Normal Distribution, Z-Score and P-Value Calculator

Q: Is the empirical rule the same as the central limit theorem?

No. The rule describes how much of a normal distribution falls in each band. The central limit theorem says the mean of many observations tends to a normal, and that is what lets you apply the rule to means.

Q: How many standard deviations contain 99% of the data?

In a normal, 2.576 standard deviations on each side (the 99% two-tailed critical z). For 90% it is 1.645 and for 99.9%, 3.291. The Inverse tab, with the central band option, computes the interval for any percentage.

Q: What is a "six sigma" value?

It is a value six standard deviations from the mean. In a normal, the tail beyond 6σ is 9.9e-10 on one side, about 1 in 1 billion. The Six Sigma methodology uses the term differently, with a 1.5σ shift of the mean, and talks about 3.4 defects per million.

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empirical rule 68 95 99.7 calculator

The 68-95-99.7 empirical rule for your mean and standard deviation

The empirical rule panel sits below the Probability, Inverse and Z-score tabs, already computed for the mean and standard deviation entered. To see a band's area on the chart, use Probability and choose "between a and b".

What the 68-95-99.7 rule says and where it falls short

In a normal distribution, the probability that a value falls within k standard deviations of the mean is erf(k / √2). For k = 1 it gives 68.27%, for k = 2 it gives 95.45% and for k = 3 it gives 99.73%. The round numbers 68, 95 and 99.7 are just an easy reminder. With mean 100 and deviation 15, the rule gives the intervals 85 to 115, 70 to 130 and 55 to 145. Outside 3 deviations 0.27% of cases remain, about 1 in 370.
Many people take the two deviations of the rule as the 95% interval, but the exact 95% interval uses 1.96 standard deviations, not 2. For an IQ with mean 100 and deviation 15, that is 70.6 to 129.4 and not 70 to 130; the difference matters when the result feeds a decision, such as a process control limit. In quality control, the 3-sigma limit leaves 0.27% false alarms in a stable, normal process.
The rule only holds for normal or near-normal distributions. With skewed or heavy-tailed data, the percentages change. Chebyshev's inequality gives the floor valid for any distribution: at least 75% of values lie within 2 deviations and at least 88.9% within 3. If your data histogram is symmetric, single-peaked and without strong outliers, the rule is a good approximation.

Checked examples

One deviation around the mean

Input: μ = 100, σ = 15
Expected output: 85 a 115: 68,27%

erf(1/√2) = 0.682689.

Two deviations

Input: μ = 100, σ = 15
Expected output: 70 a 130: 95,45%

The exact 95% runs from 70.6 to 129.4.

Three deviations

Input: μ = 100, σ = 15
Expected output: 55 a 145: 99,73%

0.27% remains outside.

Above two deviations

Input: P(X > 130)
Expected output: 2,28%

Half of the 4.55% outside ±2σ.

Between 90 and 110

Input: μ = 100, σ = 15
Expected output: 49,50%

z from −0.667 to 0.667.

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 the empirical rule the same as the central limit theorem?

No. The rule describes how much of a normal distribution falls in each band. The central limit theorem says the mean of many observations tends to a normal, and that is what lets you apply the rule to means.

How many standard deviations contain 99% of the data?

In a normal, 2.576 standard deviations on each side (the 99% two-tailed critical z). For 90% it is 1.645 and for 99.9%, 3.291. The Inverse tab, with the central band option, computes the interval for any percentage.

What is a "six sigma" value?

It is a value six standard deviations from the mean. In a normal, the tail beyond 6σ is 9.9e-10 on one side, about 1 in 1 billion. The Six Sigma methodology uses the term differently, with a 1.5σ shift of the mean, and talks about 3.4 defects per million.

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%

What the 68-95-99.7 rule says and where it falls short

In a normal distribution, the probability that a value falls within k standard deviations of the mean is erf(k / √2). For k = 1 it gives 68.27%, for k = 2 it gives 95.45% and for k = 3 it gives 99.73%. The round numbers 68, 95 and 99.7 are just an easy reminder. With mean 100 and deviation 15, the rule gives the intervals 85 to 115, 70 to 130 and 55 to 145. Outside 3 deviations 0.27% of cases remain, about 1 in 370.

Many people take the two deviations of the rule as the 95% interval, but the exact 95% interval uses 1.96 standard deviations, not 2. For an IQ with mean 100 and deviation 15, that is 70.6 to 129.4 and not 70 to 130; the difference matters when the result feeds a decision, such as a process control limit. In quality control, the 3-sigma limit leaves 0.27% false alarms in a stable, normal process.

The rule only holds for normal or near-normal distributions. With skewed or heavy-tailed data, the percentages change. Chebyshev's inequality gives the floor valid for any distribution: at least 75% of values lie within 2 deviations and at least 88.9% within 3. If your data histogram is symmetric, single-peaked and without strong outliers, the rule is a good approximation.

Checked examples

One deviation around the mean

Input: μ = 100, σ = 15
Expected output: 85 a 115: 68,27%

erf(1/√2) = 0.682689.

Two deviations

Input: μ = 100, σ = 15
Expected output: 70 a 130: 95,45%

The exact 95% runs from 70.6 to 129.4.

Three deviations

Input: μ = 100, σ = 15
Expected output: 55 a 145: 99,73%

0.27% remains outside.

Above two deviations

Input: P(X > 130)
Expected output: 2,28%

Half of the 4.55% outside ±2σ.

Between 90 and 110

Input: μ = 100, σ = 15
Expected output: 49,50%

z from −0.667 to 0.667.

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