1. Chapter 3, Part A Descriptive Statistics: Numerical Measures n Measures of Location n Measures of Variability

2. 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

3. 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 . n Perhaps the most important measure of location is the mean. n The mean provides a measure of central location.

4. 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

5. 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

6. Seventy apartments were randomly Seventy apartments were randomly sampled in a Suva. The monthly rent prices for these apartments are listed below. Sample Mean Example: Apartment Rents Example: Apartment Rents

7. Sample Mean Example: Apartment Rents Example: Apartment Rents

8. 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.

9. Median 121419262718 27 For an odd number of observations: For an odd number of observations: in ascending order 26182712142719 7 observations the median is the middle value. Median = 19

10. 121419262718 27 Median For an even number of observations: For an even number of observations: in ascending order 26182712142730 8 observations the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 19 30

11. Median Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

12. Trimmed Mean It is obtained by deleting a percentage of the It is obtained by deleting a percentage of the smallest and largest values from a data set and then smallest and largest values from a data set and then computing the mean of the remaining values. computing the mean of the remaining values. For example, the 5% trimmed mean is obtained by For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the removing the smallest 5% and the largest 5% of the data values and then computing the mean of the data values and then computing the mean of the remaining values. remaining values. Another measure, sometimes used when extreme Another measure, sometimes used when extreme values are present, is the trimmed mean. values are present, is the trimmed mean.

13. 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. Caution: If the data are bimodal or multimodal, Caution: If the data are bimodal or multimodal, Excel’s MODE function will incorrectly identify a Excel’s MODE function will incorrectly identify a single mode. single mode.

14. Mode 450 occurred most frequently (7 times) Mode = 450 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

15. 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. 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.

16. 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 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. 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.

17. 80 th Percentile i = ( p /100) n = (80/100)70 = 56 Averaging the 56 th and 57 th data values: 80th Percentile = (535 + 549)/2 = 542 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

18. 80 th Percentile “At least 80% of the items take on a items take on a value of 542 or less.” “At least 20% of the items take on a value of 542 or more.” 56/70 =.8 or 80%14/70 =.2 or 20% Example: Apartment Rents Example: Apartment Rents

19. 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

20. Third Quartile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

21. 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.

22. Measures of Variability Range Range Interquartile Range Interquartile Range Variance Variance Standard Deviation Standard Deviation Coefficient of Variation Coefficient of Variation

23. 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.

24. Range Range = largest value - smallest value Range = 615 - 425 = 190 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

25. 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.

26. Interquartile Range 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

27. 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). The variance is useful in comparing the variability The variance is useful in comparing the variability of two or more variables. of two or more variables.

28. 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. 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

29. 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.

30. The standard deviation is computed as follows: The standard deviation is computed as follows: for a sample population Standard Deviation

31. 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. 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

32. the standard deviation is about 11% of the mean Variance Variance Standard Deviation Standard Deviation Coefficient of Variation Coefficient of Variation Sample Variance, Standard Deviation, And Coefficient of Variation Example: Apartment Rents Example: Apartment Rents

33. End of Chapter 3, Part A