Z-Score & Normal Distribution Calculator (Free)

A z score calculator measures how many standard deviations a value lies from the mean: z = (x − μ) / σ. For a test score of 85 when the mean is 70 and the standard deviation is 10, z = (85 − 70) / 10 = 1.5. The normal curve puts about 93.32% of scores below 85, so it is near the 93rd percentile.

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How to use the z-score calculator

1. Find z: enter a value x, the mean μ and the standard deviation σ. You get z, the area below and above x, and the two-tailed area. Switch to A sample mean to standardize x̄ with the standard error σ/√n. Choose which area to shade on the bell curve.
2. Find x: start from a percentile (such as the 90th) or a z-score and get the matching value x = μ + zσ, plus a table of common percentiles.
3. Probability: find P(X < a), P(X > a), P(a < X < b) or the area outside a and b. Use mean 0 and SD 1 to work with z-scores directly.
4. Raw data: paste a list to get the mean, the sample or population standard deviation and the z-score of every value, with outliers flagged.
5. Z-table & p: look up the area for any z with the matching z-table row highlighted, get one- or two-tailed p-values, or turn an area or α into a critical z.

Z-score formula

z = (x − μ) / σ, where x is the value, μ is the mean and σ is the standard deviation. With sample statistics, use x̄ and s in the same places.

For the mean of a sample of size n, the spread is the standard error, so z = (x̄ − μ) / (σ / √n). To go back from z to a value, solve for x: x = μ + zσ.

The sign tells the direction and the size tells the distance. z = 0 is exactly at the mean, z = −2 is two standard deviations below it, and z = 1.5 is one and a half above it. Because z has no units, you can compare values from different scales, such as an SAT score and an ACT score.

From z-score to probability and percentile

When the data are roughly normal, the area under the standard normal curve to the left of z, written Φ(z), is the share of values below x. Multiply by 100 for the percentile. The area to the right is 1 − Φ(z), and the area between two values is Φ(z₂) − Φ(z₁).

This is the 68-95-99.7 rule. A value with |z| above 2 is unusual and one above 3 is rare, which is why z-scores are a quick way to spot outliers and data-entry errors.

Worked examples

Example 1: percentile of a test score

Scores have μ = 70 and σ = 10. For x = 85: z = 15 / 10 = 1.5. Φ(1.5) = 0.9332, so 93.32% of scores are lower and 6.68% are higher.

Example 2: the value at the 90th percentile

IQ-style scores have μ = 100 and σ = 15. The z with a left area of 0.90 is 1.2816 (invNorm(0.9) on a TI-84). Then x = 100 + 1.2816 × 15 = 119.22.

Example 3: probability between two values

With μ = 100 and σ = 15, find P(85 < X < 130). The z-scores are (85 − 100)/15 = −1 and (130 − 100)/15 = 2. So P = Φ(2) − Φ(−1) = 0.97725 − 0.15866 = 0.8186, about 81.86%.

Example 4: z for a sample mean

A population has μ = 70 and σ = 10. A sample of n = 25 has x̄ = 72. The standard error is 10 / √25 = 2, so z = (72 − 70) / 2 = 1 and P(x̄ > 72) = 1 − 0.8413 = 0.1587. A single value of 72 would only have z = 0.2: averages vary much less than individual values.

Example 5: p-value from a z test statistic

A z-test gives z = 2.1. The right tail is 1 − Φ(2.1) = 0.017864, so the two-tailed p-value is 2 × 0.017864 = 0.0357. That is below 0.05, so the result is significant at the 5% level. For t, chi-square and F tests, use the p-value calculator.

Critical z-values

The two-tailed values are the z* used in a confidence interval at the matching level.

Z-table (standard normal table)

Each entry is Φ(z), the area to the left of z. Find the row for the first decimal and the column for the second: z = 1.53 is row 1.5, column .03, giving 0.9370. For a negative z, use Φ(−z) = 1 − Φ(z); for example Φ(−1.25) = 1 − 0.8944 = 0.1056.

Common mistakes with z-scores

Checking homework and working by hand

Before calculating, estimate: a value above the mean must give a positive z and a percentile above 50. Keep the z-score to two decimals when you use a printed table, and expect your table answer to differ from the calculator in the fourth decimal. On a TI-84, normalcdf(lower, upper, μ, σ) gives areas and invNorm(area, μ, σ) goes back from an area to x. Practise a few by hand with the table above, then use the calculator to check the rest.

Z-scores practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is the z-score formula?

z = (x − μ) / σ: subtract the mean from the value and divide by the standard deviation. For a sample mean, divide by the standard error σ/√n instead.

How do you find probability from a z-score?

Look up Φ(z) in a z-table or use normalcdf. That is the probability of a value below x. Subtract from 1 for the probability above, or subtract two table values for the probability between them. For z = 1.5, P(below) = 0.9332 and P(above) = 0.0668.

How do you find a z-score on a TI-84 calculator?

The TI-84 has no z-score key, so type (x − μ)/σ directly. To go from z to an area, use 2nd VARS, normalcdf(−1E99, z, 0, 1). To go from an area to z, use invNorm(area, 0, 1); for example invNorm(0.95, 0, 1) = 1.645.

How do you find a percentile from a z-score?

Find the area to the left, Φ(z), and multiply by 100. z = 0.67 gives Φ = 0.7486, about the 75th percentile, and z = −1.25 gives 0.1056, about the 11th.

What does a z-score of 4 mean?

The value is 4 standard deviations above the mean. In a normal distribution only about 0.003% of values are that high (about 1 in 31,600), so it is either a true extreme or, more often, an outlier or data-entry error worth checking.

Is a z-score a percentage?

No. A z-score is a distance measured in standard deviations, and it can be negative or larger than 3. The percentage of values below it is a separate number, Φ(z) × 100: z = 1 corresponds to about 84%, not 1%.

Related calculators

• Standard deviation calculator: find σ or s from your data first
• Percentile calculator: percentiles of an actual data set
• P-value calculator: p-values from z, t, chi-square and F
• Confidence interval calculator: intervals for a mean with z* or t*
• Descriptive statistics calculator: mean, spread and shape in one place