1 1 Slide © 2003 Thomson/South-Western

2 2 Slide © 2003 Thomson/South-Western Chapter 3 Descriptive Statistics: Numerical Methods Part A n Measures of Location n Measures of Variability x x     % %

3 3 Slide © 2003 Thomson/South-Western Measures of Location n Mean n Median n Mode n Percentiles n Quartiles

4 4 Slide © 2003 Thomson/South-Western Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.

5 5 Slide © 2003 Thomson/South-Western Mean n The mean of a data set is the average of all the data values. n If the data are from a sample, the mean is denoted by. If the data are from a population, the mean is denoted by  (mu). If the data are from a population, the mean is denoted by  (mu).

6 6 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Mean

7 7 Slide © 2003 Thomson/South-Western Median n The median is the measure of location most often reported for annual income and property value data. n A few extremely large incomes or property values can inflate the mean.

8 8 Slide © 2003 Thomson/South-Western Median n The median of a data set is the value in the middle when the data items are arranged in ascending order. n For an odd number of observations, the median is the middle value. n For an even number of observations, the median is the average of the two middle values.

9 9 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Median Median = 50th percentile Median = 50th percentile i = ( p /100) n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = ( )/2 = 475

10 Slide © 2003 Thomson/South-Western Mode n The mode of a data set is the value that occurs with greatest frequency. n The greatest frequency can occur at two or more different values. n If the data have exactly two modes, the data are bimodal. n If the data have more than two modes, the data are multimodal.

11 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Mode 450 occurred most frequently (7 times) 450 occurred most frequently (7 times) Mode = 450 Mode = 450

12 Slide © 2003 Thomson/South-Western Percentiles n A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. n Admission test scores for colleges and universities are frequently reported in terms of percentiles.

13 Slide © 2003 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. 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 i = ( p /100) n If i is not an integer, round up. The p th percentile is the value in the i th position. If i is not an integer, round up. The p th percentile is the value in the i th position. If i is an integer, the p th percentile is the average of the values in positions i and i +1. If i is an integer, the p th percentile is the average of the values in positions i and i +1. Percentiles

14 Slide © 2003 Thomson/South-Western Example: Apartment Rents n 90th Percentile i = ( p /100) n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = ( )/2 = th Percentile = ( )/2 = 585

15 Slide © 2003 Thomson/South-Western Quartiles n Quartiles are specific percentiles n First Quartile = 25th Percentile n Second Quartile = 50th Percentile = Median n Third Quartile = 75th Percentile

16 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Third Quartile Third quartile = 75th percentile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Third quartile = 525

17 Slide © 2003 Thomson/South-Western Measures of Variability n It is often desirable to consider measures of variability (dispersion), as well as measures of location. n 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.

18 Slide © 2003 Thomson/South-Western Measures of Variability n Range n Interquartile Range n Variance n Standard Deviation n Coefficient of Variation

19 Slide © 2003 Thomson/South-Western Range n The range of a data set is the difference between the largest and smallest data values. n It is the simplest measure of variability. n It is very sensitive to the smallest and largest data values.

20 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = = 190 Range = = 190

21 Slide © 2003 Thomson/South-Western Interquartile Range n The interquartile range of a data set is the difference between the third quartile and the first quartile. n It is the range for the middle 50% of the data. n It overcomes the sensitivity to extreme data values.

22 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Interquartile Range 3rd Quartile ( Q 3) = 525 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = = 80 Interquartile Range = Q 3 - Q 1 = = 80

23 Slide © 2003 Thomson/South-Western Variance n 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 ( x for a sample,  for a population). It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample,  for a population).

24 Slide © 2003 Thomson/South-Western Variance n The variance is the average of the squared differences between each data value and the mean. n If the data set is a sample, the variance is denoted by s 2. If the data set is a population, the variance is denoted by  2. If the data set is a population, the variance is denoted by  2.

25 Slide © 2003 Thomson/South-Western Standard Deviation n The standard deviation of a data set is the positive square root of the variance. n It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. n If the data set is a sample, the standard deviation is denoted s. If the data set is a population, the standard deviation is denoted  (sigma). If the data set is a population, the standard deviation is denoted  (sigma).

26 Slide © 2003 Thomson/South-Western Coefficient of Variation n The coefficient of variation indicates how large the standard deviation is in relation to the mean. n If the data set is a sample, the coefficient of variation is computed as follows: n If the data set is a population, the coefficient of variation is computed as follows:

27 Slide © 2003 Thomson/South-Western Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

28 Slide © 2003 Thomson/South-Western End of Chapter 3, Part A