Z Score Calculator: Find Z From a Raw Score

This calculator finds the z score for a raw value when you know the population mean and standard deviation. Enter the raw score, the population mean, and the population standard deviation, and you get the z score plus the corresponding percentile (assuming a normal distribution). The z score tells you how many standard deviations the raw score is above or below the mean.

Formula

The z score is the raw score minus the population mean, divided by the population standard deviation:

z = (x − μ) / σ

• x = raw score (the value you are standardizing)
• μ (mu) = population mean
• σ (sigma) = population standard deviation

The percentile shown is the probability that a randomly selected value from a normal distribution is less than or equal to x. It is computed from the cumulative distribution function of the standard normal distribution, which uses the error function (erf).

Worked Example

Using the default values: raw score x = 85, mean μ = 70, standard deviation σ = 10.

1. Subtract the mean from the raw score: 85 − 70 = 15.
2. Divide by the standard deviation: 15 ÷ 10 = 1.5.
3. The z score is 1.50.
4. To find the percentile, look up the area to the left of z = 1.50 in a standard normal table (or use the calculator’s built-in function). That area is approximately 0.9332, so the percentile is 93.32%.

The calculator returns z = 1.50 and percentile = 93.32%, matching the manual calculation.

When to Use a Z Score

Use a z score when you have a single raw value and you know the mean and standard deviation of the entire population (not a sample). Common situations include:

• Comparing test scores from different exams with different scales.
• Identifying outliers in a dataset that is approximately normal.
• Converting a value to a standard normal variable to find probabilities.
• Standardizing data for machine learning or statistical analysis.

If you only have a sample (not the whole population), you would use the sample mean and sample standard deviation, but the z score formula still applies when the sample size is large enough (central limit theorem).

How to Read the Result

A positive z score means the raw score is above the mean; a negative z score means it is below the mean. The magnitude tells you how far in standard deviation units.

• z = 0 → the raw score equals the mean.
• z = 1 → one standard deviation above the mean.
• z = −2 → two standard deviations below the mean.

The percentile is more intuitive: a percentile of 93.32% means the raw score is higher than about 93.32% of the population. In a normal distribution, about 68% of values lie within one standard deviation of the mean (z between −1 and 1), 95% within two, and 99.7% within three.

Common Mistakes and Limitations

• Using sample standard deviation instead of population standard deviation: The z score formula assumes you know the population standard deviation. If you only have a sample, the t-score is more appropriate for small samples.
• Assuming normality: The percentile calculation is only valid if the underlying distribution is normal (or approximately normal). For non‑normal data, the z score still measures distance from the mean, but the percentile will be inaccurate.
• Mixing units: The raw score, mean, and standard deviation must be in the same units. If the mean is in inches and the raw score is in centimeters, convert first.
• Rounding too early: Round only the final z score, not intermediate steps, to avoid small errors.
• Using the wrong formula for a sample: If you are testing a sample mean (not a single value), the standard error is σ/√n, not σ.

Z Score vs. T Score

The z score uses the population standard deviation. The t score (or t statistic) is used when the population standard deviation is unknown and you have a sample. For a single value, the t score would be (x − μ) / (s), where s is the sample standard deviation. For a sample mean, the t score is (x̄ − μ) / (s/√n). The t distribution approaches the normal distribution as the sample size grows, so for large samples (n > 30), the z and t scores are nearly identical.

In practice, many people use the z score for any standardized value, but the percentile from a normal table is only exact for the population case. If you have a sample and want to estimate percentiles, use the t distribution with n−1 degrees of freedom.

Alternatives and Related Calculators

If you have a z score and want to find the raw score, use the inverse formula: x = μ + z·σ. Many calculators offer a “reverse z score� function. If you have a sample and need a confidence interval or hypothesis test, you might use a t‑score calculator instead. For comparing two groups, you might use a z‑test for proportions or a two‑sample t‑test. This calculator is specifically for a single raw value from a known normal population.

Frequently Asked Questions

What does a z score of 2 mean?

A z score of 2 means the raw score is exactly two standard deviations above the population mean. In a normal distribution, about 2.28% of values are higher than this, so the percentile is approximately 97.72%.

Can a z score be negative?

Yes. A negative z score means the raw score is below the mean. For example, if the mean is 100 and the standard deviation is 15, a raw score of 85 gives z = (85−100)/15 = −1.00.

What is the difference between z score and standard deviation?

Standard deviation measures the spread of the entire distribution in the original units (e.g., points, inches). A z score measures how many standard deviations a single value is from the mean, making it unit‑free. One standard deviation is a fixed distance; a z score is a relative position.

Do I need a normal distribution to use a z score?

To compute the z score itself, no—it is just a transformation. But to interpret the percentile, the data should be normally distributed. If the distribution is skewed, the percentile from a normal table will be misleading.

How do I find the z score for a sample mean?

For a sample mean, use the standard error: z = (x̄ − μ) / (σ/√n), where n is the sample size. This gives the z score for the sampling distribution of the mean, not for a single value.

What if I don’t know the population standard deviation?

If you have a sample, use the sample standard deviation s and the t distribution instead. For large samples (n > 30), the t distribution is very close to normal, so you can approximate with the z score using s in place of σ.

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in x | Raw score | - | 85 | 0
in mu | Population mean | - | 70 | 0
in sigma | Population standard deviation | - | 10 | 0
out z | Z score | - | (x - mu) / sigma | 2
out p | Percentile | % | 0.5 * (1 + erf((x - mu) / (sigma * sqrt(2)))) * 100 | 2