1 1 Slide Descriptive Statistics: Numerical Measures Location and Variability Chapter 3 BA 201

2 2 Slide Sample Statistics, Population Parameters, and Point Estimators 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.

3 3 Slide LOCATION

4 4 Measures of Location n n Mean n n Median n n Mode n n Percentiles n n Quartiles

5 5 Slide Mean n 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 . n n The mean provides a measure of central location.

6 6 Slide Mean SamplePopulation

7 7 Slide Sample Mean Apartment Rents

8 8 Slide Trimmed Mean It is obtained by deleting a percentage of the smallest and largest values from a data set and then computing the mean of the remaining values. For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the data values and then computing the mean of the remaining values. Another measure, sometimes used when extreme values are present, is the trimmed mean.

9 9 Slide Median Whenever a data set has extreme values, the median is the preferred measure of central location. The median of a data set is the value in the middle when the data items are arranged in ascending order.

10 Slide Median For an odd number of observations: in ascending order observations the median is the middle value. Median = 19

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

12 Slide Median Averaging the 35th and 36th data values: Median = ( )/2 = 475 Example: Apartment Rents

13 Slide 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.

14 Slide Mode 450 occurred most frequently (7 times) Mode = 450 Apartment Rents

15 Slide PRACTICE

16 Slide Practice # Compute the …   Mean   Median   Mode

17 Slide Practice #1 - Mean

18 Slide Practice #1 - Median

19 Slide Practice #1 - Mode

20 Slide Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. n 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.

21 Slide Percentiles Arrange the data in ascending order. 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 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.

22 Slide 80 th Percentile i = ( p /100) n = (80/100)70 = 56 Averaging the 56 th and 57 th data values: 80th Percentile = ( )/2 = 542 Apartment Rents

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

24 Slide Third Quartile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Apartment Rents

25 Slide PRACTICE

26 Slide Practice #2 - Percentiles th Percentile

27 Slide VARIABILITY

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

29 Slide Range The range of a data set is the difference between the largest and smallest data values.

30 Slide Range Range = largest value - smallest value Range = = 190 Apartment Rents

31 Slide 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.

32 Slide Interquartile Range 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = = 80 Apartment Rents

33 Slide PRACTICE RANGE

34 Slide Practice #3 - Range Range

35 Slide VARIANCE

36 Slide The variance is a measure of variability that utilizes all the data. Variance It is based on the difference between the value of each observation ( x i ) and the mean ( for a sample,  for a population). The variance is useful in comparing the variability of two or more variables.

37 Slide Variance The variance is computed as follows: The variance is the average of the squared differences between each data value and the mean. for a sample for a population

38 Slide Variance Sample Variance Apartment Rents

39 Slide Detailed Example - Variance = = = 11 2 = = 00 2 = = = = 11 2 = s 2 = 4/(5-1) = 1 b c d e f g a

40 Slide 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.

41 Slide The standard deviation is computed as follows: for a sample for a population Standard Deviation

42 Slide Standard Deviation Apartment Rents

43 Slide Detailed Example – Standard Deviation s 2 = 4/(5-1) = 1 a

44 Slide PRACTICE VARIANCE AND STANDARD DEVIATION

45 Slide Practice #4 – Variance

46 Slide Practice #4 – Standard Deviation

47 Slide The coefficient of variation is computed as follows: Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. for a sample for a population

48 Slide the standard deviation is about 11% of the mean Variance Standard Deviation Coefficient of Variation Sample Variance, Standard Deviation, And Coefficient of Variation Apartment Rents

49 Slide PRACTICE COEFFICIENT OF VARIATION

50 Slide Practice #5 – Coefficient of Variation s=

51 Slide