Z-Score Calculator
Find a z-score from a value, mean and standard deviation, with the percentile and full steps.
Results
| Item | Value |
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Steps
| Step | Working |
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Quick Answer
A z-score shows how many standard deviations a value is from the mean: z = (x − μ) ÷ σ. A test score of 85, with a mean of 70 and a standard deviation of 10, has a z-score of 1.5. That is about the 93rd percentile in a normal distribution.
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Table of Contents
- Z-Score Calculator
- Quick Answer
- Related Calculators
- How This Z-Score Calculator Works
- How to Use the Z-Score Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Z-Score vs Percentile
- Common Z-Scores and Percentiles
- Reading Your Result
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Z-Score Calculator Works
A z-score tells you how far a value sits from the average, measured in standard deviations. A positive score is above the mean, and a negative score is below it. The calculator also shows the share of values below yours, assuming a normal distribution.
Quick explanation: z = (x − μ) ÷ σ.
- x: your value.
- μ: the mean.
- σ: the standard deviation.
How to Use the Z-Score Calculator
- Enter the value you want to check, such as a test score.
- Enter the mean, or average, of the whole group.
- Enter the standard deviation of the group. It must be greater than zero.
- Press Calculate to see the z-score and the percentile.
- Read the steps table to see the working.
Core Formulas
- Z-score: z = (x − μ) ÷ σ
- Value from a z-score: x = μ + z × σ
- Percentile: the area under the standard normal curve to the left of z
Key Terms
- Mean (μ): the average of all values in the group.
- Standard deviation (σ): a measure of how spread out the values are around the mean.
- Z-score: how many standard deviations a value is above or below the mean.
- Normal distribution: a symmetric, bell-shaped spread of data where most values sit near the mean.
- Percentile: the percentage of values that are lower than a given value.
- Standard normal distribution: a normal distribution with a mean of 0 and a standard deviation of 1. Z-scores place values on this scale.
- Empirical rule: in a normal distribution, about 68% of values lie within 1 SD of the mean, 95% within 2, and 99.7% within 3.
Worked Example
A value is 60, the mean is 75, and the standard deviation is 5.
- 60 − 75 = −15
- −15 ÷ 5 = −3.00
- Only about 0.13% of values are below this in a normal distribution
Real-Life Examples
- Comparing two tests: you scored 78 in math (mean 70, SD 8) and 85 in English (mean 80, SD 10). The math z-score is 1.00 and the English z-score is 0.50, so the math result is stronger compared with the class.
- Height: if adult heights in a group have a mean of 69 in and an SD of 3 in, a height of 75 in has a z-score of 2.00, taller than about 97.7% of the group.
- Quality control: a part 3 standard deviations from the target size is very unusual in a normal process, so it is often flagged for checking.
Z-Score vs Percentile
- Z-score: distance from the mean in standard deviations. It can be negative.
- Percentile: the share of values below yours, from 0 to 100.
- How they connect: in a normal distribution, a z-score of 0 is the 50th percentile, and 1 is about the 84th.
- When to use each: z-scores are handy for maths and comparisons. Percentiles are easier to explain to most people.
Common Z-Scores and Percentiles
For a normal distribution. 1.96 is often used for 95% confidence.
| Z-score | Percentile (below) |
|---|---|
| -3.00 | 0.13% |
| -2.00 | 2.28% |
| -1.50 | 6.68% |
| -1.00 | 15.87% |
| -0.50 | 30.85% |
| 0.00 | 50.00% |
| +0.50 | 69.15% |
| +1.00 | 84.13% |
| +1.50 | 93.32% |
| +1.65 | 95.00% |
| +1.96 | 97.50% |
| +2.00 | 97.72% |
| +2.58 | 99.50% |
| +3.00 | 99.87% |
Reading Your Result
- Near 0: close to the average.
- About ±1: around 68% of normal data falls within this range.
- About ±2: around 95% falls within this range.
- Beyond ±3: very unusual in a normal distribution.
Common Mistakes to Avoid
- Mixing up population and sample SD. Use the one that matches your data.
- Using percentiles on skewed data. The percentile assumes a normal, bell-shaped curve.
- Swapping the order. Subtract the mean from the value, not the other way round.
- Mixing units. The value, mean and SD must all use the same unit.
- Reading a negative score as bad. It only means below the mean, which can be good, such as a fast race time.
Frequently Asked Questions
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ.
It depends on the context. For a test score, higher is better. A z-score of 1 is one standard deviation above the mean.
Yes. A negative z-score means the value is below the mean.
Use the area under the normal curve to the left of z. This calculator does that for you.
It means the value is equal to the mean.
About 1.645 in a normal distribution.
Values beyond about ±2 are often called unusual, and beyond ±3 very unusual, but the cut-off depends on the field.
Rearrange the formula: x = μ + z × σ. For z = 1.5, μ = 70 and σ = 10, x = 85.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Standard score | Definition and formula of the z-score. | https://en.wikipedia.org/wiki/Standard_score |
| Wikipedia, 68-95-99.7 rule | Share of normal data within each range. | https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule |
| Wikipedia, Normal distribution | Normal curve and percentiles. | https://en.wikipedia.org/wiki/Normal_distribution |