St. Edward’s University SLIDES BY John Loucks St. Edward’s University .

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

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

Mean Perhaps the most important measure of location is the mean. The mean provides a measure of central location. 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 m.

Sample Mean Sum of the values of the n observations Number of in the sample

Population Mean m Sum of the values of the N observations Number of observations in the population

Sample Mean Example: Apartment Rents Fifty efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below.

Sample Mean Xbar = SUM(X)/n

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.

Median For an odd number of observations: 26 18 27 12 14 27 19 in ascending order the median is the middle value. Median = 19

Median For an even number of observations: 26 18 27 12 14 27 30 19 in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5

Averaging the 25 and 26th data values: Median Example: Apartment Rents Averaging the 25 and 26th data values: Median = (216 + 220)/2 = 218 Note: Data is in ascending order.

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

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

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.

Measures of Variability Range Variance Standard Deviation Coefficient of Variation

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.

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

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 (xi) and the mean ( for a sample, m for a population). The variance is useful in comparing the variability of two or more variables.

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

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

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: for a sample for a population

Sample Mean Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below.

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Formula Worksheet A B C D E 1 Apart- ment Monthly Rent ($) 2 525 Mean =AVERAGE(B2:B71) 3 440 Median =MEDIAN(B2:B71) 4 450 Mode =MODE.SNGL(B2:B71) 5 615 Variance =VAR.S(B2:B71) 6 480 Std. Dev. =STDEV.S(B2:B71) 7 510 C.V. =E6/E2*100 Note: Rows 8-71 are not shown.

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Value Worksheet A B C D E 1 Apart- ment Monthly Rent ($) 2 525 Mean 490.80 3 440 Median 475.00 4 450 Mode 450.00 5 615 Variance 2996.16 6 480 Std. Dev. 54.74 7 510 C.V. 11.15 Note: Rows 8-71 are not shown.

Using Excel’s Descriptive Statistics Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Descriptive Statistics from the list of Analysis Tools Step 4 When the Descriptive Statistics dialog box appears: (see details on next slide)

Using Excel’s Descriptive Statistics Tool Excel’s Descriptive Statistics Dialog Box

Using Excel’s Descriptive Statistics Tool Excel Value Worksheet (Partial) Note: Rows 9-71 are not shown.

Using Excel’s Descriptive Statistics Tool Excel Value Worksheet (Partial) Note: Rows 1-8 and 17-71 are not shown.

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. 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.

Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Using Excel’s Percentile Function The formula Excel uses to compute the location (Lp) of the pth percentile is Lp = pn + (1 – p) Excel would compute the location of the 80th percentile for the apartment rent data as follows: L80 = (0.8)70 + (1 – 0.8) = 56 + .2 = 56.2 The 80th percentile would be 535 + .2(549 - 535) = 535 + 2.8 = 537.8 Excel interpolates over the interval from 0 to n.

Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Excel Formula Worksheet 80th percentile A B C D 1 Apart- ment Monthly Rent ($) 80 th Percentile 2 525 =PERCENTILE.INC(B2:B71,.8) 3 440 4 450 5 615 6 480 It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

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

Third Quartile Using Excel’s QUARTILE.INC Function Excel computes the locations of the 1st, 2nd, and 3rd quartiles by first converting the quartiles to percentiles and then using the following formula to compute the location (Lp) of the pth percentile: Lp = (p/100)n + (1 – p/100) Excel would compute the location of the 3rd quartile (75th percentile) for the rent data as follows: L75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75 The 3rd quartile would be 515 + .75(525 - 515) = 515 + 7.5 = 522.5

Third Quartile Excel Formula Worksheet 3rd quartile B C D 1 Apart- ment Monthly Rent ($) Third Quartile 2 525 =QUARTILE.INC(B2:B71,3) 3 440 4 450 5 615 6 480 It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

Using Excel’s QUARTILE.INC Function If the value of 1 in the QUARTILE.INC function is changed to 0, Excel computes the minimum value in the data set. If the value of 1 is changed to 4, Excel computes the maximum value in the data set.

Excel’s Rank and Percentile Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Rank and Percentile from the list of Analysis Tools Step 4 When the Rank and Percentile dialog box appears (see details on next slide)

Excel’s Rank and Percentile Tool Step 4 Complete the Rank and Percentile dialog box as follows:

Excel’s Rank and Percentile Tool Excel Value Worksheet Note: Rows 11-71 are not shown.

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.

Interquartile Range = Q3 - Q1 = 525 - 445 = 80 Example: Apartment Rents 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 Note: Data is in ascending order.

Trimmed Mean Another measure, sometimes used when extreme values are present, is the 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.

End of Chapter 3, Part A