Standard Score

Calculating the Standard Score

The formula for calculating the standard score is as follows:

$$z=\left(x-\mu \right)/\sigma$$

where x is an observation from the distribution, µ is the mean of the distribution, and σ is the standard deviation of the distribution. The standard score has several properties. The sum of a set of standard scores will be equal to 0, which is also the mean value when any variable is standardized. Both the variance and the standard deviation of any set of standard scores are equal to 1.

What Is the Value of the Standard Score?

The following example illustrates the utility of the standard score in social science research. Suppose a researcher is interested in better understanding who uses social media for news consumption. The researcher conducts a nationally representative survey and asks respondents to report their levels of attention (where 1 indicates low levels of attention and 10 indicates high levels of attention) to different types of news on several social media platforms. The researcher averages several of the survey items to create an index of social media news attention. The variable is continuous and ranges from 1 to 10, where higher values are associated with higher levels of reported attention. The responses are approximately normal with a mean response (µ) of 4.50 and a standard deviation (σ) of 2.10.

The researcher wants to determine how much of an outlier a response of 9.33 is on this scale. To determine this, the formula for the standard score (z) is employed:

$$z=x-\mathrm{\mu }\mathrm{\sigma }$$

To calculate the standard score for a value of 9.33, the researcher need only plug in the appropriate values as dictated by the formula. The result is

$$\begin{array}{l}z=\left(x-\mathrm{\mu }\right)/\sigma \hfill \\ z=\left(9.33-4.50\right)/2.10\hfill \\ z=2.30\hfill \end{array}$$

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