The z-score is the number of standard deviations (σ) that a given datapoint is from the assumed mean (µ). It standardizes scores allowing to compare scores with different scales. In inferential statistics the z-score is fundamental.

How many σ away?

To measure how many standard deviations (σ) a certain value is from the assumed mean, we divide the difference between this value and the mean by σ:

The z-score can be applied to compare the relative score or performance for scores on different scales. We standardize the scores, so they can be compared.

Example 

Say that as a prerequisite for applying to the University of Copenhagen we need to pass the so-called UC exam and in order to apply for Copenhagen Business School, we need to pass the CBS exam.

The exam mean scores (µ) and related standard deviations (σ) are:

• UC: µ = 160 with σ = 12
• CBS: µ = 31 with σ = 7

Say that I do 173 in the UC and 42 in the CB.

In which of the two exams did I do relatively best?

The two exam scores are on different scales, so in order to compare, I need to see the difference from my score to the mean score in relation to the standard deviation. In this way, I am calculating how many standard deviations I am from the mean. The further my score is above the mean, measured in number of standard deviations, the better the relative result.

The calculations are:

• UC: (173-160) / 12 = 1.08
• CBS: (42-31) / 7 = 1.57

It is 1.57 standard deviations above its mean, whereas my UC is 1.08 standard deviations above its mean. Therefore, I did relatively better in my CBS exam.

When σ is unknown

In the example above σ is known, as we know the parameters of the populations of UC and CBS students. However, often we don’t know the population. We therefore conduct sample statistics and calculate estimators for the parameters.

As the population standard deviation (σ) is unknown, we estimate it with the standard error (SE) which is calculated by dividing the sample standard deviation (s) with square root of the sample size (n):

Example

Say that we are statistician in a lab studying an African parrot specie population which is expected to have a total number of some 5,000. Their assumed mean weight (µ) is 420 grams.

We now find reason to believe that these parrots are losing weight and we conduct a simple random sample of 40 parrots (n=40). Our sample mean weight (x̄) is 417.90 with a sample standard deviation (s) of 8.40. The z-score calculation is:

Our z-score, also referred to as the test statistics, is -1.58. So, our sample mean (x̄) is -1.58 standard deviations away from the assumed mean (µ).

In inferential statistics

We’ve seen how the z-score allows for us to compare scores with different scales. The z-score is closely related to the significance level, the p-value and hypothesis testing and is, as mentioned, essential in inferential statistics.

z-score and the critical value

In hypothesis testing, we predefine the significance level (α) and we then calculate our test statistics and hold this up against the critical value for the respective probability distribution.

Example: If we are testing in the normal distribution, we will hold our test statistic up against the critical z-value at this given significance level (α). If our z-score falls beyond the critical z-value, we will reject the null hypothesis and determine value, that we are exploring for, as significant.

The critical z-score can also be used to calculate the critical value. For example, if we wish to see from off which value, we will qualify the test statistic as significant or not.

The Excel =STANDARDIZE function

The Excel =STANDARDIZE function can be used to calculate the z-score and the t-score. One of the function arguments is the standard deviation which refers to the population standard deviation (σ), but most often this is unknown, we will instead use the standard error (SE). Here, some different examples of how you might wish to calculate the z- or t-score:

Learning statistics

• Khan Academy video (5:04): Z-score introduction
• Statisticsfun video (4:01): How to calculate z-scores used in statistics class
• YaRrr! The Pirate’s Guide to R by, Nathaniel D. Phillips (bookdown, text): Standardization (z-score)

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