Normal Distribution, Z-Score and P-Value Calculator

Q: Is a z-score of 2 good or bad?

It depends on what is measured. z = 2 means two deviations above the mean, the 97.7th percentile: great on an exam, worrying in a cholesterol test. About 4.55% of values of a normal curve fall outside ±2.

Q: What is the difference between a z-score and a T-score?

The T-score is a rescaling of z with mean 50 and deviation 10: T = 50 + 10 · z. A z of 1.5 becomes T = 65. Do not confuse it with the t statistic of Student's t test, which is something else.

Q: Can I compute a z-score if I do not know the standard deviation?

You need it. If you have the data list, the site's standard deviation calculator gives you σ, or the sample s, to put in the field. Without a standard deviation there is no scale to measure the distance to the mean.

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More pages for this tool

how to calculate z-score

How to calculate the z-score and the percentile

In the Z-score tab, enter the mean, the standard deviation and the value. The tool returns z, the percentile and the share above, and the z field goes back to the raw value.

From raw score to percentile in two steps

The z-score measures the distance between a value and the mean in units of standard deviation: z = (x − μ) / σ. If an exam mean is 70, the standard deviation is 10 and you scored 85, then z = (85 − 70) / 10 = 1.5. The sign matters: positive is above the mean, negative below, zero is the mean itself. Because z has no unit, it lets you compare performance across exams with different scales, such as 85 on an exam with mean 70 and 640 on another with mean 500 and deviation 100 (z = 1.4).
The second step is turning z into a percentile with the normal distribution function Φ(z). With z = 1.5, Φ = 0.9332: you scored above 93.3% of candidates, if the scores follow a normal curve. For z = 2, Φ = 0.9772, the 97.7th percentile. The share above the value is the complement, 1 − Φ: 6.68% for z = 1.5 and 2.28% for z = 2. At z = −1.5 the mirror happens: Φ = 0.0668, only 6.7% fall below.
The way back is x = μ + z · σ. To find which score matches z = 1.5 on an IQ scale (mean 100, deviation 15), compute 100 + 1.5 · 15 = 122.5. Two caveats: the percentile only holds if the distribution is roughly normal, which many standardized tests guarantee by construction and most raw data, such as income, do not; and a sample z-score uses the sample standard deviation, which differs from the population one when n is small.

Checked examples

IQ of 130

Input: x = 130, μ = 100, σ = 15
Expected output: z = 2, percentil 97,72

2.28% fall above.

Score 85 with mean 70 and deviation 10

Input: x = 85, μ = 70, σ = 10
Expected output: z = 1,5, percentil 93,32

Φ(1.5) = 0.9332.

Value below the mean

Input: x = 55, μ = 70, σ = 10
Expected output: z = −1,5, percentil 6,68

Mirror of +1.5.

From z to the value

Input: z = 1,5, μ = 100, σ = 15
Expected output: x = 122,5

x = 100 + 1.5 · 15.

90th percentile of an IQ

Input: p = 90%, μ = 100, σ = 15
Expected output: x = 119,2

z = 1.2816, Inverse tab.

Full tool FAQ

How do you calculate a probability in the normal distribution?

Standardize the value with z = (x − μ) / σ and read the cumulative area Φ(z). For P(X < x) the answer is Φ(z); for P(X > x) it is 1 − Φ(z); for P(a < X < b) it is Φ(zb) − Φ(za). With μ = 100, σ = 15 and x = 115, z = 1 and P(X < 115) = 0.8413. In the calculator, pick the type, type the values and read the result with its step-by-step work.

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions

Is a z-score of 2 good or bad?

It depends on what is measured. z = 2 means two deviations above the mean, the 97.7th percentile: great on an exam, worrying in a cholesterol test. About 4.55% of values of a normal curve fall outside ±2.

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

The T-score is a rescaling of z with mean 50 and deviation 10: T = 50 + 10 · z. A z of 1.5 becomes T = 65. Do not confuse it with the t statistic of Student's t test, which is something else.

Can I compute a z-score if I do not know the standard deviation?

You need it. If you have the data list, the site's standard deviation calculator gives you σ, or the sample s, to put in the field. Without a standard deviation there is no scale to measure the distance to the mean.

Next steps

Share of values within k standard deviations of the mean
Band	Range	Rule	Exact
μ ± 1σ	85 – 115	68%	68.2689%
μ ± 2σ	70 – 130	95%	95.45%
μ ± 3σ	55 – 145	99.7%	99.73%

Share of values within k standard deviations of the mean

Band

Range

Rule

Exact

μ ± 1σ

85 – 115

68%

68.2689%

μ ± 2σ

70 – 130

95%

95.45%

μ ± 3σ

55 – 145

99.7%

99.73%

From raw score to percentile in two steps

The z-score measures the distance between a value and the mean in units of standard deviation: z = (x − μ) / σ. If an exam mean is 70, the standard deviation is 10 and you scored 85, then z = (85 − 70) / 10 = 1.5. The sign matters: positive is above the mean, negative below, zero is the mean itself. Because z has no unit, it lets you compare performance across exams with different scales, such as 85 on an exam with mean 70 and 640 on another with mean 500 and deviation 100 (z = 1.4).

The second step is turning z into a percentile with the normal distribution function Φ(z). With z = 1.5, Φ = 0.9332: you scored above 93.3% of candidates, if the scores follow a normal curve. For z = 2, Φ = 0.9772, the 97.7th percentile. The share above the value is the complement, 1 − Φ: 6.68% for z = 1.5 and 2.28% for z = 2. At z = −1.5 the mirror happens: Φ = 0.0668, only 6.7% fall below.

The way back is x = μ + z · σ. To find which score matches z = 1.5 on an IQ scale (mean 100, deviation 15), compute 100 + 1.5 · 15 = 122.5. Two caveats: the percentile only holds if the distribution is roughly normal, which many standardized tests guarantee by construction and most raw data, such as income, do not; and a sample z-score uses the sample standard deviation, which differs from the population one when n is small.

Checked examples

IQ of 130

Input: x = 130, μ = 100, σ = 15
Expected output: z = 2, percentil 97,72

2.28% fall above.

Score 85 with mean 70 and deviation 10

Input: x = 85, μ = 70, σ = 10
Expected output: z = 1,5, percentil 93,32

Φ(1.5) = 0.9332.

Value below the mean

Input: x = 55, μ = 70, σ = 10
Expected output: z = −1,5, percentil 6,68

Mirror of +1.5.

From z to the value

Input: z = 1,5, μ = 100, σ = 15
Expected output: x = 122,5

x = 100 + 1.5 · 15.

90th percentile of an IQ

Input: p = 90%, μ = 100, σ = 15
Expected output: x = 119,2

z = 1.2816, Inverse tab.

Full tool FAQ

How do you calculate a probability in the normal distribution?

What is a z-score and how do I interpret it?

How do I find the x value that matches a percentile?

What is the difference between a one-tailed and a two-tailed p-value?

What does the 68-95-99.7 empirical rule say?

How do I read the z table?

When should I use t instead of z for the p-value?

Why is P(X = x) zero?

How accurate are the calculations?

Can I use it for data that is not normal?

Frequently asked questions

Is a z-score of 2 good or bad?

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

The T-score is a rescaling of z with mean 50 and deviation 10: T = 50 + 10 · z. A z of 1.5 becomes T = 65. Do not confuse it with the t statistic of Student's t test, which is something else.