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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.

%

x

Descriptive Statistics: Numerical Methods, Part B- Measures of Relative Location and Detecting Outliers
- Exploratory Data Analysis
- Measures of Association Between Two Variables
- The Weighted Mean and

Working with Grouped Data

Measures of Relative Locationand Detecting Outliers

- z-Scores
- Chebyshev’s Theorem
- Empirical Rule
- Detecting Outliers

z-Scores

- The z-score is often called the standardized value.
- It denotes the number of standard deviations a data value xi is from the mean.
- A data value less than the sample mean will have a z-score less than zero.
- A data value greater than the sample mean will have a z-score greater than zero.
- A data value equal to the sample mean will have a z-score of zero.

Chebyshev’s Theorem

At least (1 - 1/k2) of the items in any data set will be

within k standard deviations of the mean, where k is

any value greater than 1.

- At least 75% of the items must be within

k = 2 standard deviations of the mean.

- At least 89% of the items must be within

k = 3 standard deviations of the mean.

- At least 94% of the items must be within

k = 4 standard deviations of the mean.

Example: Apartment Rents

- Chebyshev’s Theorem

Let k = 1.5 with = 490.80 and s = 54.74

At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%

of the rent values must be between

- k(s) = 490.80 - 1.5(54.74) = 409

and

+ k(s) = 490.80 + 1.5(54.74) = 573

Example: Apartment Rents

- Chebyshev’s Theorem (continued)

Actually, 86% of the rent values

are between 409 and 573.

Empirical Rule

For data having a bell-shaped distribution:

- Approximately 68% of the data values will be within onestandard deviation of the mean.

Empirical Rule

For data having a bell-shaped distribution:

- Approximately 95% of the data values will be within twostandard deviations of the mean.

Empirical Rule

For data having a bell-shaped distribution:

- Almost all (99.7%) of the items will be within threestandard deviations of the mean.

Example: Apartment Rents

- Empirical Rule

Interval% in Interval

Within +/- 1s 436.06 to 545.54 48/70 = 69%

Within +/- 2s 381.32 to 600.28 68/70 = 97%

Within +/- 3s 326.58 to 655.02 70/70 = 100%

Detecting Outliers

- An outlier is an unusually small or unusually large value in a data set.
- A data value with a z-score less than -3 or greater than +3 might be considered an outlier.
- It might be an incorrectly recorded data value.
- It might be a data value that was incorrectly included in the data set.
- It might be a correctly recorded data value that belongs in the data set !

Example: Apartment Rents

- Detecting Outliers

The most extreme z-scores are -1.20 and 2.27.

Using |z| > 3 as the criterion for an outlier,

there are no outliers in this data set.

Standardized Values for Apartment Rents

Exploratory Data Analysis

- Five-Number Summary
- Box Plot

Five-Number Summary

- Smallest Value
- First Quartile
- Median
- Third Quartile
- Largest Value

Example: Apartment Rents

- Five-Number Summary

Lowest Value = 425 First Quartile = 450

Median = 475

Third Quartile = 525 Largest Value = 615

Box Plot

- A box is drawn with its ends located at the first and third quartiles.
- A vertical line is drawn in the box at the location of the median.
- Limits are located (not drawn) using the interquartile range (IQR).
- The lower limit is located 1.5(IQR) below Q1.
- The upper limit is located 1.5(IQR) above Q3.
- Data outside these limits are considered outliers.

… continued

Box Plot (Continued)

- Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits.
- The locations of each outlier is shown with the symbol* .

Example: Apartment Rents

- Box Plot

Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5

Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5

There are no outliers.

575

600

625

450

375

400

500

525

550

425

475

Measures of Association between Two Variables

- Covariance
- Correlation Coefficient

Covariance

- The covariance is a measure of the linear association between two variables.
- Positive values indicate a positive relationship.
- Negative values indicate a negative relationship.

Covariance

- If the data sets are samples, the covariance is denoted by sxy.
- If the data sets are populations, the covariance is denoted by .

Correlation Coefficient

- The coefficient can take on values between -1 and +1.
- Values near -1 indicate a strong negative linear relationship.
- Values near +1 indicate a strong positive linear relationship.
- If the data sets are samples, the coefficient is rxy.
- If the data sets are populations, the coefficient is .

Using Excel to Compute theCovariance and Correlation Coefficient

- Formula Worksheet

Using Excel to Compute theCovariance and Correlation Coefficient

- Value Worksheet

The Weighted Mean andWorking with Grouped Data

- Weighted Mean
- Mean for Grouped Data
- Variance for Grouped Data
- Standard Deviation for Grouped Data

Weighted Mean

- When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.
- In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade.
- When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

Grouped Data

- The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data.
- To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class.
- We compute a weighted mean of the class midpoints using the class frequencies as weights.
- Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.

Mean for Grouped Data

- Sample Data
- Population Data

where:

fi = frequency of class i

Mi = midpoint of class i

Example: Apartment Rents

Given below is the previous sample of monthly rents

for one-bedroom apartments presented here as grouped

data in the form of a frequency distribution.

Example: Apartment Rents

- Mean for Grouped Data

This approximation differs by $2.41 from

the actual sample mean of $490.80.

Variance for Grouped Data

- Sample Data
- Population Data

Example: Apartment Rents

- Variance for Grouped Data
- Standard Deviation for Grouped Data

This approximation differs by only $.20

from the actual standard deviation of $54.74.

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