Key takeaways
- z = (x − μ) ÷ σ: how many standard deviations a value is from the mean.
- Positive z is above average, negative z is below, and z = 0 is exactly average.
- Divide by the standard deviation σ, not the variance σ².
- In a normal distribution, a z-score beyond ±2 is unusual.
In this guide
What is a z-score?
You scored 78 in a test where the class average was 65. That sounds good, but how good? If most students scored between 60 and 70, your 78 is excellent. If scores were all over the place, it is less special.
The z-score answers this by measuring your distance from the mean in units of standard deviation. It is also called the standard score.
Why do z-scores matter?
Raw scores from different tests cannot be compared directly. A 70 on a very hard test may be better than an 85 on an easy one. Z-scores remove the difference in average and spread, so you compare like with like.
Maths: you scored 80, mean 70, σ 5, so z = 2. English: you scored 85, mean 75, σ 10, so z = 1. Your maths result is the more impressive one, even though the raw mark is lower.
The formula
| Symbol | What it means | Example |
|---|---|---|
| z | The z-score (no unit) | 1.625 |
| x | Your value | 78 marks |
| μ (mu) | The mean of the group | 65 marks |
| σ (sigma) | The standard deviation (not 0) | 8 marks |
x, μ and σ must all use the same unit. The units cancel when you divide, which is why z has none.
How to calculate a z-score step by step
- 1Find the mean and standard deviation. The standard deviation calculator can help if you have the raw data.
- 2Subtract the mean from your value. Work out
x − μ. - 3Divide by the standard deviation. The result is z.
- 4Read the sign. Positive means above average, negative means below.
Now enter your own value, mean and standard deviation in the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Dividing by the variance. Use σ, the standard deviation, not σ². If you are given a variance of 64, σ = 8.
- Thinking a negative z-score is an error. It simply means the value is below average.
- Mixing groups. Use the mean and σ of the same group your value comes from.
Tips and tricks
| z-score | What it means |
|---|---|
| 0 | Exactly average |
| ±1 | Ordinary, well within the usual range |
| ±2 | Unusual |
| ±3 | Very rare |
- In a normal distribution, about 95% of values have z between −2 and 2.
- The z value 1.96 you see in a confidence interval is a z-score too.
Where you'll use it in real life
- Exams: comparing results from tests with different difficulty.
- Health: children's height and weight charts use z-scores to show how a child compares with others the same age.
- Quality control: flagging products that are far from the target size.
- Data cleaning: spotting outliers, values with a z-score beyond ±3.
Quick summary
A z-score tells you how many standard deviations a value is from the mean: z = (x − μ) ÷ σ. Positive means above average and negative means below. Use the calculator above to find any z-score with every step shown.
Worked example
In a maths test the class mean was 65 marks with a standard deviation of 8 marks. You scored 78. How many standard deviations above the mean are you?
- 01z = (x - μ) / σ
- 02z = (78 - 65) / 8
- 03z = 13 / 8
- 04z = 1.625
- 05The value is 1.63 standard deviation(s) above the mean
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does a negative z-score mean?
It means the value is below the mean. A z-score of −1.5 means the value is 1.5 standard deviations below the average.
What is a good z-score?
It depends on the situation. For test scores, a z-score above 1 is well above average, and above 2 is rare. In a normal distribution only about 2.5% of values have z above 2.
How do I compare two tests using z-scores?
Work out the z-score for each test with its own mean and standard deviation. The higher z-score is the better result relative to the other students.
Can a z-score be more than 3?
Yes. It just means the value is very far from the mean. In a normal distribution, values beyond ±3 are very rare, so they are often checked as possible errors or outliers.
