Z-Score to Standard Deviation: How to Convert

This calculator converts a z-score back into the original data value or standard deviation units, depending on what you need. You enter the z-score, the mean, and the standard deviation (or the raw score and mean to find the standard deviation), and it returns the missing value. The tool works both ways: given a z-score and the mean and standard deviation, it finds the raw score; given a raw score, the mean, and the z-score, it finds the standard deviation.

Formula

The z-score formula is the bridge between a raw score and its position in a distribution:

z = (x − μ) / σ

• z – the z-score (number of standard deviations from the mean)
• x – the raw score (the original data value)
• μ – the population mean (average)
• σ – the population standard deviation (spread)

Rearranged to solve for each variable:

• x = μ + z·σ – find the raw score from a z-score
• σ = (x − μ) / z – find the standard deviation from a raw score and its z-score
• μ = x − z·σ – find the mean from a raw score and its z-score

All variables are in the same units as the original data (e.g., points, cm, dollars). The z-score itself is unitless.

Worked Example

Using the default values: z = 1.5, mean = 100, standard deviation = 15.

1. Start with the formula: x = μ + z·σ
2. Substitute the numbers: x = 100 + 1.5 × 15
3. Multiply first: 1.5 × 15 = 22.5
4. Add the mean: 100 + 22.5 = 122.5

The raw score is 122.5. This matches the calculator’s first output (rounded to 2 decimals). The second output shows what the standard deviation would be if you instead entered a raw score of 122.5 with the same z and mean: (122.5 − 100) / 1.5 = 15, confirming the original value.

When to Use This Conversion

You typically need this when you have a z-score from a standardized test, a research paper, or a statistical analysis, but you want to know the actual score in the original units. For example, if an IQ test has a mean of 100 and a standard deviation of 15, a z-score of 2.0 corresponds to an IQ of 130 (100 + 2·15). Conversely, if you know a student scored 85 on a test with a mean of 70 and you know that score’s z-score is 1.5, you can find the standard deviation: (85 − 70) / 1.5 = 10.

This is also useful when you are comparing scores from different distributions. Converting raw scores to z-scores lets you compare them fairly; converting back lets you interpret a z-score in a familiar context.

How to Choose the Right Value to Calculate

The calculator has two outputs, but you only use one at a time. Decide what you are missing:

• If you have the z-score, mean, and standard deviation → you want the raw score (first output).
• If you have the raw score, mean, and z-score → you want the standard deviation (second output).

In the second case, you still enter the z-score and mean, but you also enter the raw score in the first output field (it is editable). The calculator then uses the second formula. The first output will also show a value, but ignore it if your goal is the standard deviation.

Be careful: the z-score must not be zero when you calculate the standard deviation, because dividing by zero is undefined. A z-score of zero means the raw score equals the mean, and any standard deviation would produce that same z-score, so the problem has no unique solution.

Common Mistakes and Misconceptions

• Using sample vs. population standard deviation – The z-score formula uses the population standard deviation (σ). If you only have a sample, you should use the sample standard deviation (s) as an estimate, but the formula is the same.
• Mixing up the mean and the raw score – The mean is the center of the distribution, not the score you are converting. Double-check which number is which.
• Forgetting the sign of the z-score – A negative z-score means the raw score is below the mean. If you enter a negative z, the raw score will be less than the mean. Ignoring the sign gives a wrong answer.
• Rounding too early – Keep at least two decimal places in intermediate steps to avoid rounding errors in the final answer.
• Assuming a normal distribution is required – The z-score formula is just a linear transformation; it works for any distribution. What changes is the probability interpretation, not the arithmetic.

Limits of the Formula

The formula is exact and always works arithmetically, but its usefulness depends on the context. A z-score tells you how many standard deviations a value is from the mean, but it does not tell you the percentile unless the data is normally distributed. For non-normal distributions, the same z-score can correspond to very different percentiles. Also, the formula assumes you have the true population parameters. If you use sample estimates, the result is only an approximation.

Another limit: the standard deviation cannot be negative. If your inputs produce a negative standard deviation, you likely entered the z-score with the wrong sign or swapped the mean and raw score. The calculator will show a negative number, but that is not a valid standard deviation.

Frequently Asked Questions

How do I convert a z-score to a standard deviation?
If you have a raw score (x), the mean (μ), and the z-score, the standard deviation is σ = (x − μ) / z. For example, if x = 85, μ = 70, and z = 1.5, then σ = (85 − 70) / 1.5 = 10.

Can I convert a z-score to a percentile?
Yes, but only if the data follows a normal distribution. Use a standard normal table or a calculator with the cumulative distribution function. For example, a z-score of 1.5 corresponds to about the 93.3rd percentile in a normal distribution. The z-score itself is not a percentile; it is a measure of distance from the mean.

What is the difference between a z-score and a standard deviation?
A standard deviation is a measure of spread in the original units (e.g., points, inches). A z-score is a unitless number that tells how many standard deviations a specific value is from the mean. One standard deviation is a fixed quantity; a z-score varies for each data point.

Why do I get a negative standard deviation?
That happens when the z-score and the difference (x − μ) have opposite signs. For example, if x is above the mean (positive difference) but the z-score is negative, the formula gives a negative σ. Check your inputs: the sign of the z-score should match the direction of the raw score relative to the mean.

Does this work for sample data?
Yes, if you treat the sample mean and sample standard deviation as estimates of the population parameters. The formula is the same. Just be aware that the resulting z-score is a sample z-score, and its interpretation is slightly less precise.

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in z | Z-score | - | 1.5 | 0
in m | Mean | - | 100 | 0
in s | Standard deviation | - | 15 | 0
out x | Raw score | - | m + z * s | 2
out sd_from_x | Std dev from raw score | - | (x - m) / z | 2