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A Z-Score is a statistical measure that indicates how many standard deviations a given value is from the mean of a data set. It is a way to measure the relative standing of a data point to the rest of the set. It is used to compare different sets of data in order to determine which values are typical and which values are outliers. It is also used to calculate the probability of a value occurring in a given data set.
In statistics, a z-score tells us how many standard deviations away a given value lies from the mean. We use the following formula to calculate a z-score:
z = (X – μ) / σ
where:
- X is a single raw data value
- μ is the mean
- σ is the standard deviation
A z-score for an individual value can be interpreted as follows:
- Positive z-score: The individual value is greater than the mean.
- Negative z-score: The individual value is less than the mean.
- A z-score of 0: The individual value is equal to the mean.
The larger the absolute value of the z-score, the further away an individual value lies from the mean.
The following example shows how to calculate and interpret z-scores.
Example: Calculate and Interpret Z-Scores
Suppose the scores for a certain exam are normally distributed with a mean of 80 and a standard deviation of 4.
Question 1: Find the z-score for an exam score of 87.
We can use the following steps to calculate the z-score:
- The mean is μ = 80
- The standard deviation is σ = 4
- The individual value we’re interested in is X = 87
- Thus, z = (X – μ) / σ = (87 – 80) /4 = 1.75.
This tells us that an exam score of 87 lies 1.75 standard deviations above the mean.
Question 2: Find the z-score for an exam score of 75.
We can use the following steps to calculate the z-score:
- The mean is μ = 80
- The standard deviation is σ = 4
- The individual value we’re interested in is X = 75
- Thus, z = (X – μ) / σ = (75 – 80) /4 = –1.25.
Question 3: Find the z-score for an exam score of 80.
We can use the following steps to calculate the z-score:
- The mean is μ = 80
- The standard deviation is σ = 4
- The individual value we’re interested in is X = 80
- Thus, z = (X – μ) / σ = (80 – 80) /4 = 0.
This tells us that an exam score of 80 is exactly equal to the mean.
Why Are Z-Scores Useful?
Z-scores are useful because they give us an idea of how an individual value compares to the rest of a distribution.
For example, is an exam score of 87 good? Well, that depends on the mean and standard deviation of all exam scores.
If the exam scores for the whole population are normally distributed with a mean of 90 and a standard deviation of 4, we would calculate the z-score for 87 to be:
z = (X – μ) / σ = (87 – 90) /4 = -0.75.
Since this value is negative, it tells us that an exam score of 87 is actually below the average exam score for the population. Specifically, an exam score of 87 is 0.75 standard deviations below the mean.
In a nutshell, z-scores give us an idea of how individual values compare to the mean.
How to Calculate Z-Scores in Practice
The following tutorials show step-by-step examples of how to calculate z-scores in different statistical software: