2011 Pearson Education Inc Statistics for Business and
© 2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 2 Methods for Describing Sets of Data © 2011 Pearson Education, Inc
Contents 1. Describing Qualitative Data 2. Graphical Methods for Describing Quantitative Data 3. Summation Notation 4. Numerical Measures of Central Tendency 5. Numerical Measures of Variability 6. Interpreting the Standard Deviation © 2011 Pearson Education, Inc
Contents 7. Numerical Measures of Relative Standing 8. Methods for Detecting Outliers: Box Plots and z-scores 9. Graphing Bivariate Relationships 10. The Time Series Plot 11. Distorting the Truth with Descriptive Techniques © 2011 Pearson Education, Inc
Learning Objectives 1. Describe data using graphs 2. Describe data using numerical measures © 2011 Pearson Education, Inc
2. 1 Describing Qualitative Data © 2011 Pearson Education, Inc
Key Terms A class is one of the categories into which qualitative data can be classified. The class frequency is the number of observations in the data set falling into a particular class. The class relative frequency is the class frequency divided by the total numbers of observations in the data set. The class percentage is the class relative frequency multiplied by 100. © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Summary Table 1. Lists categories & number of elements in category 2. Obtained by tallying responses in category 3. May show frequencies (counts), % or both Row Is Category Major Count Accounting 130 Economics 20 Management 50 Total 200 © 2011 Pearson Education, Inc Tally: ||||
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Bar Graph Percent Used Also Frequency Equal Bar Widths Bar Height Shows Frequency or % Vertical Bars for Qualitative Variables Zero Point © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Pie Chart 1. Shows breakdown of total quantity into categories 2. Useful for showing relative differences Majors Econ. 10% Mgmt. 25% 36° Acct. 65% 3. Angle size • (360°)(percent) (360°) (10%) = 36° © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Pareto Diagram Like a bar graph, but with the categories arranged by height in descending order from left to right. Percent Used Also Frequency Equal Bar Widths Zero Point © 2011 Pearson Education, Inc Bar Height Shows Frequency or % Vertical Bars for Qualitative Variables
Summary Bar graph: The categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency, or class percentage. Pie chart: The categories (classes) of the qualitative variable are represented by slices of a pie (circle). The size of each slice is proportional to the class relative frequency. Pareto diagram: A bar graph with the categories (classes) of the qualitative variable (i. e. , the bars) arranged by height in descending order from left to right. © 2011 Pearson Education, Inc
Thinking Challenge You’re an analyst for IRI. You want to show the market shares held by Web browsers in 2006. Construct a bar graph, pie chart, & Pareto diagram to describe the data. Browser Firefox Internet Explorer Safari Others Mkt. Share (%) 14 81 4 1 © 2011 Pearson Education, Inc
Market Share (%) Bar Graph Solution* Browser © 2011 Pearson Education, Inc
Pie Chart Solution* Market Share © 2011 Pearson Education, Inc
Market Share (%) Pareto Diagram Solution* Browser © 2011 Pearson Education, Inc
2. 2 Graphical Methods for Describing Quantitative Data © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Dot Plot 1. Horizontal axis is a scale for the quantitative variable, e. g. , percent. 2. The numerical value of each measurement is located on the horizontal scale by a dot. © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Stem-and-Leaf Display 1. Divide each observation into stem value and leaf value • Stems are listed in order in a column • Leaf value is placed in corresponding stem row to right of bar 2 144677 3 028 4 1 2. Data: 21, 24, 26, 27, 30, 32, 38, 41 © 2011 Pearson Education, Inc 26
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Frequency Distribution Table Steps 1. Determine range 2. Select number of classes • Usually between 5 & 15 inclusive 3. Compute class intervals (width) 4. Determine class boundaries (limits) 5. Compute class midpoints 6. Count observations & assign to classes © 2011 Pearson Education, Inc
Frequency Distribution Table Example Raw Data: 24, 26, 24, 21, 27 27 30, 41, 32, 38 Class Width Midpoint Frequency 15. 5 – 25. 5 20. 5 3 25. 5 – 35. 5 30. 5 5 35. 5 – 45. 5 40. 5 2 Boundaries (Lower + Upper Boundaries) / 2 © 2011 Pearson Education, Inc
Relative Frequency & % Distribution Tables Relative Frequency Distribution Percentage Distribution Class Prop. Class % 15. 5 – 25. 5 . 3 15. 5 – 25. 5 30. 0 25. 5 – 35. 5 50. 0 35. 5 – 45. 5 . 2 35. 5 – 45. 5 20. 0 © 2011 Pearson Education, Inc
Data Presentation Qualitative Data Quantitative Data Dot Plot Summary Table Bar Graph Pie Chart Stem-&-Leaf Display Pareto Diagram © 2011 Pearson Education, Inc Frequency Distribution Histogram
Histogram Class 15. 5 – 25. 5 – 35. 5 – 45. 5 Count 5 Frequency Relative Frequency Percent 4 3 Bars Touch 2 1 0 0 15. 5 25. 5 35. 5 45. 5 Lower Boundary © 2011 Pearson Education, Inc 55. 5 Freq. 3 5 2
2. 3 Summation Notation © 2011 Pearson Education, Inc
Summation Notation Most formulas we use require a summation of numbers. Sum the measurements on the variable that appears to the right of the summation symbol, beginning with the 1 st measurement and ending with the nth measurement. © 2011 Pearson Education, Inc
Summation Notation For the data © 2011 Pearson Education, Inc
2. 4 Numerical Measures of Central Tendency © 2011 Pearson Education, Inc
Thinking Challenge $400, 000 $70, 000 $50, 000 $30, 000 . . . employees cite low pay -most workers earn only $20, 000 . . . President claims average pay is $70, 000! © 2011 Pearson Education, Inc
Two Characteristics The central tendency of the set of measurements –that is, the tendency of the data to cluster, or center, about certain numerical values. Central Tendency (Location) © 2011 Pearson Education, Inc
Two Characteristics The variability of the set of measurements–that is, the spread of the data. Variation (Dispersion) © 2011 Pearson Education, Inc
Standard Notation Measure Sample Population Mean X Size n N © 2011 Pearson Education, Inc
Mean 1. 2. 3. 4. Most common measure of central tendency Acts as ‘balance point’ Affected by extreme values (‘outliers’) Denoted x where n x x i i 1 n x 1 x 2 … x n © 2011 Pearson Education, Inc n
Mean Example Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 n x x i i 1 n x 1 x 2 x 3 x 4 x 5 6 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 8. 30 6 © 2011 Pearson Education, Inc x 6
Median 1. Measure of central tendency 2. Middle value in ordered sequence • • If n is odd, middle value of sequence If n is even, average of 2 middle values 3. Position of median in sequence n 1 Positioning Point 2 4. Not affected by extreme values © 2011 Pearson Education, Inc
Median Example Odd-Sized Sample • Raw Data: 24. 1 22. 6 21. 5 23. 7 22. 6 • Ordered: 21. 5 22. 6 23. 7 24. 1 • Position: 1 2 3 4 5 n 1 5 1 Positioning Point 3. 0 2 2 Median 22. 6 © 2011 Pearson Education, Inc
Median Example Even-Sized Sample • Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 • Ordered: 4. 9 6. 3 7. 7 8. 9 10. 3 11. 7 • Position: 1 2 3 4 5 6 n 1 6 1 Positioning Point 3. 5 2 2 7. 7 8. 9 Median 8. 30 2 © 2011 Pearson Education, Inc
Mode 1. Measure of central tendency 2. Value that occurs most often 3. Not affected by extreme values 4. May be no mode or several modes 5. May be used for quantitative or qualitative data © 2011 Pearson Education, Inc
Mode Example • No Mode Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 • One Mode Raw Data: 6. 3 4. 9 8. 9 6. 3 4. 9 • More Than 1 Mode Raw Data: 21 28 28 41 43 © 2011 Pearson Education, Inc 43
Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. Describe the stock prices in terms of central tendency. © 2011 Pearson Education, Inc
Central Tendency Solution* Mean n x x i i 1 n x 1 x 2 … x 8 8 17 16 21 18 13 16 12 11 8 15. 5 © 2011 Pearson Education, Inc
Central Tendency Solution* Median • Raw Data: 17 16 21 18 13 16 12 11 • Ordered: 11 12 13 16 16 17 18 21 n 1 8 1 Positioning Point • Position: 1 2 3 4 5 64. 5 7 2 2 8 16 Median 16 2 © 2011 Pearson Education, Inc
Central Tendency Solution* Mode Raw Data: 17 16 21 18 13 16 12 11 Mode = 16 © 2011 Pearson Education, Inc
Summary of Central Tendency Measures Measure Mean Median Mode Formula x i / n (n+1) Position 2 none Description Balance Point Middle Value When Ordered Most Frequent © 2011 Pearson Education, Inc
Shape 1. Describes how data are distributed 2. Measures of Shape • Skew = Symmetry Left-Skewed Mean Median Symmetric Mean = Median © 2011 Pearson Education, Inc Right-Skewed Median Mean
2. 5 Numerical Measures of Variability © 2011 Pearson Education, Inc
Range 1. Measure of dispersion 2. Difference between largest & smallest observations Range = xlargest – xsmallest 3. Ignores how data are distributed 7 8 9 10 Range = 10 – 7 = 3 © 2011 Pearson Education, Inc
Variance & Standard Deviation 1. Measures of dispersion 2. Most common measures 3. Consider how data are distributed 4. Show variation about mean (x or μ) x = 8. 3 4 6 8 10 12 © 2011 Pearson Education, Inc
Standard Notation Measure Mean Sample Population x s Standard Deviation 2 Variance s Size n © 2011 Pearson Education, Inc 2 N
Sample Variance Formula n – 1 in denominator! © 2011 Pearson Education, Inc
Sample Standard Deviation Formula © 2011 Pearson Education, Inc
Variance Example Raw Data: 10. 3 4. 9 n s 2 (x i x ) i 1 8. 9 11. 7 6. 3 n 2 n 1 where x s x i i 1 n 8. 3 10. 3 8. 3 ) (4. 9 8. 3 ) … (7. 7 8. 3 ) ( 2 2 7. 7 2 6 1 6. 368 © 2011 Pearson Education, Inc 2
Thinking Challenge • You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. • What are the variance and standard deviation of the stock prices? © 2011 Pearson Education, Inc
Variation Solution* Sample Variance Raw Data: 17 16 21 18 13 16 12 11 n s 2 (x i x ) i 1 n 2 n 1 where x s i 1 n 15. 5 17 15. 5 ) (16 15. 5 ) … (11 15. 5 ) ( 2 2 x i 11. 14 2 8 1 © 2011 Pearson Education, Inc 2
Variation Solution* Sample Standard Deviation © 2011 Pearson Education, Inc
Summary of Variation Measures Measure Range Formula X largest – X smallest Description Total Spread Standard Deviation (Sample) Dispersion about Sample Mean Standard Deviation (Population) Dispersion about Population Mean Variance (Sample) Squared Dispersion about Sample Mean © 2011 Pearson Education, Inc
2. 6 Interpreting the Standard Deviation © 2011 Pearson Education, Inc
Interpreting Standard Deviation: Chebyshev’s Theorem • Applies to any shape data set • No useful information about the fraction of data in the interval x – s to x + s • At least 3/4 of the data lies in the interval x – 2 s to x + 2 s • At least 8/9 of the data lies in the interval x – 3 s to x + 3 s • In general, for k > 1, at least 1 – 1/k 2 of the data lies in the interval x – ks to x + ks © 2011 Pearson Education, Inc
Interpreting Standard Deviation: Chebyshev’s Theorem No useful information At least 3/4 of the data At least 8/9 of the data © 2011 Pearson Education, Inc
Chebyshev’s Theorem Example • Previously we found the mean closing stock price of new stock issues is 15. 5 and the standard deviation is 3. 34. • Use this information to form an interval that will contain at least 75% of the closing stock prices of new stock issues. © 2011 Pearson Education, Inc
Chebyshev’s Theorem Example At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean. x = 15. 5 s = 3. 34 (x – 2 s, x + 2 s) = (15. 5 – 2∙ 3. 34, 15. 5 + 2∙ 3. 34) = (8. 82, 22. 18) © 2011 Pearson Education, Inc
Interpreting Standard Deviation: Empirical Rule • Applies to data sets that are mound shaped and symmetric • Approximately 68% of the measurements lie in the interval • Approximately 95% of the measurements lie in the interval • Approximately 99. 7% of the measurements lie in the interval © 2011 Pearson Education, Inc
Interpreting Standard Deviation: Empirical Rule x – 3 s x – 2 s x–s x x+s x +2 s x + 3 s Approximately 68% of the measurements Approximately 95% of the measurements Approximately 99. 7% of the measurements © 2011 Pearson Education, Inc
Empirical Rule Example Previously we found the mean closing stock price of new stock issues is 15. 5 and the standard deviation is 3. 34. If we can assume the data is symmetric and mound shaped, calculate the percentage of the data that lie within the intervals x + s, x + 2 s, x + 3 s. © 2011 Pearson Education, Inc
Empirical Rule Example • According to the Empirical Rule, approximately 68% of the data will lie in the interval (x – s, x + s), (15. 5 – 3. 34, 15. 5 + 3. 34) = (12. 16, 18. 84) • Approximately 95% of the data will lie in the interval (x – 2 s, x + 2 s), (15. 5 – 2∙ 3. 34, 15. 5 + 2∙ 3. 34) = (8. 82, 22. 18) • Approximately 99. 7% of the data will lie in the interval (x – 3 s, x + 3 s), (15. 5 – 3∙ 3. 34, 15. 5 + 3∙ 3. 34) = (5. 48, 25. 52) © 2011 Pearson Education, Inc
2. 7 Numerical Measures of Relative Standing © 2011 Pearson Education, Inc
Numerical Measures of Relative Standing: Percentiles • Describes the relative location of a measurement compared to the rest of the data • The pth percentile is a number such that p% of the data falls below it and (100 – p)% falls above it • Median = 50 th percentile © 2011 Pearson Education, Inc
Percentile Example • You scored 560 on the GMAT exam. This score puts you in the 58 th percentile. • What percentage of test takers scored lower than you did? • What percentage of test takers scored higher than you did? © 2011 Pearson Education, Inc
Percentile Example • What percentage of test takers scored lower than you did? 58% of test takers scored lower than 560. • What percentage of test takers scored higher than you did? (100 – 58)% = 42% of test takers scored higher than 560. © 2011 Pearson Education, Inc
Numerical Measures of Relative Standing: z–Scores • Describes the relative location of a measurement compared to the rest of the data • Sample z–score Population z–score • Measures the number of standard deviations away from the mean a data value is located © 2011 Pearson Education, Inc
Z–Score Example • The mean time to assemble a product is 22. 5 minutes with a standard deviation of 2. 5 minutes. • Find the z–score for an item that took 20 minutes to assemble. • Find the z–score for an item that took 27. 5 minutes to assemble. © 2011 Pearson Education, Inc
Z–Score Example x = 20, μ = 22. 5 σ = 2. 5 z = x σ– μ = 20 – 22. 5 = – 1. 0 2. 5 x = 27. 5, μ = 22. 5 σ = 2. 5 z = x σ– μ = 27. 5 – 22. 5 = 2. 0 2. 5 © 2011 Pearson Education, Inc
Interpretation of z–Scores for Mound-Shaped Distributions of Data 1. Approximately 68% of the measurements will have a z-score between – 1 and 1. 2. Approximately 95% of the measurements will have a z-score between – 2 and 2. 3. Approximately 99. 7% of the measurements will have a z-score between – 3 and 3. (see the figure on the next slide) © 2011 Pearson Education, Inc
Interpretation of z–Scores © 2011 Pearson Education, Inc
2. 8 Methods for Detecting Outliers: Box Plots and z-Scores © 2011 Pearson Education, Inc
Outlier An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers typically are attributable to one of the following causes: 1. The measurement is observed, recorded, or entered into the computer incorrectly. 2. The measurement comes from a different population. 3. The measurement is correct but represents a rare (chance) event. © 2011 Pearson Education, Inc
Quartiles Measure of noncentral tendency Split ordered data into 4 quarters 25% Q 1 25% Q 2 25% Q 3 Lower quartile QL is 25 th percentile. Middle quartile m is the median. Upper quartile QU is 75 th percentile. Interquartile range: IQR = QU – QL © 2011 Pearson Education, Inc
Quartile (Q 2) Example • Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 • Ordered: 4. 9 6. 3 7. 7 8. 9 10. 3 11. 7 • Position: 1 2 3 4 5 6 Q 2 is the median, the average of the two middle scores (7. 7 + 8. 9)/2 = 8. 8 © 2011 Pearson Education, Inc
Quartile (Q 1) Example • Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 • Ordered: 4. 9 6. 3 7. 7 8. 9 10. 3 11. 7 • Position: 1 2 3 4 5 6 QL is median of bottom half = 6. 3 © 2011 Pearson Education, Inc
Quartile (Q 3) Example • Raw Data: 10. 3 4. 9 8. 9 11. 7 6. 3 7. 7 • Ordered: 4. 9 6. 3 7. 7 8. 9 10. 3 11. 7 • Position: 1 2 3 4 5 6 QU is median of bottom half = 10. 3 © 2011 Pearson Education, Inc
Interquartile Range 1. Measure of dispersion 2. Also called midspread 3. Difference between third & first quartiles • Interquartile Range = Q 3 – Q 1 4. Spread in middle 50% 5. Not affected by extreme values © 2011 Pearson Education, Inc
Thinking Challenge • You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. • What are the quartiles, Q 1 and Q 3, and the interquartile range? © 2011 Pearson Education, Inc
Quartile Solution* Q 1 Raw Data: Ordered: Position: 17 16 21 18 13 16 12 11 11 12 13 16 16 17 18 21 1 2 3 4 5 6 7 8 QL is the median of the bottom half, the average of the two middle scores (12 + 13)/2 = 12. 5 © 2011 Pearson Education, Inc
Quartile Solution* Q 3 Raw Data: Ordered: Position: 17 16 21 18 13 16 12 11 11 12 13 16 16 17 18 21 1 2 3 4 5 6 7 8 QU is the median of the bottom half, the average of the two middle scores (17 + 18)/2 = 17. 5 © 2011 Pearson Education, Inc
Interquartile Range Solution* Interquartile Range Raw Data: 17 16 21 18 13 16 12 11 Ordered: 11 12 13 16 16 17 18 21 Position: 1 2 3 4 5 6 7 8 Interquartile Range = Q 3 – Q 1 = 17. 5 – 12. 5 = 5 © 2011 Pearson Education, Inc
Box Plot 1. Graphical display of data using 5 -number summary Xsmallest Q 1 Median Q 3 4 6 8 10 © 2011 Pearson Education, Inc Xlargest 12
Box Plot 1. Draw a rectangle (box) with the ends (hinges) drawn at the lower and upper quartiles (QL and QU). The median data is shown by a line or symbol (such as “+”). 2. The points at distances 1. 5(IQR) from each hinge define the inner fences of the data set. Line (whiskers) are drawn from each hinge to the most extreme measurements inside the inner fence. © 2011 Pearson Education, Inc
Box Plot 3. A second pair of fences, the outer fences, are defined at a distance of 3(IQR) from the hinges. One symbol (*) represents measurements falling between the inner and outer fences, and another (0) represents measurements beyond the outer fences. 4. Symbols that represent the median and extreme data points vary depending on software used. You may use your own symbols if you are constructing a box plot by hand. © 2011 Pearson Education, Inc
Shape & Box Plot Left-Skewed Q 1 Median Q 3 Symmetric Q 1 Median Q 3 © 2011 Pearson Education, Inc Right-Skewed Q 1 Median Q 3
Detecting Outliers Box Plots: Observations falling between the inner and outer fences are deemed suspect outliers. Observations falling beyond the outer fence are deemed highly suspect outliers. z-scores: Observations with z-scores greater than 3 in absolute value are considered outliers. (For some highly skewed data sets, observations with z-scores greater than 2 in absolute value may be outliers. ) © 2011 Pearson Education, Inc
2. 9 Graphing Bivariate Relationships © 2011 Pearson Education, Inc
Graphing Bivariate Relationships • Describes a relationship between two quantitative variables • Plot the data in a scattergram (or scatterplot) y y y x Positive relationship x Negative relationship © 2011 Pearson Education, Inc x No relationship
Scattergram Example • You’re a marketing analyst for Hasbro Toys. You gather the following data: Ad $ (x) Sales (Units) (y) 1 1 2 1 3 2 4 2 5 4 • Draw a scattergram of the data © 2011 Pearson Education, Inc
Scattergram Example Sales 4 3 2 1 0 0 1 2 3 Advertising © 2011 Pearson Education, Inc 4 5
2. 10 The Time Series Plot © 2011 Pearson Education, Inc
Time Series Plot • Used to graphically display data produced over time • Shows trends and changes in the data over time • Time recorded on the horizontal axis • Measurements recorded on the vertical axis • Points connected by straight lines © 2011 Pearson Education, Inc
Time Series Plot Example • The following data shows the average retail price of regular gasoline in New York City for 8 weeks in 2006. • Draw a time series plot for this data. Date Average Price Oct 16, 2006 Oct 23, 2006 $2. 219 $2. 173 Oct 30, 2006 Nov 6, 2006 Nov 13, 2006 Nov 20, 2006 Nov 27, 2006 Dec 4, 2006 $2. 177 $2. 158 $2. 185 $2. 208 $2. 236 $2. 298 © 2011 Pearson Education, Inc
Time Series Plot Example Price Date © 2011 Pearson Education, Inc
2. 11 Distorting the Truth with Descriptive Statistics © 2011 Pearson Education, Inc
Errors in Presenting Data 1. Use area to equate to value 2. No relative basis in comparing data batches 3. Compress the vertical axis 4. No zero point on the vertical axis 5. Gap in the vertical axis 6. Use of misleading wording 7. Knowing central tendency without knowing variability © 2011 Pearson Education, Inc
Reader Equates Area to Value Bad Presentation Good Presentation Minimum Wage 1960: $1. 00 4 1970: $1. 60 $ 2 1980: $3. 10 0 1990: $3. 80 1960 © 2011 Pearson Education, Inc 1970 1980 1990
No Relative Basis Bad Presentation 300 Freq. Good Presentation A’s by Class 30% 200 20% 100 10% 0 0% FR SO JR SR % A’s by Class FR SO JR SR © 2011 Pearson Education, Inc
Compressing Vertical Axis Bad Presentation 200 $ Good Presentation Quarterly Sales 50 100 25 0 0 Q 1 Q 2 Q 3 Q 4 $ Quarterly Sales Q 1 © 2011 Pearson Education, Inc Q 2 Q 3 Q 4
No Zero Point on Vertical Axis Bad Presentation 45 $ Good Presentation Monthly Sales 60 42 40 39 20 36 0 J M M J S N Monthly Sales $ J M M J © 2011 Pearson Education, Inc S N
Gap in the Vertical Axis Bad Presentation © 2011 Pearson Education, Inc
Changing the Wording Changing the title of the graph can influence the reader. We’re not doing so well. Still in prime years! © 2011 Pearson Education, Inc
Knowing only central tendency Knowing ONLY the central tendency might lead one to purchase Model A. Knowing the variability as well may change one’s decision! © 2011 Pearson Education, Inc
Key Ideas Describing Qualitative Data 1. 2. 3. 4. Identify category classes Determine class frequencies Class relative frequency = (class freq)/n Graph relative frequencies © 2011 Pearson Education, Inc
Key Ideas Graphing Quantitative Data 1 Variable 1. Identify class intervals 2. Determine class interval frequencies 3. Class relative frequency = (class interval frequencies)/n 4. Graph class interval relative frequencies © 2011 Pearson Education, Inc
Key Ideas Graphing Quantitative Data 2 Variables Scatterplot © 2011 Pearson Education, Inc
Key Ideas Numerical Description of Quantitative Date Central Tendency Mean Median Mode © 2011 Pearson Education, Inc
Key Ideas Numerical Description of Quantitative Date Variation Range Variance Standard Deviation Interquartile range © 2011 Pearson Education, Inc
Key Ideas Numerical Description of Quantitative Date Relative standing Percentile score z-score © 2011 Pearson Education, Inc
Key Ideas Rules for Detecting Quantitative Outliers Interval Chebyshev’s Rule Empirical Rule At least 0% At least 57% At least 89% ≈ 68% ≈ 95% All © 2011 Pearson Education, Inc
Key Ideas Rules for Detecting Quantitative Outliers Method Box plot: z-score Suspect Values between inner and outer fences 2 < |z| < 3 © 2011 Pearson Education, Inc Highly Suspect Values beyond outer fences 2 < |z| < 3
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