Z score calculator with the normal curve

A z-score says how many standard deviations a value is from the mean: z = (x − μ) ÷ σ. A score of 130 when the mean is 100 and the standard deviation is 15 has z = (130 − 100) ÷ 15 = 2, two standard deviations above the mean. Values below the mean have negative z-scores. This z score calculator is part of our Probability Calculator: set the mean and standard deviation, type your value, and it shows the z-score with its working and the probability of being below or above it.

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How to find a z-score in the calculator

1. Open the Probability Calculator. It opens on the Normal distribution, which is the one you want.
2. Under Distribution, type the Mean μ and the Std deviation σ.
3. Under Probability, choose a region in Shade: P(X < a) for the area below your value, or P(X > a) for the area above it.
4. Type your value in Bound a.
5. Read the probability, and under it the z-score with its working, such as z = (130.00 − 100.00)/15.00 = 2.00. The curve shades the area you asked for.

For a range, choose P(a < X < b) or P(X < a or X > b) and fill in Bound b too: the calculator then gives both z-scores. It works from a mean and standard deviation, not from a list of raw data, and it doesn't go backwards from a probability to a z-score. Example 5 shows how to handle that.

The z score formula

Subtracting the mean measures how far x is from the centre. Dividing by σ turns that distance into a number of standard deviations, so values from different scales can be compared. A z-score of 0 is exactly at the mean, positive is above it and negative is below. For a sample, use x̄ and s in place of μ and σ.

Z score to probability

When the data is normally distributed, the z-score fixes the probability. The area under the standard normal curve to the left of z is the probability of a value below it, which is also the value's percentile. The empirical rule gives the rough picture: about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.

Worked examples you can check in the calculator

Example 1: a value above the mean

Test scores are normally distributed with mean 100 and standard deviation 15. What is the z-score of 130, and what fraction of people score less?

With μ = 100, σ = 15, Shade P(X < a) and a = 130, the calculator gives P(X < 130) = 0.9772 and z = 2.00. So 130 is at about the 98th percentile, and only 2.28% score higher.

Example 2: a negative z-score

Heights in a group are normal with mean 170 cm and standard deviation 8 cm. For 158 cm, z = (158 − 170) ÷ 8 = −1.5: one and a half standard deviations below the mean. The calculator gives P(X < 158) = 0.0668, so about 6.7% of the group are shorter.

Example 3: comparing two results

A student scores 78 in a test with mean 70 and standard deviation 5, and 85 in another with mean 80 and standard deviation 10. Which was the better result? The z-scores are (78 − 70) ÷ 5 = 1.6 and (85 − 80) ÷ 10 = 0.5. The 78 is further above its class, so relative to everyone else it was the better result, even though 85 is the higher mark.

Example 4: between two values

With mean 100 and standard deviation 15, what share of scores are between 85 and 130? Choose P(a < X < b) with a = 85 and b = 130. The calculator shows z = −1.00 and z = 2.00, and the probability 0.8186: the area to the left of z = 2 minus the area to the left of z = −1.

Example 5: from a probability back to a value

What score puts someone in the top 10%? Here you know the probability and want the value. The calculator doesn't solve this directly, so choose P(X > a) and move Bound a until the probability is as close to 0.1 as it gets: at a = 119.2 it reads 0.1003. Or use a z-table: the z with 10% to its right is 1.2816, so x = μ + zσ = 100 + 1.2816 × 15 = 119.2.

Try a normal distribution question, then press Mark to check your answer in the Probability course.

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Z-scores from your own data

If you have a list of values rather than a mean and standard deviation, find those first. The standard deviation calculator guide shows how, and the calculator there gives both. Then z = (x − x̄) ÷ s for each value. A z-score beyond about ±3 is unusual and worth checking as a possible outlier.

Common mistakes

• Dividing by the variance. Divide by the standard deviation σ, not σ².
• Subtracting the wrong way. It is x − μ, so values below the mean come out negative.
• Reading the wrong tail. P(X > a) is 1 minus the area to the left. Check which side the question wants.
• Using the normal table for data that isn't normal. A z-score can always be worked out, but its probability only comes from the normal curve when the data is roughly bell-shaped.
• Rounding z too early. Small changes in z can move the probability noticeably in the middle of the curve.

Read next

Maths · StatisticsNormal distribution calculator: probabilities and curveFind normal distribution probabilities below, above, between or outside values for any mean and standard deviation, with the shaded curve and examples.