Z-Value Calculator: Formula & How to Find It

This calculator works out the z-value (also called the z-score or standard score) for a single data point. You enter the raw score, the population mean, and the population standard deviation, and you get back the z-value, which tells you how many standard deviations that score sits above or below the mean. The result is a dimensionless number, so it has no units.

Formula

The z-value formula is written as:

z = (x − μ) / σ

Where:

• x is the raw score you are standardizing (same unit as the data, e.g., points, cm, dollars).
• μ (mu) is the population mean (same unit as x).
• σ (sigma) is the population standard deviation (same unit as x).
• z is the z-value, a pure number with no unit.

The formula simply measures the distance from the raw score to the mean, then divides that distance by the typical spread of the data. A positive z means the score is above the mean; a negative z means it is below the mean.

Worked Example

Using the default values from the calculator above: a raw score of 85, a population mean of 70, and a population standard deviation of 10.

1. Subtract the mean from the raw score: 85 − 70 = 15.
2. Divide that difference by the standard deviation: 15 ÷ 10 = 1.5.
3. The z-value is 1.5.

This means the score of 85 is 1.5 standard deviations above the mean of 70. The calculator rounds to two decimal places, so it shows 1.50, which is the same thing.

When to Use the Z-Value

The z-value is used in three main situations:

• Comparing scores from different distributions: If one test has a mean of 50 and a standard deviation of 5, and another has a mean of 200 and a standard deviation of 20, you cannot compare raw scores directly. Converting both to z-values puts them on the same scale.
• Finding probabilities: Once you have the z-value, you can look it up in a standard normal table (or use a calculator) to find the probability of getting a score at or below that value, assuming the data follows a normal distribution.
• Identifying outliers: As a rough rule, z-values below −3 or above +3 are often considered unusual, since about 99.7% of data in a normal distribution falls within 3 standard deviations of the mean.

To read the result: a z-value of 0 means the score equals the mean. A z of 1 means it is one standard deviation above the mean, which corresponds to about the 84th percentile in a normal distribution. A z of −1.5 means it is 1.5 standard deviations below the mean, around the 7th percentile.

Common Mistakes

• Using the sample standard deviation instead of the population standard deviation. The z-value formula requires the population σ. If you only have a sample, you need the sample standard deviation and a different formula (the t-statistic).
• Swapping the mean and the raw score. The order matters: it is always x minus μ, not μ minus x. Swapping them flips the sign of the z-value.
• Forgetting units. The subtraction and division must be done with consistent units. If x is in centimeters and μ is in meters, convert first.
• Rounding too early. If you round the difference or the standard deviation before dividing, you can lose precision. Keep full precision until the final step.
• Assuming normality. The z-value itself can be calculated for any data, but using it to find probabilities assumes the underlying distribution is normal (or approximately so).

Tips

• Always write down the formula before plugging in numbers. It helps catch sign errors.
• Check the sign of your answer: if the raw score is above the mean, z should be positive; if below, negative.
• Use the population standard deviation, not the variance. The variance is σ², so take the square root first if needed.
• When comparing two scores from different tests, convert both to z-values first, then compare the z-values directly.
• For a quick sanity check: a z-value between −2 and +2 covers about 95% of a normal distribution, so values outside that range are relatively rare.

Troubleshooting

Your z-value looks too large or too small. Check that you entered the standard deviation, not the variance. If you entered the variance, take its square root first. Also check that the mean and raw score are in the same units.

You get a negative z-value but expected a positive one. You likely subtracted in the wrong order. The formula is (x − μ), so if x is below the mean, the result is negative by definition.

You are unsure whether to use the population or sample standard deviation. The z-value always uses the population standard deviation. If your data is a sample and you have the sample standard deviation (denoted s), you need the t-statistic instead: t = (x − μ) / (s / √n), where n is the sample size.

The calculator gives an error. Make sure the standard deviation is not zero. A zero standard deviation means all values are identical, so the z-value is undefined.

Your result does not match a textbook example. Check whether the textbook used the population standard deviation or the sample standard deviation. Some examples use the sample standard deviation and call it a z-score, which is technically incorrect but common in introductory courses.

Frequently Asked Questions

What is the difference between a z-value and a z-score?

They are the same thing. Z-value, z-score, standard score, and normal score all refer to the result of the formula (x − μ) / σ.

Can the z-value be greater than 3?

Yes. The z-value has no upper or lower limit. Values beyond ±3 are possible but rare in a normal distribution, occurring less than 0.3% of the time.

Do I need the data to be normally distributed to calculate a z-value?

No. You can calculate a z-value for any single data point as long as you know the mean and standard deviation. However, using the z-value to find probabilities or percentiles does require the normal distribution assumption.

What does a z-value of 0 mean?

A z-value of 0 means the raw score is exactly equal to the population mean. It is neither above nor below average.

How do I convert a z-value back to a raw score?

Rearrange the formula: x = μ + z × σ. For example, if μ = 70, σ = 10, and z = 1.5, then x = 70 + 1.5 × 10 = 85.

Is the z-value the same as the standard deviation?

No. The standard deviation (σ) measures the spread of the entire distribution and has the same units as the data. The z-value measures the position of a single score relative to that spread and has no units.

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in x | Raw score | - | 85 | 0
in mu | Population mean (μ) | - | 70 | 0
in sigma | Population standard deviation (σ) | - | 10 | 0
out z | Z-value | - | (x - mu) / sigma | 2