By Saul McLeod, published 2019

365体育登录A z-score describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The z-score is positive if the value lies above the mean, and negative if it lies below the mean.

365体育登录It is also known as a standard score, because it allows comparison of scores on different kinds of variables by standardizing the distribution. A standard normal distribution (SND) is a normally shaped distribution with a mean of 0 and a standard deviation (SD) of 1 (see Fig. 1).

*Figure 1.* A standard normal distribution (SND).

It is useful to standardized the values (raw scores) of a normal distribution365体育登录 by converting them into z-scores because:

365体育登录 (a) it allows researchers to calculate the probability of a score occurring within a standard normal distribution;365体育登录 (b) and enables us to compare two scores that are from different samples (which may have different means and standard deviations).

As the formula shows, the z-score is simply the raw score minus the sample mean, divided by the sample standard deviation.

*Figure 2.* Z-score formula.

The value of the z-score tells you how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean.

A positive z-score indicates the raw score is higher than the mean average. For example, if a z-score is equal to +1, it is 1 standard deviation above the mean.

A negative z-score reveals the raw score is below the mean average. For example, if a z-score is equal to -2, it is 2 standard deviations below the mean.

Another way to interpret z-scores is by creating a standard normal distribution (also known as the z-score distribution or probability distribution).

Fig 3 illustrates the important features of any standard normal distribution (SND).

*Figure 3.* A standard normal distribution (SND) and normal distribution.

- The SND (i.e. z-distribution) is always the same shape as the raw score distribution. For example, if the distribution of raw scores if normally distributed, so is the distribution of z-scores.
- The mean of any SND always = 0.
- The standard deviation of any SND always = 1. Therefore, one standard deviation of the raw score (whatever raw value this is) converts into 1 z-score unit.

365体育登录The SND allows researchers to calculate the probability of randomly obtaining a score from the distribution (i.e. sample). For example, there is a 68% probability of randomly selecting a score between -1 and +1 standard deviations from the mean (see Fig. 4).

*Figure 4.* Proportion of a standard normal distribution (SND) in percentages.

The probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean is 95% (see Fig. 4). If there is less than a 5% chance of a raw score being selected randomly, then this is a statistically significant result.

Learn how to use a z-score tableSometimes we know a z-score and want to find the corresponding raw score. The formula for calculating a z-score in a sample into a raw score is given below:

365体育登录X = (z)(SD) + mean

365体育登录 As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean.

365体育登录Check your answer makes sense: If we have a negative z-score the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean.

To calculate the z-score of a specific value, x, first you must calculate the mean of the sample by using the AVERAGE formula.

For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula =AVERAGE(A1:A20) returns the average of those numbers.

365体育登录Next, you mush calculate the standard deviation of the sample by using the STDEV.S formula. For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers.

Now to calculate the z-score type the following formula in an empty cell: = (x – mean) / [standard deviation].

365体育登录To make things easier, instead of writing the mean and SD values in the formula you could use the cell values corresponding to these values. For example, = (A12 – B1) / [C1].

365体育登录Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), type the following into a blank cell: = NORMSDIST( and input the z-score you calculated).

365体育登录To find the probability of LARGER z-score, which is the probability of observing a value greater than x (the area under the curve to the RIGHT of x), type: =1 - NORMSDIST (and input the z-score you calculated).

Further Information

McLeod, S. A. (2019, May 17). *Z-score: definition, calculation and interpretation*365体育登录. Simply Psychology. http://morrisonfoto.com/z-score.html

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