Standard Scores

A standard score is a score that uses the same metric as other standard scores so that they can be compared to one another easily. Any standard score can be defined as a converted score (based on the raw score) with a set mean and standard deviation. Popular examples are scores on IQ tests and the two types that we will cover here, are called z scores and T scores.

What both of these, as well as other standard scores, have in common is that they share the same mean and standard deviation. It is because of this that standard scores from different distributions can be compared to one another. However, the one restriction is that only the same type of standard scores can be compared. One can compare z scores from one distribution with z scores from another, but one cannot compare z scores from one distribution with T scores from another.

For example, here is a list of 10 raw scores and their corresponding z scores. All the information needed to compute the z scores is contained in Table 1.

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Table 1 Raw Scores and Standard Scores
	Raw Score	z Score
	87	0.66
	78	−0.07
	93	1.15
	69	−0.81
	95	1.31
	57	−1.79
	87	0.66
	71	−0.64
	69	−0.81
	68	−0.89
Mean	77.40
s	12.58
Note: s = standard deviation.

The z scores are standard scores with a mean of 0 and a standard deviation of 1. The formula for computing a z score is

where

z is the standard z score,

X is the raw score,

X¯ is the mean of the distribution, and

s is the standard deviation for the distribution.

For example, where the mean is 77.4 and the standard deviation is 12.58, the z score for a corresponding raw score of 69 is -.81 (almost one standard score below the mean). Here's the formula:

Another popular type of standard score is the T score. T scores are standard scores with a mean of 50 and a standard deviation of 10. The formula for computing a T score is

Table 2 Raw Scores, z Scores, and T Scores
	Raw Score	z Score	T Score
	87	0.66	56.61
	78	−0.07	49.27
	93	1.15	61.50
	69	−0.81	41.92
	95	1.31	63.13
	57	−1.79	32.14
	87	0.66	56.61
	71	−0.64	43.56
	69	−0.81	41.92
	68	−0.89	41.11
Mean	77.40
s	12.58
Note: s = standard deviation.

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