Statistics for Business and Economics 6 th Edition

Chapter Goals After completing this chapter, you should be able to: n n n Compute and interpret the mean, median, and mode for a set of data Find the range, variance, standard deviation, and coefficient of variation and know what these values mean Apply the empirical rule to describe the variation of population values around the mean Explain the weighted mean and when to use it Explain how a least squares regression line estimates a linear relationship between two variables Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 2

Chapter Topics n Measures of central tendency, variation, and shape n n n Mean, median, mode, geometric mean Quartiles Range, interquartile range, variance and standard deviation, coefficient of variation Symmetric and skewed distributions Population summary measures n n Mean, variance, and standard deviation The empirical rule and Bienaymé-Chebyshev rule Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 3

Chapter Topics (continued) n n n Five number summary and box-and-whisker plots Covariance and coefficient of correlation Pitfalls in numerical descriptive measures and ethical considerations Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 4

Describing Data Numerically Central Tendency Variation Arithmetic Mean Range Median Interquartile Range Mode Variance Standard Deviation Coefficient of Variation Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 5

Measures of Central Tendency Overview Central Tendency Mean Median Mode Arithmetic average Midpoint of ranked values Most frequently observed value Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 6

Arithmetic Mean n The arithmetic mean (mean) is the most common measure of central tendency n For a population of N values: Population values Population size n For a sample of size n: Observed values Sample size Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 7

Arithmetic Mean (continued) n n n The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 0 1 2 3 4 5 6 7 8 9 10 Mean = 4 8

Median n In an ordered list, the median is the “middle” number (50% above, 50% below) 0 1 2 3 4 5 6 7 8 9 10 Median = 3 n Not affected by extreme values Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 9

Finding the Median n The location of the median: n n n If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers Note that is not the value of the median, only the position of the median in the ranked data Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 10

Mode n n n A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 0 1 2 3 4 5 6 No Mode 11

Review Example n Five houses on a hill by the beach House Prices: $2,

Review Example: Summary Statistics House Prices: $2, 000 500, 000 300, 000 100, 000 n n Sum 3, 000 n Mean: ($3, 000/5) = $600, 000 Median: middle value of ranked data = $300, 000 Mode: most frequent value = $100, 000 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 13

Which measure of location is the “best”? n n Mean is generally used, unless extreme values (outliers) exist Then median is often used, since the median is not sensitive to extreme values. n Example: Median home prices may be reported for a region – less sensitive to outliers Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 14

Shape of a Distribution n Describes how data are distributed n Measures of shape n Symmetric or skewed Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median < Mean Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 15

Measures of Variability Variation Range n Interquartile Range Variance Standard Deviation Coefficient of Variation Measures of variation give information on the spread or variability of the data values. Same center, different variation Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 16

Range n n Simplest measure of variation Difference between the largest and the smallest observations: Range = Xlargest – Xsmallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 17

Disadvantages of the Range n Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 n 7 8 9 10 11 12 Range = 12 - 7 = 5 Sensitive to outliers 1, 1, 1, 2, 2, 3, 3, 4, 5 Range = 5 - 1 = 4 1, 1, 1, 2, 2, 3, 3, 4, 120 Range = 120 - 1 = 119 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 18

Interquartile Range n n n Can eliminate some outlier problems by using the interquartile range Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data Interquartile range = 3 rd quartile – 1 st quartile IQR = Q 3 – Q 1 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 19

Interquartile Range Example: X minimum Q 1 25% 12 Median (Q 2) 25% 30 25% 45 X Q 3 maximum 25% 57 70 Interquartile range = 57 – 30 = 27 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 20

Quartiles n Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% Q 1 n n n 25% Q 2 25% Q 3 The first quartile, Q 1, is the value for which 25% of the observations are smaller and 75% are larger Q 2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 21

Quartile Formulas Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q 1 = 0. 25(n+1) Second quartile position: Q 2 = 0. 50(n+1) (the median position) Third quartile position: Q 3 = 0. 75(n+1) where n is the number of observed values Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 22

Quartiles n Example: Find the first quartile Sample Ranked Data: 11 12 13 16 16 17 18 21 22 (n = 9) Q 1 = is in the 0. 25(9+1) = 2. 5 position of the ranked data so use the value half way between the 2 nd and 3 rd values, so Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Q 1 = 12. 5 23

Population Variance n Average of squared deviations of values from the mean n Population variance: Where = population mean N = population size xi = ith value of the variable x Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 24

Sample Variance n Average (approximately) of squared deviations of values from the mean n Sample variance: Where = arithmetic mean n = sample size Xi = ith value of the variable X Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 25

Population Standard Deviation n Most commonly used measure of variation Shows variation about the mean Has the same units as the original data n Population standard deviation: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 26

Sample Standard Deviation n Most commonly used measure of variation Shows variation about the mean Has the same units as the original data n Sample standard deviation: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 27

Calculation Example: Sample Standard Deviation Sample Data (xi) : 10 12 14 n=8 15 17 18 18 24 Mean = x = 16 A measure of the “average” scatter around the mean Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 28

Measuring variation Small standard deviation Large standard deviation Statistics for Business and Economics, 6

Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Mean = 15. 5 s = 3. 338 20 21 Mean = 15. 5 s = 0. 926 20 21 Mean = 15. 5 s = 4. 570 Data B 11 12 13 14 15 16 17 18 19 Data C 11 12 13 14 15 16 17 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 18 19 30

Advantages of Variance and Standard Deviation n n Each value in the data set is used in the calculation Values far from the mean are given extra weight (because deviations from the mean are squared) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 31

Chebyshev’s Theorem n For any population with mean μ and standard deviation σ , and k > 1 , the percentage of observations that fall within the interval [μ + kσ] Is at least Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 32

Chebyshev’s Theorem (continued) n Regardless of how the data are distributed, at least (1 - 1/k 2) of the values will fall within k standard deviations of the mean (for k > 1) n Examples: At 2 least (1 - 1/1 ) = 0% ……. . . k=1 within (μ ± 1σ) (1 - 1/22) = 75% …. . . . k=2 (μ ± 2σ) (1 - 1/32) = 89% ………. k=3 (μ ± 3σ) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 33

The Empirical Rule n n If the data distribution is bell-shaped, then the interval: contains about 68% of the values in the population or the sample 68% Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 34

The Empirical Rule n n contains about 95% of the values in the population or the sample contains about 99. 7% of the values in the population or the sample 95% Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 99. 7% 35

Coefficient of Variation n Measures relative variation n Always in percentage (%) n Shows variation relative to mean n Can be used to compare two or more sets of data measured in different units Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 36

Comparing Coefficient of Variation n n Stock A: n Average price last year = $50 n Standard deviation = $5 Stock B: n n Average price last year = $100 Standard deviation = $5 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Both stocks have the same standard deviation, but stock B is less variable relative to its price 37

Using Microsoft Excel n Descriptive Statistics can be obtained from Microsoft® Excel n Use menu choice: tools / data analysis / descriptive statistics n Enter details in dialog box Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 38

Using Excel n Use menu choice: tools / data analysis / descriptive statistics Statistics

Using Excel (continued) n n n Enter dialog box details Check box for summary

Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2, 000 500, 000 300, 000 100, 000 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 41

Weighted Mean n The weighted mean of a set of data is n n Where wi is the weight of the ith observation Use when data is already grouped into n classes, with wi values in the ith class Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 42

Approximations for Grouped Data Suppose a data set contains values m 1, m 2, . . . , mk, occurring with frequencies f 1, f 2, . . . f. K n For a population of N observations the mean is n For a sample of n observations, the mean is Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 43

Approximations for Grouped Data Suppose a data set contains values m 1, m 2, . . . , mk, occurring with frequencies f 1, f 2, . . . f. K n For a population of N observations the variance is n For a sample of n observations, the variance is Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 44

The Sample Covariance n The covariance measures the strength of the linear relationship between two variables n The population covariance: n The sample covariance: n n Only concerned with the strength of the relationship No causal effect is implied Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 45

Interpreting Covariance n Covariance between two variables: Cov(x, y) > 0 x and y tend to move in the same direction Cov(x, y) < 0 x and y tend to move in opposite directions Cov(x, y) = 0 x and y are independent Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 46

Coefficient of Correlation n Measures the relative strength of the linear relationship between two variables n Population correlation coefficient: n Sample correlation coefficient: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 47

Features of Correlation Coefficient, r n Unit free n Ranges between – 1 and 1 n n n The closer to – 1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 48

Scatter Plots of Data with Various Correlation Coefficients Y Y r = -1 X

Using Excel to Find the Correlation Coefficient n n n Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Select Tools/Data Analysis Choose Correlation from the selection menu Click OK. . . 50

Using Excel to Find the Correlation Coefficient (continued) n n Input data range and select appropriate options Click OK to get output Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 51

Interpreting the Result n n n r =. 733 There is a relatively strong positive linear relationship between test score #1 and test score #2 Students who scored high on the first tended to score high on second test Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 52

Obtaining Linear Relationships n An equation can be fit to show the best linear relationship between two variables: Y = β 0 + β 1 X Where Y is the dependent variable and X is the independent variable Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 53

Least Squares Regression n Estimates for coefficients β 0 and β 1 are found to minimize the sum of the squared residuals The least-squares regression line, based on sample data, is Where b 1 is the slope of the line and b 0 is the yintercept: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 54

Chapter Summary n Described measures of central tendency n n Illustrated the shape of the distribution n Symmetric, skewed Described measures of variation n n Mean, median, mode Range, interquartile range, variance and standard deviation, coefficient of variation Discussed measures of grouped data Calculated measures of relationships between variables n covariance and correlation coefficient Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 55