Descriptive Statistics

Analyzing Distributions

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Business Analytics

Lecture # 05

TOPICS to be COVERED

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z-Scores

• A z-score allows us to measure the relative location of a value in the data set.
• More specifically, a z-score helps us determine how far a particular value is from the mean relative to the data set’s standard deviation.
• The z-score is often called the standardized value.

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© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

• How many standard deviations a value is from the mean.��In this example, the value 1.7 is 2 standard deviations away from the mean of 1.4, so 1.7 has a z-score of 2.�Similarly 1.85 has a z-score of 3.

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• For example, z₁ =1.2 indicates that x₁ is 1.2 standard deviations greater than the sample mean.
• Similarly, z₂ = − 0.5 indicates that x₂ is 0.5, or 1/2, standard deviation less than the sample mean.
• A z-score of zero indicates that the value of the observation is equal to the mean.

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The z-scores for the class size data are computed in Table 2.13.

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Figure 2.20: Calculating z-Scores for the Home Sales Data in Excel

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Empirical Rule

• When the distribution of data exhibits a symmetric bell-shaped distribution, as shown in Figure 2.21, the empirical rule can be used to determine the percentage of data values that are within a specified number of standard deviations of the mean.

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Empirical Rule

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• For data having a bell-shaped distribution:
• Within 1 standard deviation:
• Approximately 68% of the data values
• Within 2 standard deviations:
• Approximately 95% of the data values
• Within 3 standard deviations:
• Almost all the data values

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The height of adult males in the United States has a bell-shaped distribution similar to that shown in Figure 2.21, with a mean of approximately 69.5 inches and standard deviation of approximately 3 inches.

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Using the empirical rule, we can draw the following conclusions.

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• Approximately 68% of adult males in the United States have heights between 69.5 - 3 = 66.5 and 69.5 + 3 = 72.5 inches.

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• Approximately 95% of adult males in the United States have heights between 63.5 and 75.5 inches.

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• Almost all adult males in the United States have heights between 60.5 and 78.5 inches.

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Identifying Outliers

• Outliers: Extreme values in a data set
• It can be identified using standardized values (z-scores)
• Any data value with a z-score less than –3 or greater than +3 is an outlier

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Box Plots

• A box plot is a graphical summary of the distribution of data. A box plot is developed from the quartiles for a data set. Figure 2.22 is a box plot for the home sales data.

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Figure 2.23: Box Plots Comparing Home Sale Prices in Different Communities

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EXAMPLE

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Thank You !

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