation Final value for i each term of the sum Quantity changing in](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-53.jpg)
![Example Let x 1, x 2, x 3, x 4, x 5 denote a Example Let x 1, x 2, x 3, x 4, x 5 denote a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-54.jpg)
![Then the symbol denotes the sum of these 3 numbers = 153 + 213 Then the symbol denotes the sum of these 3 numbers = 153 + 213](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-55.jpg)
![Then the symbol denotes the sum of these 5 numbers x 1 + x Then the symbol denotes the sum of these 5 numbers x 1 + x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-56.jpg)
![Measures of Central Location (Mean) Measures of Central Location (Mean)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-57.jpg)
![Mean Let x 1, x 2, x 3, … xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-58.jpg)
![Example Again let x 1, x 2, x 3 , x 4, x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-59.jpg)
![Then the mean of the 5 numbers is: Then the mean of the 5 numbers is:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-60.jpg)
![Interpretation of the Mean Let x 1, x 2, x 3, … xn denote Interpretation of the Mean Let x 1, x 2, x 3, … xn denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-61.jpg)
![x 1 x 3 x 4 x 2 xn x 1 x 3 x 4 x 2 xn](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-62.jpg)
![In the Example 7 0 10 10 13 21 15 20 In the Example 7 0 10 10 13 21 15 20](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-63.jpg)
![The mean, , is also approximately the center of gravity of a histogram The mean, , is also approximately the center of gravity of a histogram](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-64.jpg)
![The Median Let x 1, x 2, x 3, … xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-65.jpg)
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-66.jpg)
![Example Again let x 1, x 2, x 3 , x 4, x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-67.jpg)
![The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-68.jpg)
![Example 2 Let x 1, x 2, x 3 , x 4, x 5 Example 2 Let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-69.jpg)
![Median = average of two “middle” observations = Median = average of two “middle” observations =](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-70.jpg)
![Example The data on N = 23 students Variables • Verbal IQ • Math Example The data on N = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-71.jpg)
![Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-72.jpg)
![Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-73.jpg)
![Summary Summary](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-74.jpg)
![Some Comments • The mean is the centre of gravity of a set of Some Comments • The mean is the centre of gravity of a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-75.jpg)
![• The median splits the area under a histogram in two parts of • The median splits the area under a histogram in two parts of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-76.jpg)
![• For symmetric distributions the mean and the median will be approximately the • For symmetric distributions the mean and the median will be approximately the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-77.jpg)
![• For Positively skewed distributions the mean exceeds the median • For Negatively • For Positively skewed distributions the mean exceeds the median • For Negatively](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-78.jpg)
![• An outlier is a “wild” observation in the data • Outliers occur • An outlier is a “wild” observation in the data • Outliers occur](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-79.jpg)
![• The mean is altered to a significant degree by the presence of • The mean is altered to a significant degree by the presence of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-80.jpg)
![Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges) Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-81.jpg)
![Definition The P× 100 Percentile is a point , x. P , underneath a Definition The P× 100 Percentile is a point , x. P , underneath a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-82.jpg)
![Definition (Quartiles) The first Quartile , Q 1 , is the 25 Percentile , Definition (Quartiles) The first Quartile , Q 1 , is the 25 Percentile ,](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-83.jpg)
![The second Quartile , Q 2 , is the 50 th Percentile , x The second Quartile , Q 2 , is the 50 th Percentile , x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-84.jpg)
![• The second Quartile , Q 2 , is also the median and • The second Quartile , Q 2 , is also the median and](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-85.jpg)
![The third Quartile , Q 3 , is the 75 th Percentile , x The third Quartile , Q 3 , is the 75 th Percentile , x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-86.jpg)
![The Quartiles – Q 1, Q 2, Q 3 divide the population into 4 The Quartiles – Q 1, Q 2, Q 3 divide the population into 4](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-87.jpg)
![Computing Percentiles and Quartiles – Method 1 • The first step is to order Computing Percentiles and Quartiles – Method 1 • The first step is to order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-88.jpg)
![Example The data on n = 23 students Variables • Verbal IQ • Math Example The data on n = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-89.jpg)
![The position, k, of the 75 th Percentile. k = P × (n+1) =. The position, k, of the 75 th Percentile. k = P × (n+1) =.](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-90.jpg)
![When the position k is an not an integer but an integer(m) + a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-91.jpg)
![When the position k is an not an integer but an integer(m) + a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-92.jpg)
![When the position k is an not an integer but an integer(m) + a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-93.jpg)
![Example The data Verbal IQ on n = 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-94.jpg)
![x 0. 75 = 75 th percentile = 18 th observation in size =105 x 0. 75 = 75 th percentile = 18 th observation in size =105](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-95.jpg)
![An Alternative method for computing Quartiles – Method 2 • Sometimes this method will An Alternative method for computing Quartiles – Method 2 • Sometimes this method will](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-96.jpg)
![Let x 1, x 2, x 3, … xn denote a set of n Let x 1, x 2, x 3, … xn denote a set of n](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-97.jpg)
![Example Consider the 5 numbers: 10 15 21 7 13 Arranged in increasing order: Example Consider the 5 numbers: 10 15 21 7 13 Arranged in increasing order:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-98.jpg)
![The lower mid-hinge (the first quartile) is the “median” of the lower half of The lower mid-hinge (the first quartile) is the “median” of the lower half of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-99.jpg)
![Consider the five number in increasing order: Lower Half 7 Upper Half 10 13 Consider the five number in increasing order: Lower Half 7 Upper Half 10 13](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-100.jpg)
![Computing the median and the quartile using the first method: Position of the median: Computing the median and the quartile using the first method: Position of the median:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-101.jpg)
![• Both methods result in the same value • This is not always • Both methods result in the same value • This is not always](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-102.jpg)
![Example The data Verbal IQ on n = 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-103.jpg)
![Computing the median and the quartile using the first method: Position of the median: Computing the median and the quartile using the first method: Position of the median:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-104.jpg)
![• Many programs compute percentiles, quartiles etc. • Each may use different methods. • Many programs compute percentiles, quartiles etc. • Each may use different methods.](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-105.jpg)
![Measures of Central Location Mean Median Measures of Central Location Mean Median](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-106.jpg)
![Mean Let x 1, x 2, x 3, … xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-107.jpg)
![The Median Let x 1, x 2, x 3, … xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-108.jpg)
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-109.jpg)
![Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges) Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-110.jpg)
![Definition The P× 100 Percentile is a point , x. P , underneath a Definition The P× 100 Percentile is a point , x. P , underneath a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-111.jpg)
![Computing Percentiles and Quartiles – Method 1 • The first step is to order Computing Percentiles and Quartiles – Method 1 • The first step is to order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-112.jpg)
![When the position k is an integer the percentile is the kth observation (in When the position k is an integer the percentile is the kth observation (in](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-113.jpg)
![An Alternative method for computing Quartiles – Method 2 • Sometimes this method will An Alternative method for computing Quartiles – Method 2 • Sometimes this method will](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-114.jpg)
![Let x 1, x 2, x 3, … xn denote a set of n Let x 1, x 2, x 3, … xn denote a set of n](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-115.jpg)
![The lower mid-hinge (the first quartile) is the “median” of the lower half of The lower mid-hinge (the first quartile) is the “median” of the lower half of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-116.jpg)
![Box-Plots Box-Whisker Plots • A graphical method of of displaying data • An alternative Box-Plots Box-Whisker Plots • A graphical method of of displaying data • An alternative](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-117.jpg)
![To Draw a Box Plot • Compute the Hinge (Median, Q 2) and the To Draw a Box Plot • Compute the Hinge (Median, Q 2) and the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-118.jpg)
![Example The data Verbal IQ on n = 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-119.jpg)
![The Box Plot is then drawn • Drawing above an axis a “box” from The Box Plot is then drawn • Drawing above an axis a “box” from](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-120.jpg)
![Lower Whisker min Upper Whisker Box Q 1 Q 2 Q 3 max Lower Whisker min Upper Whisker Box Q 1 Q 2 Q 3 max](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-121.jpg)
![Example The data Verbal IQ on n = 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-122.jpg)
![Box Plot of Verbal IQ 70 80 90 100 110 120 130 Box Plot of Verbal IQ 70 80 90 100 110 120 130](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-123.jpg)
![130 120 110 100 90 80 70 Box Plot can also be drawn vertically 130 120 110 100 90 80 70 Box Plot can also be drawn vertically](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-124.jpg)
![Box-Whisker plots (Verbal IQ, Math IQ) Box-Whisker plots (Verbal IQ, Math IQ)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-125.jpg)
![Box-Whisker plots (Initial RA, Final RA ) Box-Whisker plots (Initial RA, Final RA )](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-126.jpg)
![Summary Information contained in the box plot 25% 25% Middle 50% of population 25% Summary Information contained in the box plot 25% 25% Middle 50% of population 25%](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-127.jpg)
![Next topic: Numerical Measures of Variability Next topic: Numerical Measures of Variability](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-128.jpg)
- Slides: 128
![Numerical Measures Numerical Measures](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-1.jpg)
Numerical Measures
![Numerical Measures Measures of Central Tendency Location Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-2.jpg)
Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Measure of Variability (Dispersion, Spread) Measures of Shape
![Measures of Central Tendency Location Mean Median Mode Central Location Measures of Central Tendency (Location) • Mean • Median • Mode Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-3.jpg)
Measures of Central Tendency (Location) • Mean • Median • Mode Central Location
![Measures of Noncentral Location Quartiles MidHinges Percentiles Non Central Location Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-4.jpg)
Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location
![Measure of Variability Dispersion Spread Variance standard deviation Range InterQuartile Range Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-5.jpg)
Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range Variability
![Measures of Shape Skewness Kurtosis Measures of Shape • Skewness • Kurtosis](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-6.jpg)
Measures of Shape • Skewness • Kurtosis
![Summation Notation Summation Notation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-7.jpg)
Summation Notation
![Summation Notation Let x 1 x 2 x 3 xn denote a set Summation Notation Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-8.jpg)
Summation Notation Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the symbol denotes the sum of these n numbers x 1 + x 2 + x 3 + …+ xn
![Example Let x 1 x 2 x 3 x 4 x 5 denote a Example Let x 1, x 2, x 3, x 4, x 5 denote a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-9.jpg)
Example Let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![Then the symbol denotes the sum of these 5 numbers x 1 x Then the symbol denotes the sum of these 5 numbers x 1 + x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-10.jpg)
Then the symbol denotes the sum of these 5 numbers x 1 + x 2 + x 3 + x 4 + x 5 = 10 + 15 + 21 + 7 + 13 = 66
![Meaning of parts of summation notation Final value for i each term of the Meaning of parts of summation notation Final value for i each term of the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-11.jpg)
Meaning of parts of summation notation Final value for i each term of the sum Quantity changing in each term of the sum Starting value for i
![Example Again let x 1 x 2 x 3 x 4 x 5 denote Example Again let x 1, x 2, x 3, x 4, x 5 denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-12.jpg)
Example Again let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![Then the symbol denotes the sum of these 3 numbers 153 213 Then the symbol denotes the sum of these 3 numbers = 153 + 213](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-13.jpg)
Then the symbol denotes the sum of these 3 numbers = 153 + 213 + 73 = 3375 + 9261 + 343 = 12979
![Measures of Central Location Mean Measures of Central Location (Mean)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-14.jpg)
Measures of Central Location (Mean)
![Mean Let x 1 x 2 x 3 xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-15.jpg)
Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean of the n numbers is defined as:
![Example Again let x 1 x 2 x 3 x 4 x 5 denote Example Again let x 1, x 2, x 3, x 4, x 5 denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-16.jpg)
Example Again let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![Then the mean of the 5 numbers is Then the mean of the 5 numbers is:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-17.jpg)
Then the mean of the 5 numbers is:
![Interpretation of the Mean Let x 1 x 2 x 3 xn denote Interpretation of the Mean Let x 1, x 2, x 3, … xn denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-18.jpg)
Interpretation of the Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean, , is the centre of gravity of those the n numbers. That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point.
![x 1 x 3 x 4 x 2 xn x 1 x 3 x 4 x 2 xn](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-19.jpg)
x 1 x 3 x 4 x 2 xn
![In the Example 7 0 10 10 13 21 15 20 In the Example 7 0 10 10 13 21 15 20](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-20.jpg)
In the Example 7 0 10 10 13 21 15 20
![The mean is also approximately the center of gravity of a histogram The mean, , is also approximately the center of gravity of a histogram](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-21.jpg)
The mean, , is also approximately the center of gravity of a histogram
![Measures of Central Location Median Measures of Central Location (Median)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-22.jpg)
Measures of Central Location (Median)
![The Median Let x 1 x 2 x 3 xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-23.jpg)
The Median Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the median of the n numbers is defined as the number that splits the numbers into two equal parts. To evaluate the median we arrange the numbers in increasing order.
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-24.jpg)
If the number of observations is odd there will be one observation in the middle. This number is the median. If the number of observations is even there will be two middle observations. The median is the average of these two observations
![Example Again let x 1 x 2 x 3 x 4 x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-25.jpg)
Example Again let x 1, x 2, x 3 , x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![The numbers arranged in order are 7 10 13 15 21 Unique Middle observation The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-26.jpg)
The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation – the median
![Example 2 Let x 1 x 2 x 3 x 4 x 5 Example 2 Let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-27.jpg)
Example 2 Let x 1, x 2, x 3 , x 4, x 5 , x 6 denote the 6 denote numbers: 23 41 12 19 64 8 Arranged in increasing order these observations would be: 8 12 19 23 41 64 Two “Middle” observations
![Median average of two middle observations Median = average of two “middle” observations =](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-28.jpg)
Median = average of two “middle” observations =
![Example The data on N 23 students Variables Verbal IQ Math Example The data on N = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-29.jpg)
Example The data on N = 23 students Variables • Verbal IQ • Math IQ • Initial Reading Achievement Score • Final Reading Achievement Score
![Data Set 3 The following table gives data on Verbal IQ Math IQ Initial Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-30.jpg)
Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program Student Verbal IQ Math IQ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 86 104 86 105 118 96 90 95 105 84 94 119 82 80 109 111 89 99 94 99 95 102 94 103 92 100 115 102 87 100 96 80 87 116 91 93 124 119 94 117 93 110 97 104 93 Initial Reading Acheivement 1. 1 1. 5 2. 0 1. 9 1. 4 1. 5 1. 4 1. 7 1. 6 1. 7 1. 2 1. 0 1. 8 1. 4 1. 6 1. 4 1. 5 1. 7 1. 6 Final Reading Acheivement 1. 7 1. 9 2. 0 3. 5 2. 4 1. 8 2. 0 1. 7 3. 1 1. 8 1. 7 2. 5 3. 0 1. 8 2. 6 1. 4 2. 0 1. 3 3. 1 1. 9
![Computing the Median Stem leaf Diagrams Median middle observation 12 th observation Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-31.jpg)
Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation
![Summary Summary](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-32.jpg)
Summary
![Numerical Measures Numerical Measures](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-33.jpg)
Numerical Measures
![Numerical Measures Measures of Central Tendency Location Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-34.jpg)
Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Measure of Variability (Dispersion, Spread) Measures of Shape
![Measures of Central Tendency Location Mean Median Mode Central Location Measures of Central Tendency (Location) • Mean • Median • Mode Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-35.jpg)
Measures of Central Tendency (Location) • Mean • Median • Mode Central Location
![Measures of Noncentral Location Quartiles MidHinges Percentiles Non Central Location Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-36.jpg)
Measures of Non-central Location • Quartiles, Mid-Hinges • Percentiles Non - Central Location
![Measure of Variability Dispersion Spread Variance standard deviation Range InterQuartile Range Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-37.jpg)
Measure of Variability (Dispersion, Spread) • Variance, standard deviation • Range • Inter-Quartile Range Variability
![Measures of Shape Skewness Kurtosis Measures of Shape • Skewness • Kurtosis](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-38.jpg)
Measures of Shape • Skewness • Kurtosis
![Measures of Central Location Mean Median Measures of Central Location Mean Median](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-39.jpg)
Measures of Central Location Mean Median
![Mean Let x 1 x 2 x 3 xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-40.jpg)
Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean of the n numbers is defined as:
![The Median Let x 1 x 2 x 3 xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-41.jpg)
The Median Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the median of the n numbers is defined as the number that splits the numbers into two equal parts. To evaluate the median we arrange the numbers in increasing order.
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-42.jpg)
If the number of observations is odd there will be one observation in the middle. This number is the median. If the number of observations is even there will be two middle observations. The median is the average of these two observations
![Some Comments The mean is the centre of gravity of a set of Some Comments • The mean is the centre of gravity of a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-43.jpg)
Some Comments • The mean is the centre of gravity of a set of observations. The balancing point. • The median splits the obsevations equally in two parts of approximately 50%
![The median splits the area under a histogram in two parts of • The median splits the area under a histogram in two parts of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-44.jpg)
• The median splits the area under a histogram in two parts of 50% • The mean is the balancing point of a histogram 50% median
![For symmetric distributions the mean and the median will be approximately the • For symmetric distributions the mean and the median will be approximately the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-45.jpg)
• For symmetric distributions the mean and the median will be approximately the same value 50% Median &
![For Positively skewed distributions the mean exceeds the median For Negatively • For Positively skewed distributions the mean exceeds the median • For Negatively](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-46.jpg)
• For Positively skewed distributions the mean exceeds the median • For Negatively skewed distributions the median exceeds the mean 50% median
![An outlier is a wild observation in the data Outliers occur • An outlier is a “wild” observation in the data • Outliers occur](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-47.jpg)
• An outlier is a “wild” observation in the data • Outliers occur because – of errors (typographical and computational) – Extreme cases in the population
![The mean is altered to a significant degree by the presence of • The mean is altered to a significant degree by the presence of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-48.jpg)
• The mean is altered to a significant degree by the presence of outliers • Outliers have little effect on the value of the median • This is a reason for using the median in place of the mean as a measure of central location • Alternatively the mean is the best measure of central location when the data is Normally distributed (Bell-shaped)
![Review Review](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-49.jpg)
Review
![Summarizing Data Graphical Methods Summarizing Data Graphical Methods](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-50.jpg)
Summarizing Data Graphical Methods
![Histogram StemLeaf Diagram Grouped Freq Table Histogram Stem-Leaf Diagram Grouped Freq Table](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-51.jpg)
Histogram Stem-Leaf Diagram Grouped Freq Table
![Numerical Measures Measures of Central Tendency Location Measures of Non Central Location Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-52.jpg)
Numerical Measures • • Measures of Central Tendency (Location) Measures of Non Central Location Measure of Variability (Dispersion, Spread) Measures of Shape The objective is to reduce the data to a small number of values that completely describe the data and certain aspects of the data.
![Summation Notation Final value for i each term of the sum Quantity changing in Summation Notation Final value for i each term of the sum Quantity changing in](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-53.jpg)
Summation Notation Final value for i each term of the sum Quantity changing in each term of the sum Starting value for i
![Example Let x 1 x 2 x 3 x 4 x 5 denote a Example Let x 1, x 2, x 3, x 4, x 5 denote a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-54.jpg)
Example Let x 1, x 2, x 3, x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![Then the symbol denotes the sum of these 3 numbers 153 213 Then the symbol denotes the sum of these 3 numbers = 153 + 213](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-55.jpg)
Then the symbol denotes the sum of these 3 numbers = 153 + 213 + 73 = 3375 + 9261 + 343 = 12979
![Then the symbol denotes the sum of these 5 numbers x 1 x Then the symbol denotes the sum of these 5 numbers x 1 + x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-56.jpg)
Then the symbol denotes the sum of these 5 numbers x 1 + x 2 + x 3 + x 4 + x 5 = 10 + 15 + 21 + 7 + 13 = 66
![Measures of Central Location Mean Measures of Central Location (Mean)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-57.jpg)
Measures of Central Location (Mean)
![Mean Let x 1 x 2 x 3 xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-58.jpg)
Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean of the n numbers is defined as:
![Example Again let x 1 x 2 x 3 x 4 x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-59.jpg)
Example Again let x 1, x 2, x 3 , x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![Then the mean of the 5 numbers is Then the mean of the 5 numbers is:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-60.jpg)
Then the mean of the 5 numbers is:
![Interpretation of the Mean Let x 1 x 2 x 3 xn denote Interpretation of the Mean Let x 1, x 2, x 3, … xn denote](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-61.jpg)
Interpretation of the Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean, , is the centre of gravity of those the n numbers. That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point.
![x 1 x 3 x 4 x 2 xn x 1 x 3 x 4 x 2 xn](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-62.jpg)
x 1 x 3 x 4 x 2 xn
![In the Example 7 0 10 10 13 21 15 20 In the Example 7 0 10 10 13 21 15 20](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-63.jpg)
In the Example 7 0 10 10 13 21 15 20
![The mean is also approximately the center of gravity of a histogram The mean, , is also approximately the center of gravity of a histogram](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-64.jpg)
The mean, , is also approximately the center of gravity of a histogram
![The Median Let x 1 x 2 x 3 xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-65.jpg)
The Median Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the median of the n numbers is defined as the number that splits the numbers into two equal parts. To evaluate the median we arrange the numbers in increasing order.
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-66.jpg)
If the number of observations is odd there will be one observation in the middle. This number is the median. If the number of observations is even there will be two middle observations. The median is the average of these two observations
![Example Again let x 1 x 2 x 3 x 4 x 5 Example Again let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-67.jpg)
Example Again let x 1, x 2, x 3 , x 4, x 5 denote a set of 5 denote the set of numbers in the following table. i 1 2 3 4 5 xi 10 15 21 7 13
![The numbers arranged in order are 7 10 13 15 21 Unique Middle observation The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-68.jpg)
The numbers arranged in order are: 7 10 13 15 21 Unique “Middle” observation – the median
![Example 2 Let x 1 x 2 x 3 x 4 x 5 Example 2 Let x 1, x 2, x 3 , x 4, x 5](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-69.jpg)
Example 2 Let x 1, x 2, x 3 , x 4, x 5 , x 6 denote the 6 denote numbers: 23 41 12 19 64 8 Arranged in increasing order these observations would be: 8 12 19 23 41 64 Two “Middle” observations
![Median average of two middle observations Median = average of two “middle” observations =](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-70.jpg)
Median = average of two “middle” observations =
![Example The data on N 23 students Variables Verbal IQ Math Example The data on N = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-71.jpg)
Example The data on N = 23 students Variables • Verbal IQ • Math IQ • Initial Reading Achievement Score • Final Reading Achievement Score
![Data Set 3 The following table gives data on Verbal IQ Math IQ Initial Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-72.jpg)
Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program Student Verbal IQ Math IQ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 86 104 86 105 118 96 90 95 105 84 94 119 82 80 109 111 89 99 94 99 95 102 94 103 92 100 115 102 87 100 96 80 87 116 91 93 124 119 94 117 93 110 97 104 93 Initial Reading Acheivement 1. 1 1. 5 2. 0 1. 9 1. 4 1. 5 1. 4 1. 7 1. 6 1. 7 1. 2 1. 0 1. 8 1. 4 1. 6 1. 4 1. 5 1. 7 1. 6 Final Reading Acheivement 1. 7 1. 9 2. 0 3. 5 2. 4 1. 8 2. 0 1. 7 3. 1 1. 8 1. 7 2. 5 3. 0 1. 8 2. 6 1. 4 2. 0 1. 3 3. 1 1. 9
![Computing the Median Stem leaf Diagrams Median middle observation 12 th observation Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-73.jpg)
Computing the Median Stem leaf Diagrams Median = middle observation =12 th observation
![Summary Summary](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-74.jpg)
Summary
![Some Comments The mean is the centre of gravity of a set of Some Comments • The mean is the centre of gravity of a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-75.jpg)
Some Comments • The mean is the centre of gravity of a set of observations. The balancing point. • The median splits the observations equally in two parts of approximately 50%
![The median splits the area under a histogram in two parts of • The median splits the area under a histogram in two parts of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-76.jpg)
• The median splits the area under a histogram in two parts of 50% • The mean is the balancing point of a histogram 50% median
![For symmetric distributions the mean and the median will be approximately the • For symmetric distributions the mean and the median will be approximately the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-77.jpg)
• For symmetric distributions the mean and the median will be approximately the same value 50% Median &
![For Positively skewed distributions the mean exceeds the median For Negatively • For Positively skewed distributions the mean exceeds the median • For Negatively](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-78.jpg)
• For Positively skewed distributions the mean exceeds the median • For Negatively skewed distributions the median exceeds the mean 50% median
![An outlier is a wild observation in the data Outliers occur • An outlier is a “wild” observation in the data • Outliers occur](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-79.jpg)
• An outlier is a “wild” observation in the data • Outliers occur because – of errors (typographical and computational) – Extreme cases in the population
![The mean is altered to a significant degree by the presence of • The mean is altered to a significant degree by the presence of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-80.jpg)
• The mean is altered to a significant degree by the presence of outliers • Outliers have little effect on the value of the median • This is a reason for using the median in place of the mean as a measure of central location • Alternatively the mean is the best measure of central location when the data is Normally distributed (Bell-shaped)
![Measures of NonCentral Location Percentiles Quartiles Hinges Midhinges Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-81.jpg)
Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)
![Definition The P 100 Percentile is a point x P underneath a Definition The P× 100 Percentile is a point , x. P , underneath a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-82.jpg)
Definition The P× 100 Percentile is a point , x. P , underneath a distribution that has a fixed proportion P of the population (or sample) below that value P× 100 % x. P
![Definition Quartiles The first Quartile Q 1 is the 25 Percentile Definition (Quartiles) The first Quartile , Q 1 , is the 25 Percentile ,](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-83.jpg)
Definition (Quartiles) The first Quartile , Q 1 , is the 25 Percentile , x 0. 25 25 % x 0. 25
![The second Quartile Q 2 is the 50 th Percentile x The second Quartile , Q 2 , is the 50 th Percentile , x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-84.jpg)
The second Quartile , Q 2 , is the 50 th Percentile , x 0. 50 50 % x 0. 50
![The second Quartile Q 2 is also the median and • The second Quartile , Q 2 , is also the median and](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-85.jpg)
• The second Quartile , Q 2 , is also the median and the 50 th percentile
![The third Quartile Q 3 is the 75 th Percentile x The third Quartile , Q 3 , is the 75 th Percentile , x](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-86.jpg)
The third Quartile , Q 3 , is the 75 th Percentile , x 0. 75 75 % x 0. 75
![The Quartiles Q 1 Q 2 Q 3 divide the population into 4 The Quartiles – Q 1, Q 2, Q 3 divide the population into 4](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-87.jpg)
The Quartiles – Q 1, Q 2, Q 3 divide the population into 4 equal parts of 25%. 25 % Q 1 25 % Q 2 Q 3
![Computing Percentiles and Quartiles Method 1 The first step is to order Computing Percentiles and Quartiles – Method 1 • The first step is to order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-88.jpg)
Computing Percentiles and Quartiles – Method 1 • The first step is to order the observations in increasing order. • We then compute the position, k, of the P× 100 Percentile. k = P × (n+1) Where n = the number of observations
![Example The data on n 23 students Variables Verbal IQ Math Example The data on n = 23 students Variables • Verbal IQ • Math](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-89.jpg)
Example The data on n = 23 students Variables • Verbal IQ • Math IQ • Initial Reading Achievement Score • Final Reading Achievement Score We want to compute the 75 th percentile and the 90 th percentile
![The position k of the 75 th Percentile k P n1 The position, k, of the 75 th Percentile. k = P × (n+1) =.](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-90.jpg)
The position, k, of the 75 th Percentile. k = P × (n+1) =. 75 × (23+1) = 18 The position, k, of the 90 th Percentile. k = P × (n+1) =. 90 × (23+1) = 21. 6 When the position k is an integer the percentile is the kth observation (in order of magnitude) in the data set. For example the 75 th percentile is the 18 th (in size) observation
![When the position k is an not an integer but an integerm a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-91.jpg)
When the position k is an not an integer but an integer(m) + a fraction(f). i. e. k=m+f then the percentile is x. P = (1 -f) × (mth observation in size) + f × (m+1 st observation in size) In the example the position of the 90 th percentile is: k = 21. 6 Then x. 90 = 0. 4(21 st observation in size) + 0. 6(22 nd observation in size)
![When the position k is an not an integer but an integerm a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-92.jpg)
When the position k is an not an integer but an integer(m) + a fraction(f). i. e. k=m+f then the percentile is x. P = (1 -f) × (mth observation in size) + f × (m+1 st observation in size) mth obs (m+1)st obs xp = (1 - f) ( mth obs) + f [(m+1)st obs]
![When the position k is an not an integer but an integerm a When the position k is an not an integer but an integer(m) + a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-93.jpg)
When the position k is an not an integer but an integer(m) + a fraction(f). i. e. k = m + f mth obs (m+1)st obs xp = (1 - f) ( mth obs) + f [(m+1)st obs] Thus the position of xp is 100 f% through the interval between the mth observation and the (m +1)st observation
![Example The data Verbal IQ on n 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-94.jpg)
Example The data Verbal IQ on n = 23 students arranged in increasing order is: 80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 104 105 109 111 118 119
![x 0 75 75 th percentile 18 th observation in size 105 x 0. 75 = 75 th percentile = 18 th observation in size =105](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-95.jpg)
x 0. 75 = 75 th percentile = 18 th observation in size =105 (position k = 18) x 0. 90 = 90 th percentile = 0. 4(21 st observation in size) + 0. 6(22 nd observation in size) = 0. 4(111)+ 0. 6(118) = 115. 2 (position k = 21. 6)
![An Alternative method for computing Quartiles Method 2 Sometimes this method will An Alternative method for computing Quartiles – Method 2 • Sometimes this method will](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-96.jpg)
An Alternative method for computing Quartiles – Method 2 • Sometimes this method will result in the same values for the quartiles. • Sometimes this method will result in the different values for the quartiles. • For large samples the two methods will result in approximately the same answer.
![Let x 1 x 2 x 3 xn denote a set of n Let x 1, x 2, x 3, … xn denote a set of n](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-97.jpg)
Let x 1, x 2, x 3, … xn denote a set of n numbers. The first step in Method 2 is to arrange the numbers in increasing order. From the arranged numbers we compute the median. This is also called the Hinge
![Example Consider the 5 numbers 10 15 21 7 13 Arranged in increasing order Example Consider the 5 numbers: 10 15 21 7 13 Arranged in increasing order:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-98.jpg)
Example Consider the 5 numbers: 10 15 21 7 13 Arranged in increasing order: 7 10 13 15 21 Median (Hinge) The median (or Hinge) splits the observations in half
![The lower midhinge the first quartile is the median of the lower half of The lower mid-hinge (the first quartile) is the “median” of the lower half of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-99.jpg)
The lower mid-hinge (the first quartile) is the “median” of the lower half of the observations (excluding the median). The upper mid-hinge (the third quartile) is the “median” of the upper half of the observations (excluding the median).
![Consider the five number in increasing order Lower Half 7 Upper Half 10 13 Consider the five number in increasing order: Lower Half 7 Upper Half 10 13](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-100.jpg)
Consider the five number in increasing order: Lower Half 7 Upper Half 10 13 15 21 Upper Mid-Hinge Median (Hinge) Upper Mid-Hinge (First Quartile) 13 (Third Quartile) (7+10)/2 =8. 5 (15+21)/2 = 18
![Computing the median and the quartile using the first method Position of the median Computing the median and the quartile using the first method: Position of the median:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-101.jpg)
Computing the median and the quartile using the first method: Position of the median: k = 0. 5(5+1) = 3 Position of the first Quartile: k = 0. 25(5+1) = 1. 5 Position of the third Quartile: k = 0. 75(5+1) = 4. 5 7 Q 1 = 8. 5 10 13 Q 2 = 13 15 21 Q 3 = 18
![Both methods result in the same value This is not always • Both methods result in the same value • This is not always](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-102.jpg)
• Both methods result in the same value • This is not always true.
![Example The data Verbal IQ on n 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-103.jpg)
Example The data Verbal IQ on n = 23 students arranged in increasing order is: 80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 104 105 109 111 118 119 Lower Mid-Hinge (First Quartile) Median (Hinge) 89 96 Upper Mid-Hinge (Third Quartile) 105
![Computing the median and the quartile using the first method Position of the median Computing the median and the quartile using the first method: Position of the median:](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-104.jpg)
Computing the median and the quartile using the first method: Position of the median: k = 0. 5(23+1) = 12 Position of the first Quartile: k = 0. 25(23+1) = 6 Position of the third Quartile: k = 0. 75(23+1) = 18 80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 104 105 109 111 118 119 Q 1 = 89 Q 2 = 96 Q 3 = 105
![Many programs compute percentiles quartiles etc Each may use different methods • Many programs compute percentiles, quartiles etc. • Each may use different methods.](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-105.jpg)
• Many programs compute percentiles, quartiles etc. • Each may use different methods. • It is important to know which method is being used. • The different methods result in answers that are close when the sample size is large.
![Measures of Central Location Mean Median Measures of Central Location Mean Median](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-106.jpg)
Measures of Central Location Mean Median
![Mean Let x 1 x 2 x 3 xn denote a set of Mean Let x 1, x 2, x 3, … xn denote a set of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-107.jpg)
Mean Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the mean of the n numbers is defined as:
![The Median Let x 1 x 2 x 3 xn denote a set The Median Let x 1, x 2, x 3, … xn denote a set](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-108.jpg)
The Median Let x 1, x 2, x 3, … xn denote a set of n numbers. Then the median of the n numbers is defined as the number that splits the numbers into two equal parts. To evaluate the median we arrange the numbers in increasing order.
![If the number of observations is odd there will be one observation in the If the number of observations is odd there will be one observation in the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-109.jpg)
If the number of observations is odd there will be one observation in the middle. This number is the median. If the number of observations is even there will be two middle observations. The median is the average of these two observations
![Measures of NonCentral Location Percentiles Quartiles Hinges Midhinges Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-110.jpg)
Measures of Non-Central Location • • Percentiles Quartiles (Hinges, Mid-hinges)
![Definition The P 100 Percentile is a point x P underneath a Definition The P× 100 Percentile is a point , x. P , underneath a](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-111.jpg)
Definition The P× 100 Percentile is a point , x. P , underneath a distribution that has a fixed proportion P of the population (or sample) below that value P× 100 % x. P
![Computing Percentiles and Quartiles Method 1 The first step is to order Computing Percentiles and Quartiles – Method 1 • The first step is to order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-112.jpg)
Computing Percentiles and Quartiles – Method 1 • The first step is to order the observations in increasing order. • We then compute the position, k, of the P× 100 Percentile. k = P × (n+1) Where n = the number of observations
![When the position k is an integer the percentile is the kth observation in When the position k is an integer the percentile is the kth observation (in](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-113.jpg)
When the position k is an integer the percentile is the kth observation (in order of magnitude) in the data set. When the position k is an not an integer but an integer(m) + a fraction(f). i. e. k = m + f then the percentile is x. P = (1 -f) × (mth observation in size) + f × (m+1 st observation in size)
![An Alternative method for computing Quartiles Method 2 Sometimes this method will An Alternative method for computing Quartiles – Method 2 • Sometimes this method will](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-114.jpg)
An Alternative method for computing Quartiles – Method 2 • Sometimes this method will result in the same values for the quartiles. • Sometimes this method will result in the different values for the quartiles. • For large samples the two methods will result in approximately the same answer.
![Let x 1 x 2 x 3 xn denote a set of n Let x 1, x 2, x 3, … xn denote a set of n](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-115.jpg)
Let x 1, x 2, x 3, … xn denote a set of n numbers. The first step in Method 2 is to arrange the numbers in increasing order. From the arranged numbers we compute the median. This is also called the Hinge
![The lower midhinge the first quartile is the median of the lower half of The lower mid-hinge (the first quartile) is the “median” of the lower half of](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-116.jpg)
The lower mid-hinge (the first quartile) is the “median” of the lower half of the observations (excluding the median). The upper mid-hinge (the third quartile) is the “median” of the upper half of the observations (excluding the median).
![BoxPlots BoxWhisker Plots A graphical method of of displaying data An alternative Box-Plots Box-Whisker Plots • A graphical method of of displaying data • An alternative](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-117.jpg)
Box-Plots Box-Whisker Plots • A graphical method of of displaying data • An alternative to the histogram and stem-leaf diagram
![To Draw a Box Plot Compute the Hinge Median Q 2 and the To Draw a Box Plot • Compute the Hinge (Median, Q 2) and the](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-118.jpg)
To Draw a Box Plot • Compute the Hinge (Median, Q 2) and the Mid-hinges (first & third quartiles – Q 1 and Q 3 ) • We also compute the largest and smallest of the observations – the max and the min.
![Example The data Verbal IQ on n 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-119.jpg)
Example The data Verbal IQ on n = 23 students arranged in increasing order is: 80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 104 105 109 111 118 119 min = 80 Q 1 = 89 Q 2 = 96 Q 3 = 105 max = 119
![The Box Plot is then drawn Drawing above an axis a box from The Box Plot is then drawn • Drawing above an axis a “box” from](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-120.jpg)
The Box Plot is then drawn • Drawing above an axis a “box” from Q 1 to Q 3. • Drawing vertical line in the box at the median, Q 2 • Drawing whiskers at the lower and upper ends of the box going down to the min and up to max.
![Lower Whisker min Upper Whisker Box Q 1 Q 2 Q 3 max Lower Whisker min Upper Whisker Box Q 1 Q 2 Q 3 max](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-121.jpg)
Lower Whisker min Upper Whisker Box Q 1 Q 2 Q 3 max
![Example The data Verbal IQ on n 23 students arranged in increasing order Example The data Verbal IQ on n = 23 students arranged in increasing order](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-122.jpg)
Example The data Verbal IQ on n = 23 students arranged in increasing order is: min = 80 Q 1 = 89 Q 2 = 96 Q 3 = 105 max = 119
![Box Plot of Verbal IQ 70 80 90 100 110 120 130 Box Plot of Verbal IQ 70 80 90 100 110 120 130](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-123.jpg)
Box Plot of Verbal IQ 70 80 90 100 110 120 130
![130 120 110 100 90 80 70 Box Plot can also be drawn vertically 130 120 110 100 90 80 70 Box Plot can also be drawn vertically](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-124.jpg)
130 120 110 100 90 80 70 Box Plot can also be drawn vertically
![BoxWhisker plots Verbal IQ Math IQ Box-Whisker plots (Verbal IQ, Math IQ)](https://slidetodoc.com/presentation_image_h2/414e4e19db4c41a75609ea7acadeceef/image-125.jpg)
Box-Whisker plots (Verbal IQ, Math IQ)
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Box-Whisker plots (Initial RA, Final RA )
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Summary Information contained in the box plot 25% 25% Middle 50% of population 25%
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Next topic: Numerical Measures of Variability
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