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Descriptive Measures Other Specifications Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 12, 2011

Z Score

Sometimes you'll hear people say that such-and-such an observation or result is ''2.2 standard deviations below the mean'' or ''1.6 standard deviations above the mean.'' The Z score, symbolized z, is a quantitative measure of the position of a particular element with respect to the mean. The Z score of an element is equal to the number of standard deviations that the element differs from the mean, either positively or negatively.

For a specific element x in a data set, the value of z depends on both the mean (μ) and the standard deviation (σ) and can be found from this formula:

              z = (xμ)/σ

If x is below the mean, then z is a negative number. If x is above the mean, then z is a positive number. If x is equal to the mean, then z = 0.

In the graphical distributions of Fig. 4-6, z > 0 for the point x shown. This is true for both curves. We can't tell from the graph alone exactly what the Z score is for x with respect to either curve, but we can at least see that it's positive in both cases.

Other Specifications

Interquartile Range

Sometimes it's useful to know the ''central half '' of the data in a set. The interquartile range, abbreviated IQR, is an expression of this. The IQR is equal to the value of the 3rd quartile point minus the value of the 1st quartile point. If a quartile point occurs between two integers, it can be considered as the average of the two integers (the smaller one plus 0.5).

Consider again the hypothetical 40-question test taken by 1000 students. The quartile points are shown in Fig. 4-2A.

Fig. 4-2A. At A, positions of quartiles in the test results described in the text.

The 1st quartile occurs between scores of 16 and 17; the 3rd quartile occurs between scores of 31 and 32. Therefore:

    IQR = 31 – 16
            = 15

Descriptive Measures Other Specification Practice Problems

Practice 1

Suppose a different 40-question test is given to 1000 students, and the results are much more closely concentrated than those from the test depicted in Fig. 4-2A. How would the IQR of this test compare with the IQR of the previous test?

Quartiles and Deciles

Fig. 4-2A. At A, positions of quartiles in the test results described in the text.

Solution 1

The IQR would be smaller, because the 1st and 3rd quartiles would be closer together.

Practice 2

Recall the empirical rule from the previous chapter. It states that all normal distributions have the following three characteristics:

  • Approximately 68% of the data points are within the range ±σ of μ.
  • Approximately 95% of the data points are within the range ±2σ of μ.
  • Approximately 99.7% of the data points are within the range ±3σ of μ.

Re-state this principle in terms of Z scores.

Solution 2

As defined above, the Z score of an element is the number of standard deviations that the element departs from the mean, either positively or negatively. All normal distributions have the following three characteristics:

  • Approximately 68% of the data points have Z scores between –1 and +1.
  • Approximately 95% of the data points have Z scores between –2 and +2.
  • Approximately 99.7% of the data points have Z scores between –3 and +3.

Practice problems for these concepts can be found at:

Descriptive Measures Practice Test

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