1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2009 Thomson South-Western. All Rights Reserved Chapter 3, Part B Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Exploratory Data Analysis n Measures of Association Between Two Variables n The Weighted Mean and Working with Grouped Data Working with Grouped Data

3 3 Slide © 2009 Thomson South-Western. All Rights Reserved Measures of Distribution Shape, Relative Location, and Detecting Outliers n Distribution Shape n z-Scores n Chebyshev’s Theorem n Empirical Rule n Detecting Outliers

4 4 Slide © 2009 Thomson South-Western. All Rights Reserved Distribution Shape: Skewness n An important measure of the shape of a distribution is called skewness. n The formula for computing skewness for a data set is somewhat complex. n Skewness can be easily computed using statistical software.

5 5 Slide © 2009 Thomson South-Western. All Rights Reserved Distribution Shape: Skewness n Symmetric (not skewed) Skewness is zero. Skewness is zero. Mean and median are equal. Mean and median are equal. Relative Frequency Skewness = 0 Skewness = 0

6 6 Slide © 2009 Thomson South-Western. All Rights Reserved Relative Frequency Distribution Shape: Skewness n Moderately Skewed Left Skewness is negative. Skewness is negative. Mean will usually be less than the median. Mean will usually be less than the median. Skewness = .31 Skewness = .31

7 7 Slide © 2009 Thomson South-Western. All Rights Reserved Distribution Shape: Skewness n Moderately Skewed Right Skewness is positive. Skewness is positive. Mean will usually be more than the median. Mean will usually be more than the median. Relative Frequency Skewness =.31 Skewness =.31

8 8 Slide © 2009 Thomson South-Western. All Rights Reserved Distribution Shape: Skewness n Highly Skewed Right Skewness is positive (often above 1.0). Skewness is positive (often above 1.0). Mean will usually be more than the median. Mean will usually be more than the median. Relative Frequency Skewness = 1.25 Skewness = 1.25

9 9 Slide © 2009 Thomson South-Western. All Rights Reserved Seventy efficiency apartments were randomly 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. Distribution Shape: Skewness n Example: Apartment Rents

10 Slide © 2009 Thomson South-Western. All Rights Reserved Distribution Shape: Skewness n Example: Apartment Rents

11 Slide © 2009 Thomson South-Western. All Rights Reserved Relative Frequency Skewness =.92 Skewness =.92 Distribution Shape: Skewness

12 Slide © 2009 Thomson South-Western. All Rights Reserved The z-score is often called the standardized value. The z-score is often called the standardized value. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. z-Scores

13 Slide © 2009 Thomson South-Western. All Rights Reserved z-Scores A data value less than the sample mean will have a A data value less than the sample mean will have a z-score less than zero. z-score less than zero. A data value greater than the sample mean will have A data value greater than the sample mean will have a z-score greater than zero. a z-score greater than zero. A data value equal to the sample mean will have a A data value equal to the sample mean will have a z-score of zero. z-score of zero. An observation’s z-score is a measure of the relative An observation’s z-score is a measure of the relative location of the observation in a data set. location of the observation in a data set.

14 Slide © 2009 Thomson South-Western. All Rights Reserved n z-Score of Smallest Value (425) z-Scores Standardized Values for Apartment Rents

15 Slide © 2009 Thomson South-Western. All Rights Reserved Chebyshev’s Theorem At least (1 - 1/ z 2 ) of the items in any data set will be 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 within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. At least (1 - 1/ z 2 ) of the items in any data set will be 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 within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1.

16 Slide © 2009 Thomson South-Western. All Rights Reserved At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 75%75% z = 2 standard deviations z = 2 standard deviations Chebyshev’s Theorem At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean.89%89% z = 3 standard deviations z = 3 standard deviations At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 94%94% z = 4 standard deviations z = 4 standard deviations

17 Slide © 2009 Thomson South-Western. All Rights Reserved For example: Chebyshev’s Theorem Let z = 1.5 with = and s = At least (1  1/(1.5) 2 ) = 1  0.44 = 0.56 or 56% of the rent values must be between - z ( s ) =  1.5(54.74) = z ( s ) =  1.5(54.74) = 409and + z ( s ) = (54.74) = z ( s ) = (54.74) = 573 (Actually, 86% of the rent values are between 409 and 573.) are between 409 and 573.)

18 Slide © 2009 Thomson South-Western. All Rights Reserved Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean.68.26%68.26% +/- 1 standard deviation of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %95.44% +/- 2 standard deviations of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean.99.72%99.72% +/- 3 standard deviations

19 Slide © 2009 Thomson South-Western. All Rights Reserved Empirical Rule x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

20 Slide © 2009 Thomson South-Western. All Rights Reserved Detecting Outliers An outlier is an unusually small or unusually large An outlier is an unusually small or unusually large value in a data set. value in a data set. A data value with a z-score less than -3 or greater A data value with a z-score less than -3 or greater than +3 might be considered an outlier. than +3 might be considered an outlier. It might be: It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the a data value that was incorrectly included in the data set data set a correctly recorded data value that belongs in a correctly recorded data value that belongs in the data set the data set

21 Slide © 2009 Thomson South-Western. All Rights Reserved Detecting Outliers The most extreme z-scores are and 2.27 The most extreme z-scores are and 2.27 Using | z | > 3 as the criterion for an outlier, there are Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. no outliers in this data set. Standardized Values for Apartment Rents

22 Slide © 2009 Thomson South-Western. All Rights Reserved Exploratory Data Analysis n Five-Number Summary n Box Plot

23 Slide © 2009 Thomson South-Western. All Rights Reserved Five-Number Summary 1 Smallest Value Smallest Value First Quartile First Quartile Median Median Third Quartile Third Quartile Largest Value Largest Value

24 Slide © 2009 Thomson South-Western. All Rights Reserved Five-Number Summary Lowest Value = 425 First Quartile = 445 Median = 475 Third Quartile = 525 Largest Value = 615

25 Slide © 2009 Thomson South-Western. All Rights Reserved A box is drawn with its ends located at the first and A box is drawn with its ends located at the first and third quartiles. third quartiles. Box Plot A vertical line is drawn in the box at the location of A vertical line is drawn in the box at the location of the median (second quartile). the median (second quartile). Q1 = 445 Q3 = 525 Q2 = 475

26 Slide © 2009 Thomson South-Western. All Rights Reserved Box Plot n Limits are located (not drawn) using the interquartile range (IQR). n Data outside these limits are considered outliers. n The locations of each outlier is shown with the symbol *. … continued

27 Slide © 2009 Thomson South-Western. All Rights Reserved Box Plot Lower Limit: Q (IQR) = (80) = 325 Upper Limit: Q (IQR) = (80) = 645 The lower limit is located 1.5(IQR) below Q 1. The lower limit is located 1.5(IQR) below Q 1. The upper limit is located 1.5(IQR) above Q 3. The upper limit is located 1.5(IQR) above Q 3. There are no outliers (values less than 325 or There are no outliers (values less than 325 or greater than 645) in the apartment rent data. greater than 645) in the apartment rent data.

28 Slide © 2009 Thomson South-Western. All Rights Reserved Box Plot n 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 = 615

29 Slide © 2009 Thomson South-Western. All Rights Reserved Measures of Association Between Two Variables n Covariance n Correlation Coefficient

30 Slide © 2009 Thomson South-Western. All Rights Reserved Covariance Positive values indicate a positive relationship. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Negative values indicate a negative relationship. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables.

31 Slide © 2009 Thomson South-Western. All Rights Reserved Covariance The covariance is computed as follows: The covariance is computed as follows: forsamples forpopulations

32 Slide © 2009 Thomson South-Western. All Rights Reserved Correlation Coefficient Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation.

33 Slide © 2009 Thomson South-Western. All Rights Reserved The correlation coefficient is computed as follows: The correlation coefficient is computed as follows: forsamplesforpopulations Correlation Coefficient

34 Slide © 2009 Thomson South-Western. All Rights Reserved Correlation Coefficient Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1.

35 Slide © 2009 Thomson South-Western. All Rights Reserved A golfer is interested in investigating the A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score Average Driving Distance (yds.) Average 18-Hole Score Covariance and Correlation Coefficient

36 Slide © 2009 Thomson South-Western. All Rights Reserved Covariance and Correlation Coefficient xy Average Std. Dev Total

37 Slide © 2009 Thomson South-Western. All Rights Reserved n Sample Covariance n Sample Correlation Coefficient Covariance and Correlation Coefficient

38 Slide © 2009 Thomson South-Western. All Rights Reserved The Weighted Mean and Working with Grouped Data n Weighted Mean n Mean for Grouped Data n Variance for Grouped Data n Standard Deviation for Grouped Data

39 Slide © 2009 Thomson South-Western. All Rights Reserved Weighted Mean When the mean is computed by giving each data When the mean is computed by giving each data value a weight that reflects its importance, it is value a weight that reflects its importance, it is referred to as a weighted mean. referred to as a weighted mean. In the computation of a grade point average (GPA), In the computation of a grade point average (GPA), the weights are the number of credit hours earned for the weights are the number of credit hours earned for each grade. each grade. When data values vary in importance, the analyst When data values vary in importance, the analyst must choose the weight that best reflects the must choose the weight that best reflects the importance of each value. importance of each value.

40 Slide © 2009 Thomson South-Western. All Rights Reserved Weighted Mean where: x i = value of observation i x i = value of observation i w i = weight for observation i w i = weight for observation i

41 Slide © 2009 Thomson South-Western. All Rights Reserved Grouped Data The weighted mean computation can be used to The weighted mean computation can be used to obtain approximations of the mean, variance, and obtain approximations of the mean, variance, and standard deviation for the grouped data. standard deviation for the grouped data. To compute the weighted mean, we treat the To compute the weighted mean, we treat the midpoint of each class as though it were the mean midpoint of each class as though it were the mean of all items in the class. of all items in the class. We compute a weighted mean of the class midpoints We compute a weighted mean of the class midpoints using the class frequencies as weights. using the class frequencies as weights. Similarly, in computing the variance and standard Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. deviation, the class frequencies are used as weights.

42 Slide © 2009 Thomson South-Western. All Rights Reserved Mean for Grouped Data where: f i = frequency of class i f i = frequency of class i M i = midpoint of class i M i = midpoint of class i n For sample data n For population data

43 Slide © 2009 Thomson South-Western. All Rights Reserved 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. Sample Mean for Grouped Data

44 Slide © 2009 Thomson South-Western. All Rights Reserved Sample Mean for Grouped Data This approximation differs by $2.41 from the actual sample mean of $

45 Slide © 2009 Thomson South-Western. All Rights Reserved Variance for Grouped Data n For sample data n For population data

46 Slide © 2009 Thomson South-Western. All Rights Reserved Sample Variance for Grouped Data continued

47 Slide © 2009 Thomson South-Western. All Rights Reserved s 2 = 208,234.29/(70 – 1) = 3, This approximation differs by only $.20 from the actual standard deviation of $ Sample Variance for Grouped Data n Sample Variance n Sample Standard Deviation

48 Slide © 2009 Thomson South-Western. All Rights Reserved End of Chapter 3, Part B