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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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Chapter 3 Descriptive Statistics: Numerical Methods, Part A

- Measures of Location
- Measures of Variability

%

x

Measures of Location

- Mean
- Median
- Mode
- Percentiles
- Quartiles

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.

Mean

- The mean of a data set is the average of all the data values.
- If the data are from a sample, the mean is denoted by

.

- If the data are from a population, the mean is denoted by m (mu).

Median

- 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

- The median of a data set is the value in the middle when the data items are arranged in ascending order.
- For an odd number of observations, the median is the middle value.
- For an even number of observations, the median is the average of the two middle values.

Example: Apartment Rents

- Median

Median = 50th percentile

i = (p/100)n = (50/100)70 = 35.5 Averaging the 35th and 36th data values:

Median = (475 + 475)/2 = 475

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.

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.

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.
- Arrange the data in ascending order.
- Compute index i, the position of the pth percentile.

i = (p/100)n

- If i is not an integer, round up. The pth percentile is the value in the ith position.
- If i is an integer, the pth percentile is the average of the values in positions i and i+1.

Example: Apartment Rents

- 90th Percentile

i = (p/100)n = (90/100)70 = 63

Averaging the 63rd and 64th data values:

90th Percentile = (580 + 590)/2 = 585

Quartiles

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

Example: Apartment Rents

- Third Quartile

Third quartile = 75th percentile

i = (p/100)n = (75/100)70 = 52.5 = 53

Third quartile = 525

Using Excel to ComputePercentiles and Quartiles

- Unsorted Monthly Rent ($)

Note: Rows 7-71 are not shown.

Using Excel to ComputePercentiles and Quartiles

- Sorting Data

Step 1 Select any cell containing data in column B

Step 2 Select the Data pull-down menu

Step 3 Choose the Sort option

Step 4 When the Sort dialog box appears:

In the Sort by box, make sure that Monthly Rent ($) appears and that Ascending is selected

In the My list has box, make sure that Header row is selected

Click OK

Using Excel to ComputePercentiles and Quartiles

- Sorted Monthly Rent ($)

Note: Rows 7-71 are not shown.

Using Excel to ComputePercentiles and Quartiles

- Formula Worksheet for 90th Percentile’s Index

Note: Rows 7-71 are not shown.

Using Excel to ComputePercentiles and Quartiles

- Value Worksheet for 90th Percentile’s Index

Note: Rows 7-71 are not shown.

Using Excel to ComputePercentiles and Quartiles

- Value Worksheet for 3rd Quartile’s Index

Note: Rows 7-71 are not shown.

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
- Interquartile 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.

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.

Example: Apartment Rents

- Interquartile Range

3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

Interquartile Range = Q3 - Q1 = 525 - 445 = 80

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 (x for a sample, m for a population).

Variance

- The variance is the average of the squared differences between each data value and the mean.
- If the data set is a sample, the variance is denoted by s2.
- If the data set is a population, the variance is denoted by 2.

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 comparable, than the variance, to the mean.
- 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).

Coefficient of Variation

- The coefficient of variation indicates how large the standard deviation is in relation to the mean.
- If the data set is a sample, the coefficient of variation is computed as follows:
- If the data set is a population, the coefficient of variation is computed as follows:

Example: Apartment Rents

- Variance
- Standard Deviation
- Coefficient of Variation

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation

- Formula Worksheet

Note: Rows 8-71 are not shown.

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation

- Value Worksheet

Note: Rows 8-71 are not shown.

Using Excel’sDescriptive Statistics Tool

Step 1 Select the Tools pull-down menu

Step 2 Choose the Data Analysis option

Step 3 Choose Descriptive Statistics from the list of

Analysis Tools

… continued

Using Excel’sDescriptive Statistics Tool

Step 4 When the Descriptive Statistics dialog box appears:

Enter B1:B71 in the Input Range box Select Grouped By Columns

Select Labels in First Row

Select Output Range

Enter D1 in the Output Range box

Select Summary Statistics

Click OK

Using Excel’sDescriptive Statistics Tool

- Value Worksheet (Partial)

Note: Rows 1-8 and 17-71 are not shown.

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