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2 Chapter 3 Descriptive Statistics: Numerical Measures Measures of Location Measures of Variability Measures of Distribution Shape, Relative Location, and Detecting Outliers Measures of Association Between Two Variables Weighted Mean Slide 2

3 Measures of Location Mean Median Mode Percentiles Quartiles If the measures are computed for data from a sample, they are called sample statistics. If the measures are computed for data from a population, they are called population parameters. A sample statistic is referred to as the point estimator of the corresponding population parameter. Slide 3

4 Mean The mean of a data set is the average of all the data values. x The sample mean population mean m. is the point estimator of the Slide 4

5 Sample Mean x x n x i Sum of the values of the n observations Number of observations in the sample Slide 5

6 Population Mean m m N x i Sum of the values of the N observations Number of observations in the population Slide 6

7 Median The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. The median is the measure of location most often reported for annual income and property value data. A few extremely large incomes or property values can inflate the mean. Slide 7

8 Median For an odd number of observations: observations in ascending order the median is the middle value. Median = 26 Slide 8

9 Median For an even number of observations: observations in ascending order the median is the average of the middle two values. Median = ( )/2 = 26.5 Slide 9

10 Mean VS Median The mean IS affected by outliers (extreme observations) The median IS NOT affected by outliers Slide 10

11 Mode The mode of a data set is the value that occurs with greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data are bimodal. If the data have more than two modes, the data are multimodal. Slide 11

12 Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universities are frequently reported in terms of percentiles. Slide 12

13 Percentiles The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more. Slide 13

14 Percentiles Arrange the data in ascending order. Compute index i, the position of the pth percentile. i = (p/100)n If i is not an integer, round up. The pth percentile is the value in the ith position. If i is an integer, the pth percentile is the average of the values in positions i and i+1. Slide 14

15 Note on Excel s Percentile Function The formula that Excel uses is different from the one used in the textbook! In order to find the observation where the median occurs, Excel uses the following formula: L p = (p/100)n + (1 p/100) Once the observation is identified Excel will: 1. If L p is a whole number (e.g. 12), Excel s result will be the same as the textbook s. 2. If Lp is not a whole number (e.g. 12.3) Excel s result will be different from the textbook s. Slide 15

16 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile Slide 16

17 Measures of Variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each. Slide 17

18 Range Interquartile Range Variance Standard Deviation Coefficient of Variation Measures of Variability Slide 18

19 Range The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values. Slide 19

20 Interquartile Range The interquartile range of a data set is the difference between the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. Slide 20

21 Variance The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation (x i ) and the mean ( for a sample, m for a population). x Slide 21

22 Variance The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows: s 2 ( x i x ) n1 2 2 ( xi m) N 2 for a sample for a population Slide 22

23 Standard Deviation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily interpreted than the variance. Slide 23

24 Standard Deviation The standard deviation is computed as follows: s s 2 2 for a sample for a population Slide 24

25 Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: s x 100 % for a sample 100 % m for a population Slide 25

26 Measures of Distribution Shape, Relative Location, and Detecting Outliers Distribution Shape z-scores Chebyshev s Theorem Empirical Rule Detecting Outliers Slide 26

27 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. Excel s SKEW function can be used to compute the skewness of a data set. Slide 27

28 Relative Frequency Distribution Shape: Skewness Symmetric (not skewed) Skewness is zero. Mean and median are equal Skewness = 0 Slide 28

29 Relative Frequency Distribution Shape: Skewness Moderately Skewed Left Skewness is negative. Mean will usually be less than the median Skewness =.31 Slide 29

30 Relative Frequency Distribution Shape: Skewness Moderately Skewed Right Skewness is positive. Mean will usually be more than the median Skewness =.31 Slide 30

31 z-scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value x i is from the mean. z i x i s x Slide 31

32 z-scores An observation s z-score 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 greater than zero. A data value equal to the sample mean will have a z-score of zero. Slide 32

33 Chebyshev 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. Slide 33

34 Chebyshev s Theorem At least within 75% of the data values must be z = 2 standard deviations of the mean. At least within 89% of the data values must be z = 3 standard deviations of the mean. At least within 94% of the data values must be z = 4 standard deviations of the mean. Slide 34

35 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. Slide 35

36 Empirical Rule 99.72% 95.44% 68.26% m 3 m 1 m 2 m m + 1 m + 3 m + 2 x Slide 36

37 Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 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 Slide 37

38 Covariance Correlation Coefficient Measures of Association Between Two Variables Slide 38

39 Covariance The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Slide 39

40 Covariance The covariance coefficient is computed as follows: s xy ( xi x )( yi y) n 1 for samples xy ( x m )( y m ) i x i y N for populations Slide 40

41 Correlation Coefficient The coefficient can take on values between -1 and +1. Values near -1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. Slide 41

42 Correlation Coefficient The correlation coefficient is computed as follows: r xy s xy s s x y xy x xy y for samples for populations Slide 42

43 Correlation Coefficient Correlation is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it does not mean that one variable is the cause of the other. Slide 43

44 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. Slide 44

45 Weighted Mean x wx i i wi where: x i = value of observation i w i = weight for observation i Slide 45

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