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

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Slides Prepared by

JOHN S. LOUCKS

St. Edward’s University

- Measures of Location
- Measures of Variability

%

x

- Mean
- Median
- Mode
- Percentiles
- Quartiles

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.

- 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).

- Mean

- 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.

- 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.

- 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

- 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.

- Mode
450 occurred most frequently (7 times)

Mode = 450

- Formula Worksheet

Note: Rows 7-71 are not shown.

- Value Worksheet

Note: Rows 7-71 are not shown.

- 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.
- 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.

- 90th Percentile
i = (p/100)n = (90/100)70 = 63

Averaging the 63rd and 64th data values:

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

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

- Third Quartile
Third quartile = 75th percentile

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

Third quartile = 525

- Unsorted Monthly Rent ($)

Note: Rows 7-71 are not shown.

- 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

- Sorted Monthly Rent ($)

Note: Rows 7-71 are not shown.

- Formula Worksheet for 90th Percentile’s Index

Note: Rows 7-71 are not shown.

- Value Worksheet for 90th Percentile’s Index

Note: Rows 7-71 are not shown.

- Value Worksheet for 3rd Quartile’s Index

Note: Rows 7-71 are not shown.

- 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.

- Range
- Interquartile Range
- Variance
- Standard Deviation
- Coefficient of Variation

- 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
Range = largest value - smallest value

Range = 615 - 425 = 190

- 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
3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

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

- 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).

- 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.

- 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).

- 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:

- Variance
- Standard Deviation
- Coefficient of Variation

- Formula Worksheet

Note: Rows 8-71 are not shown.

- Value Worksheet

Note: Rows 8-71 are not shown.

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

Step 4 When the Descriptive Statistics dialog box appears:

Enter B1:B71 in the Input Range boxSelect Grouped By Columns

Select Labels in First Row

Select Output Range

Enter D1 in the Output Range box

Select Summary Statistics

Click OK

- Value Worksheet (Partial)

Note: Rows 9-71 are not shown.

- Value Worksheet (Partial)

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