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Chapter 12: Problem 78

Describe how to determine the \(z\)-score for a data item in a normal distribution.

Short Answer

Expert verified
The z-score for a data item in a normal distribution is found by the formula \(z = \frac{{X - \mu}}{\sigma}\), where X is the data item value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. The z-score indicates how many standard deviations a data item is from the mean.

Step by step solution

01

Understanding the Z-score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. In a normal distribution, the mean is zero, so the z-score represents the distance between a datapoint and the mean, given in units of the standard deviation.
02

Identifying the Required Variables

To compute the z-score for a particular data item in a data set, three pieces of information are necessary: the value of the data item (X), the mean (\(\mu\)) of the data set, and the standard deviation (\(\sigma\)) of the data set.
03

Applying the Z-score Formula

The formula for finding the z-score is \(z = \frac{{X - \mu}}{\sigma}\). In this formula, X represents the value of the data item, \(\mu\) is the mean of the data set, and \(\sigma\) is the standard deviation of the data set. Plug in the values and perform the calculation to obtain the z-score.

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