Descriptive Statistics Frequency Tables Visual Displays Measures of
Descriptive Statistics Frequency Tables Visual Displays Measures of Center 1
Overview v Descriptive Statistics summarize or describe the important characteristics of a known set of population data v Inferential Statistics use sample data to make inferences (or generalizations) about a population 2
Important Characteristics of Data 1. Center: A representative or average value that indicates where the middle of the data set is located 2. Variation: A measure of the amount that the values vary among themselves 3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed) 4. Outliers: Sample values that lie very far away from the vast majority of other sample values 5. Time: Changing characteristics of the data over time 3
Summarizing Data With Frequency Tables v Frequency Table lists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class 4
Table 2 -1 Qwerty Keyboard Word Ratings 2 2 5 1 2 6 3 3 4 2 4 0 5 7 7 5 6 6 8 10 7 2 2 10 5 8 2 5 4 2 6 1 7 2 3 8 1 5 2 14 2 2 6 3 1 7 5
Table 2 -3 Frequency Table of Qwerty Word Ratings Rating Frequency 0 -2 20 3 -5 14 6 -8 15 9 - 11 2 12 - 14 1 6
Lower Class Limits are the smallest numbers that can actually belong to different classes Rating Lower Class Limits Frequency 0 -2 20 3 -5 14 6 -8 15 9 - 11 2 12 - 14 1 7
Upper Class Limits are the largest numbers that can actually belong to different classes Rating Upper Class Limits Frequency 0 -2 20 3 -5 14 6 -8 15 9 - 11 2 12 - 14 1 8
Class Boundaries number separating classes Rating Frequency - 0. 5 Class Boundaries 2. 5 5. 5 8. 5 11. 5 0 -2 20 3 -5 14 6 -8 15 9 - 11 2 12 - 14 1 14. 5 9
Class Midpoints midpoints of the classes Rating Class Midpoints Frequency 0 - 1 2 20 3 - 4 5 14 6 - 7 8 15 9 - 10 11 2 12 - 13 14 1 10
Class Width is the difference between two consecutive lower class limits or two consecutive class boundaries Rating Class Width Frequency 3 0 -2 20 3 3 -5 14 3 6 -8 15 3 9 - 11 2 3 12 - 14 1 11
Inputting a Frequency Table into the TI 83 Calculator • • Push Stat Select Edit Input Class Marks in List 1 Input Frequencies in List 2 12
Relative Frequency Table relative frequency = class frequency sum of all frequencies 13
Relative Frequency Table Rating Frequency Relative Rating Frequency 0 -2 20 0 -2 38. 5% 20/52 = 38. 5% 3 -5 14 3 -5 26. 9% 14/52 = 26. 9% 6 -8 15 6 -8 28. 8% 9 - 11 2 9 - 11 3. 8% 12 - 14 1. 9% etc. Total frequency = 52 14
Cumulative Frequency Table Rating Frequency Rating Cumulative Frequency 0 -2 20 Less than 3 20 3 -5 14 Less than 6 34 6 -8 15 Less than 9 49 9 - 11 2 Less than 12 51 12 - 14 1 Less than 15 52 Cumulative Frequencies 15
Frequency Tables Rating Frequency Rating Relative Frequency Rating Cumulative Frequency 0 -2 20 0 -2 38. 5% Less than 3 20 3 -5 14 3 -5 26. 9% Less than 6 34 6 -8 15 6 -8 28. 8% Less than 9 49 9 - 11 2 9 - 11 3. 8% Less than 12 51 12 - 14 1. 9% Less than 15 52 16
Histogram of Qwerty Word Ratings Rating Frequency 0 -2 20 3 -5 14 6 -8 15 9 - 11 2 12 - 14 1 17
Relative Frequency Histogram of Qwerty Word Ratings Relative Rating Frequency 0 -2 38. 5% 3 -5 26. 9% 6 -8 28. 8% 9 - 11 3. 8% 12 - 14 1. 9% 18
Histogram and Relative Frequency Histogram 19
Dot Plot 20
Stem-and Leaf Plot Stem Raw Data (Test Grades) 67 72 89 85 88 90 75 89 99 100 6 7 8 9 10 Leaves 7 25 5899 09 0 21
Scatter Diagram 20 TAR • 10 • • 0 0. 0 • • • • • 0. 5 1. 0 1. 5 NICOTINE 22
Measures of Center a value at the center or middle of a data set 23
Definitions Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values 24
Notation denotes the addition of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population 25
Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample values x x = n µ is pronounced ‘mu’ and denotes the mean of all values population µ = in a x N Calculators can calculate the mean of data 26
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 12 • Find the mean for the sample data above, flossings per week. 27
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 12 28
Finding a Mean Using the TI 83 Calculator • • • Push Stat Select Edit Enter data into L 1 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1 (L 1) The first output item is the mean 29
Definitions v Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude v often denoted by x~ (pronounced ‘x-tilde’) v is not affected by an extreme value 30
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 12 • Find the median of the data set above. 31
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 1200 • How would changing the 12 to a 1200 change the median? 1200 32
6, 6, 7, 8, 8, 9, 10, 12, 16, 24 • Find the median of the data set above, months between cleanings. • With an even number of numbers (10), you will need to find the midpoint between the two middle values. 6, 6, 7, 8, 8, 9, 10, 12, 16, 24 33
Finding a Median Using the TI 83 Calculator • • • Push Stat Select Edit Enter data into L 1 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1 (L 1) Arrow down to “Med” to see the value for median 34
Definitions v Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data 35
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 12 • Find the mode for the data above , if it exists. 36
Examples a. 5 5 5 3 1 5 1 4 3 5 ïMode is 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 ïBimodal c. 1 2 3 6 7 8 9 10 ïNo Mode 2 and 6 37
Definitions v Midrange the value midway between the highest and lowest values in the original data set Midrange = highest score + lowest score 2 38
2, 3, 4, 4, 4, 5, 6, 7, 9, 10 12 • Find the mode for the data above , if it exists. 39
Finding a Midrange Using the TI 83 Calculator • • • Push Stat Select Edit Enter data into L 1 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1 (L 1) Arrow down to see the “min” and “max” values for the midrange calculation 40
Round-off Rule for Measures of Center Carry one more decimal place than is present in the original set of values 41
Mean from a Frequency Table use class midpoint of classes for variable x (f • x) x = f x = class midpoint f = frequency f=n 42
Finding a Grouped Mean Using the TI 83 Calculator • • • Push Stat Select Edit Enter class marks into L 1 and frequencies into L 2 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1, 2 nd 2 (L 1, L 2) The first output item is the mean 43
Definitions v Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half. v Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. 44
Skewness Mode = Mean = Median SYMMETRIC Mean Mode Median SKEWED LEFT (negatively) Mean Mode Median SKEWED RIGHT (positively) 45
Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank 6. 5 6. 6 6. 7 6. 8 7. 1 7. 3 7. 4 7. 7 Bank of Providence 4. 2 5. 4 5. 8 6. 2 6. 7 7. 7 8. 5 9. 3 10. 0 Jefferson Valley Bank of Providence Mean 7. 15 Median 7. 20 Mode 7. 7 Midrange 7. 10 46
Dotplots of Waiting Times 47
Measures of Variation Range highest value lowest value 48
Measures of Variation Standard Deviation a measure of variation of the scores about the mean (average deviation from the mean) 49
Sample Standard Deviation Formula S= (x - x) n-1 2 calculators can compute the sample standard deviation of data 50
Population Standard Deviation = (x - µ) 2 N calculators can compute the population standard deviation of data 51
Finding a Standard Deviation Using the TI 83 Calculator • • • Push Stat Select Edit Enter data into L 1 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1 (L 1) The 4 th output item is the sample s. d. and the 5 th output item is the population s. d. 52
Measures of Variation Variance standard deviation squared } Notation s 2 2 use square key on calculator 53
Variance 2 s = 2 = (x - x ) 2 n-1 (x - µ) N 2 Sample Variance Population Variance 54
Round-off Rule for measures of variation Carry one more decimal place than is present in the original set of values. Round only the final answer, never in the middle of a calculation. 55
Finding a Grouped Standard Deviation Using the TI 83 Calculator • • • Push Stat Select Edit Enter class marks into L 1 and frequencies into L 2 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1, 2 nd 2 (L 1, L 2) The 4 th output item is the sample s. d and the 5 th output item is the population s. d 56
Estimation of Standard Deviation Range Rule of Thumb x - 2 s x (minimum usual value) Range 4 s x + 2 s (maximum usual value) or s Range 4 highest value - lowest value = 4 57
Usual Sample Values minimum ‘usual’ value (mean) - 2 (standard deviation) minimum x - 2(s) maximum ‘usual’ value (mean) + 2 (standard deviation) maximum x + 2(s) 58
The Empirical Rule (applies to bell-shaped distributions) 68% within 1 standard deviation 34% x-s 34% x x+s 59
The Empirical Rule (applies to bell-shaped distributions) 95% within 2 standard deviations 68% within 1 standard deviation 34% 13. 5% x - 2 s 13. 5% x-s x x+s x + 2 s 60
The Empirical Rule (applies to bell-shaped distributions) 99. 7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation 34% 2. 4% 0. 1% 13. 5% x - 3 s x - 2 s 13. 5% x-s x x+s x + 2 s x + 3 s 61
Chebyshev’s Theorem v applies to distributions of any shape. v the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at 2 least 1 - 1/K , where K is any positive number greater than 1. v at least 3/4 (75%) of all values lie within 2 standard deviations of the mean. v at least 8/9 (89%) of all values lie within 3 standard deviations of the mean. 62
Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations. 63
Measures of Position v z Score (or standard score) the number of standard deviations that a given value x is above or below the mean 64
Measures of Position z score Sample x x z= s Population x µ z= Round to 2 decimal places 65
Interpreting Z Scores Unusual Values -3 Ordinary Values -2 -1 0 Unusual Values 1 2 3 Z 66
Quartiles Q 1, Q 2, Q 3 divides ranked scores into four equal parts 25% (minimum) 25% 25% Q 1 Q 2 Q 3 (maximum) (median) 67
Deciles D 1, D 2, D 3, D 4, D 5, D 6, D 7, D 8, D 9 divides ranked data into ten equal parts 10% 10% D 1 D 2 D 3 10% 10% D 4 D 5 10% 10% D 6 D 7 D 8 D 9 68
Quartiles Q 1 = P 25 Q 2 = P 50 Q 3 = P 75 Deciles D 1 = P 10 D 2 = P 20 D 3 = P 30 • • • D 9 = P 90 69
Viewing Quartiles Using the TI 83 Calculator • • • Push Stat Select Edit Enter data into L 1 2 nd mode (quit) Push Stat >Calc Select 1 (1 -Var. Stats) 2 nd 1 (L 1) Arrow down to view the values for Q 2 and Q 3 (the median “med” is Q 2) 70
Exploratory Data Analysis the process of using statistical tools (such as graphs, measures of center, and measures of variation) to investigate the data sets in order to understand their important characteristics 71
Outliers v a value located very far away from almost all of the other values v an extreme value v can have a dramatic effect on the mean, standard deviation, and on the scale of the histogram so that the true nature of the distribution is totally obscured 72
Boxplots (Box-and-Whisker Diagram) Reveals the: v center of the data v spread of the data v distribution of the data v presence of outliers Excellent for comparing two or more data sets 73
Boxplots 5 - number summary v Minimum v first quartile Q 1 v Median (Q 2) v third quartile Q 3 v Maximum 74
Boxplots 2 4 6 14 0 0 6 8 10 12 14 Boxplot of Qwerty Word Ratings 75
Boxplots Bell-Shaped Uniform Skewed 76
Exploring v Measures of center: mean, median, and mode v Measures of variation: standard deviation and range v Measures of spread & relative location: minimum values, maximum value, and quartiles v Unusual values: outliers v Distribution: histograms, stem-leaf plots, and boxplots 77
- Slides: 77