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1 s Prepared by JOHN S. LOUCKS St. Edward s s University 1

2 Chapter 3 Descriptive Statistics: Numerical Measures Part B Measures of Distribution Shape, Relative Location, and Detecting Outliers Exploratory Data Analysis Measures of Association Between Two Variables The Weighted Mean and Working with Grouped Data 2

3 Measures of Distribution Shape, Relative Location, and Detecting Outliers Distribution Shape z-scores Chebyshev s s Theorem Empirical Rule Detecting Outliers 3

4 Distribution Shape: Skewness An important measure of the shape of a distribution is called skewness. The formula for computing skewness for a data set is somewhat complex. Skewness can be easily computed using statistical software. 4

5 Distribution Shape: Skewness Symmetric (not skewed) Skewness is zero. Mean and median are equal. Relative Frequency Skewness = 0 5

6 Distribution Shape: Skewness Moderately Skewed Left Skewness is negative. Mean will usually be less than the median. Relative Frequency Skewness =.31 6

7 Distribution Shape: Skewness Moderately Skewed Right Skewness is positive. Mean will usually be more than the median. Relative Frequency Skewness =.31 7

8 Distribution Shape: Skewness Highly Skewed Right Skewness is positive (often above 1.0). Mean will usually be more than the median. Relative Frequency Skewness =

9 Distribution Shape: Skewness Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed in ascending order on the next slide. 9

10 Distribution Shape: Skewness

11 Distribution Shape: Skewness.35 Skewness =.92 Relative Frequency

12 z-scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value x ii is from the mean. z i = x i s x 12

13 z-scores An observation s s z-score z is a measure of the relative location of the observation in a data set. A data value less than the sample mean will have a z-score less than zero. A data value greater than the sample mean will have a z-score z greater than zero. A data value equal to the sample mean will have a z-score of zero. 13

14 z-scores z-score of Smallest Value (425) z x i x = = = 1.20 s Standardized Values for Apartment Rents

15 Chebyshev s s Theorem At least (1-1/z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is any value greater than 1. 15

16 Chebyshev s s Theorem At least 75% of the data values must be within z = 2 standard deviations of the mean. At least 89% of the data values must be within z = 3 standard deviations of the mean. At least 94% of the data values must be within z = 4 standard deviations of the mean. 16

17 Chebyshev s s Theorem For example: x Let z = 1.5 with = and s = At least (1 1/(1.5) 2 ) = = 0.56 or 56% of the rent values must be between x - z(s) ) = (54.74) = 409 and x + z(s) ) = (54.74) = 573 (Actually, 86% of the rent values are between 409 and 573.) 17

18 Empirical Rule For data having a bell-shaped distribution: 68.26% of the values of a normal random variable are within +/- 1 standard deviation of its mean % of the values of a normal random variable are within +/- 2 standard deviations of its mean % of the values of a normal random variable are within +/- 3 standard deviations of its mean. 18

19 Empirical Rule 99.72% 95.44% 68.26% μ 3σ μ 1σ μ 2σ μ μ + 1 1σ μ + 3 3σ μ + 2 2σ x 19

20 Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score z less than -33 or greater than +3 might be considered an outlier. It might be: an incorrectly recorded data value a data value that was incorrectly included in the data set a correctly recorded data value that belongs in the data set 20

21 Detecting Outliers The most extreme z-scores z are and 2.27 Using z > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

22 Exploratory Data Analysis Five-Number Summary Box Plot 22

23 Five-Number Summary 1 Smallest Value First Quartile Median Third Quartile Largest Value 23

24 Five-Number Summary Lowest Value = 425 First Quartile = 445 Median = 475 Third Quartile = 525 Largest Value =

25 Box Plot A box is drawn with its ends located at the first and third quartiles. A vertical line is drawn in the box at the location of the median (second quartile) Q1 = 445 Q3 = 525 Q2 =

26 Box Plot Limits are located (not drawn) using the interquartile range (IQR). Data outside these limits are considered outliers. The locations of each outlier is shown with the symbol *. continued 26

27 Box Plot The lower limit is located 1.5(IQR) below Q1. Lower Limit: Q1-1.5(IQR) = (75) = The upper limit is located 1.5(IQR) above Q3. Upper Limit: Q (IQR) = (75) = There are no outliers (values less than or greater than 637.5) in the apartment rent data. 27

28 Box Plot Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits Smallest value inside limits = 425 Largest value inside limits =

29 Measures of Association Between Two Variables Covariance Correlation Coefficient 29

30 Covariance The covariance is is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. 30

31 Covariance The correlation coefficient is computed as follows: s xy = ( x i x )( y i y ) n 1 for samples σ xy = ( x μ )( y μ ) i x i y N for populations 31

32 Correlation Coefficient The coefficient can take on values between -1 1 and +1. Values near -1 1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. 32

33 Correlation Coefficient The correlation coefficient is computed as follows: r xy s xy = ρ ss x y xy = σ xy σσ x y for samples for populations 33

34 Correlation Coefficient Correlation is is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it it does not mean that one variable is is the cause of the other. 34

35 Covariance and Correlation Coefficient A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score. Average Driving Distance (yds( yds.) Average 18-Hole Score

36 Covariance and Correlation Coefficient x y ( x x ) ( y y ) ( x x )( y y ) i i i i Average Std. Dev Total

37 Covariance and Correlation Coefficient Sample Covariance s xy ( x )( ) i x yi y = = = 7.08 n Sample Correlation Coefficient r xy s xy 7.08 = = = ss (8.2192)(.8944) x y

38 The Weighted Mean and Working with Grouped Data Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data 38

39 Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. 39

40 Weighted Mean x wx = w = i i wi where: x i = value of observation i w i = weight for observation i 40

41 Grouped Data The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. 41

42 Mean for Grouped Data Sample Data x fm = n = i i Population Data μ = f M i N i where: f i = frequency of class i M i = midpoint of class i 42

43 Sample Mean for Grouped Data Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the form of a frequency distribution. Rent ($) Frequency

44 Sample Mean for Grouped Data Rent ($) f i Total 70 M i f i M i ,525 x = = This approximation differs by $2.41 from the actual sample mean of $

45 Variance for Grouped Data For sample data s 2 = f i ( M i x ) n 1 2 For population data 2 σ = f i ( M i μ ) N 2 45

46 Sample Variance for Grouped Data Rent ($) f i Total 70 M i M i - x (M i - x) f i (M i - x) continued 46

47 Sample Variance for Grouped Data Sample Variance s 2 = 208,234.29/(70 1) = 3, Sample Standard Deviation s = 3, = This approximation differs by only $.20 from the actual standard deviation of $

48 End of Chapter 3, Part B 48

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