1 1 Slide © 2006 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2006 Thomson/South-Western Chapter 3 Descriptive Statistics: Numerical Measures Part A n Measures of Location n Measures of Variability

3 3 Slide © 2006 Thomson/South-Western Measures of Location If the measures are computed for data from a sample, for data from a sample, they are called sample statistics. If the measures are computed for data from a population, 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. n Mean n Median n Mode n Percentiles n Quartiles

4 4 Slide © 2006 Thomson/South-Western Mean n The mean of a data set is the average of all the data values. The sample mean is the point estimator of the population mean . The sample mean is the point estimator of the population mean .

5 5 Slide © 2006 Thomson/South-Western Sample Mean Number of observations in the sample Number of observations in the sample Sum of the values of the n observations Sum of the values of the n observations

6 6 Slide © 2006 Thomson/South-Western Population Mean  Number of observations in the population Number of observations in the population Sum of the values of the N observations Sum of the values of the N observations

7 7 Slide © 2006 Thomson/South-Western Seventy efficiency apartments 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. Sample Mean Example: Apartment Rents Example: Apartment Rents

8 8 Slide © 2006 Thomson/South-Western Sample Mean

9 9 Slide © 2006 Thomson/South-Western Sample Mean

10 Slide © 2006 Thomson/South-Western Median Whenever a data set has extreme values, the median Whenever a data set has extreme values, the median is the preferred measure of central location. is the preferred measure of central location. A few extremely large incomes or property values A few extremely large incomes or property values can inflate the mean. can inflate the mean. The median is the measure of location most often The median is the measure of location most often reported for annual income and property value data. reported for annual income and property value data. The median of a data set is the value in the middle The median of a data set is the value in the middle when the data items are arranged in ascending order. when the data items are arranged in ascending order.

11 Slide © 2006 Thomson/South-Western Median For an odd number of observations: For an odd number of observations: in ascending order observations the median is the middle value. Median = 19

12 Slide © 2006 Thomson/South-Western Median For an even number of observations: For an even number of observations: in ascending order observations the median is the average of the middle two values. Median = ( )/2 =

13 Slide © 2006 Thomson/South-Western Median Averaging the 35th and 36th data values: Median = ( )/2 = 475

14 Slide © 2006 Thomson/South-Western Mode The mode of a data set is the value that occurs with The mode of a data set is the value that occurs with greatest frequency. greatest frequency. The greatest frequency can occur at two or more The greatest frequency can occur at two or more different values. different values. If the data have exactly two modes, the data are If the data have exactly two modes, the data are bimodal. bimodal. If the data have more than two modes, the data are If the data have more than two modes, the data are multimodal. multimodal.

15 Slide © 2006 Thomson/South-Western Mode 450 occurred most frequently (7 times) Mode = 450

16 Slide © 2006 Thomson/South-Western Percentiles A percentile provides information about how the A percentile provides information about how the data are spread over the interval from the smallest data are spread over the interval from the smallest value to the largest value. value to the largest value. Admission test scores for colleges and universities Admission test scores for colleges and universities are frequently reported in terms of percentiles. are frequently reported in terms of percentiles.

17 Slide © 2006 Thomson/South-Western n The p th 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. Percentiles

18 Slide © 2006 Thomson/South-Western Percentiles Arrange the data in ascending order. Arrange the data in ascending order. Compute index i, the position of the p th percentile. Compute index i, the position of the p th percentile. i = ( p /100) n If i is not an integer, round up. The p th percentile If i is not an integer, round up. The p th percentile is the value in the i th position. is the value in the i th position. If i is an integer, the p th percentile is the average If i is an integer, the p th percentile is the average of the values in positions i and i +1. of the values in positions i and i +1.

19 Slide © 2006 Thomson/South-Western 90 th Percentile i = ( p /100) n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = ( )/2 = 585

20 Slide © 2006 Thomson/South-Western 90 th Percentile “At least 90% of the items of the items take on a value take on a value of 585 or less.” of 585 or less.” “At least 10% of the items of the items take on a value take on a value of 585 or more.” of 585 or more.” 63/70 =.9 or 90%7/70 =.1 or 10%

21 Slide © 2006 Thomson/South-Western Quartiles Quartiles are specific percentiles. Quartiles are specific percentiles. First Quartile = 25th Percentile First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile Third Quartile = 75th Percentile

22 Slide © 2006 Thomson/South-Western Third Quartile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525

23 Slide © 2006 Thomson/South-Western Measures of Variability It is often desirable to consider measures of variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we For example, in choosing supplier A or supplier B we might consider not only the average delivery time for might consider not only the average delivery time for each, but also the variability in delivery time for each. each, but also the variability in delivery time for each.

24 Slide © 2006 Thomson/South-Western Measures of Variability n Range n Interquartile Range n Variance n Standard Deviation n Coefficient of Variation

25 Slide © 2006 Thomson/South-Western Range The range of a data set is the difference between the The range of a data set is the difference between the largest and smallest data values. largest and smallest data values. It is the simplest measure of variability. It is the simplest measure of variability. It is very sensitive to the smallest and largest data It is very sensitive to the smallest and largest data values. values.

26 Slide © 2006 Thomson/South-Western Range Range = largest value - smallest value Range = = 190

27 Slide © 2006 Thomson/South-Western Interquartile Range The interquartile range of a data set is the difference The interquartile range of a data set is the difference between the third quartile and the first quartile. between the third quartile and the first quartile. It is the range for the middle 50% of the data. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. It overcomes the sensitivity to extreme data values.

28 Slide © 2006 Thomson/South-Western Interquartile Range 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = = 80

29 Slide © 2006 Thomson/South-Western The variance is a measure of variability that utilizes The variance is a measure of variability that utilizes all the data. all the data. Variance It is based on the difference between the value of It is based on the difference between the value of each observation ( x i ) and the mean ( for a sample, each observation ( x i ) and the mean ( for a sample,  for a population).  for a population).

30 Slide © 2006 Thomson/South-Western Variance The variance is computed as follows: The variance is computed as follows: The variance is the average of the squared The variance is the average of the squared differences between each data value and the mean. differences between each data value and the mean. for a sample population

31 Slide © 2006 Thomson/South-Western Standard Deviation The standard deviation of a data set is the positive The standard deviation of a data set is the positive square root of the variance. square root of the variance. It is measured in the same units as the data, making It is measured in the same units as the data, making it more easily interpreted than the variance. it more easily interpreted than the variance.

32 Slide © 2006 Thomson/South-Western The standard deviation is computed as follows: The standard deviation is computed as follows: for a sample population Standard Deviation

33 Slide © 2006 Thomson/South-Western The coefficient of variation is computed as follows: The coefficient of variation is computed as follows: Coefficient of Variation The coefficient of variation indicates how large the The coefficient of variation indicates how large the standard deviation is in relation to the mean. standard deviation is in relation to the mean. for a sample population

34 Slide © 2006 Thomson/South-Western the standard deviation is about 11% of the mean n Variance n Standard Deviation n Coefficient of Variation Variance, Standard Deviation, And Coefficient of Variation