Normal Distribution Calculator

What is a normal distribution?

A normal distribution is a symmetric, bell-shaped probability distribution in which values cluster around the mean and become rarer the further they are from it. It is described completely by two numbers: the mean (μ), which sets the centre, and the standard deviation (σ), which sets the spread. Many measurements – heights, measurement errors, test scores, manufacturing tolerances – are approximately normal, and averages of large samples are close to normal even when the underlying data are not (the central limit theorem).

The probability density function

The height of the curve at any point is given by f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)²/(2σ²)). The height itself is not a probability; probabilities are areas under the curve. The total area is exactly 1, and the curve is symmetric, so 50% of the area lies on each side of the mean.

Z-scores and the standard normal distribution

A z-score tells you how many standard deviations a value lies from the mean: z = (x − μ) ÷ σ. Converting to z-scores turns any normal distribution into the standard normal distribution, which has mean 0 and standard deviation 1. That is why a single z-table works for every normal problem.

Example: a test has a mean of 75 and a standard deviation of 10. A score of 85 has z = (85 − 75) ÷ 10 = 1. The area to the left of z = 1 is 0.8413, so the score is better than about 84% of test-takers.

If you already have a z-score

Set μ = 0 and σ = 1 – the standard normal distribution calculator mode – and enter the z-score directly.

How to find probabilities

• Below a value: P(X < x) = Φ(z). For the test, P(X < 85) = 0.8413.
• Above a value: P(X > x) = 1 − Φ(z). P(X > 85) = 0.1587.
• Between two values: Φ(z₂) − Φ(z₁). P(70 < X < 90) = 0.9332 − 0.3085 = 0.6247.
• Outside two values: 1 − P(between) = 0.3753.

For a continuous distribution, the probability of any exact value is zero, so P(X < 85) and P(X ≤ 85) are the same.

Inverse normal: finding a value from a probability

Sometimes you know the percentile and need the value. To find the score needed to beat 90% of test-takers, find the z with 0.90 of the area to its left – z ≈ 1.2816 – then convert back: x = μ + z × σ = 75 + 1.2816 × 10 ≈ 87.8. The calculator's inverse mode does this for any percentile.

The empirical rule (68-95-99.7)

The empirical rule states that about 68%, 95% and 99.7% of data fall within one, two and three standard deviations of the mean. More precisely the areas are 68.27%, 95.45% and 99.73%. The rule is a quick sanity check: with μ = 75 and σ = 10, roughly two-thirds of scores lie between 65 and 85, and almost all between 45 and 105.

Z-table: common values

Critical values used in statistics

These values appear in confidence intervals (mean ± 1.96 × standard error for 95%) and hypothesis tests.

Real-life examples

Delivery times

A courier's delivery times are roughly normal with a mean of 3 days and a standard deviation of 0.5 days. The chance a parcel arrives within 4 days is P(X < 4): z = 2, so about 97.7%.

Filling bottles on a production line

Bottles are filled to a mean of 500 g with a standard deviation of 4 g. The share within the 490–510 g tolerance is Φ(2.5) − Φ(−2.5) ≈ 98.76%, so about 12 bottles in 1,000 fall outside – useful for setting quality-control limits.

Adult heights

If adult heights in a group have a mean of 170 cm and a standard deviation of 7 cm, about 85% are between 160 and 180 cm, and about 1.6% are taller than 185 cm.

Normal approximation to the binomial

When a binomial distribution has many trials, it is close to normal with mean np and standard deviation √(np(1 − p)). A common rule of thumb is that np and n(1 − p) should both be at least 10. Because the binomial is discrete, add or subtract 0.5 (a continuity correction): for 100 coin flips, P(at most 55 heads) ≈ P(X < 55.5) with μ = 50 and σ = 5, giving z = 1.1 and a probability of about 86.4%.

Is my data normal?

Before using these probabilities on real data, check the shape. A histogram should look roughly symmetric and bell-shaped; a normal Q–Q plot should be close to a straight line; formal tests such as Shapiro–Wilk can help with small samples. Skewed data (incomes, waiting times) or data with hard limits (percentages near 0 or 100) are often not normal, and probabilities from a normal model can then be badly wrong in the tails.

How the calculator computes probabilities

There is no simple algebraic formula for the area under the bell curve, so calculators use the error function: Φ(z) = ½ × [1 + erf(z ÷ √2)]. Modern numerical methods give this to many decimal places, which is more precise than a printed z-table with four decimals. The inverse, Φ⁻¹(p), is computed with a rational approximation refined by iteration.

Comparing scores from different tests

Z-scores let you compare results on different scales. A score of 62 on a test with mean 50 and SD 10 has z = 1.2; a score of 80 on a test with mean 70 and SD 8 has z = 1.25. The second is slightly better relative to its group – about the 89th percentile versus the 88th.

IQ scores are designed to be normal with mean 100 and SD 15, so an IQ of 130 (z = 2) is above about 97.7% of the population, and only about 2.3% score higher.

Sampling distributions and standard error

The mean of a random sample is itself approximately normal, with standard deviation σ ÷ √n – the standard error. If individual values have σ = 12, the mean of a sample of 36 has a standard error of 12 ÷ 6 = 2. So about 95% of sample means fall within ±4 of the true mean. This is the basis of confidence intervals and many statistical tests.

Normal or t-distribution?

When you estimate the standard deviation from a small sample, use the t-distribution instead of the normal for confidence intervals and tests. The t-distribution has heavier tails; with about 30 or more observations it is very close to normal.

Normal distribution in quality control

In manufacturing, control limits are often set at the mean ± 3σ. If a process is normal and stable, only about 0.27% of items fall outside those limits by chance, so a point outside them signals that something has changed. 'Six Sigma' programmes aim for processes so consistent that specification limits sit six standard deviations from the mean; allowing for a conventional 1.5σ drift, that corresponds to about 3.4 defects per million opportunities.

Working backwards: finding σ or μ

If you know a percentile, you can solve for a missing parameter. Suppose 95% of a machine's parts must fall between 490 and 510 mm and the process is centred at 500. Two-tailed 95% corresponds to z = ±1.96, so the largest acceptable standard deviation is 10 ÷ 1.96 ≈ 5.1 mm. Similarly, if a test's mean is unknown but the standard deviation is 10 and 20% of students score below 60, then z = −0.8416 and the mean is 60 + 0.8416 × 10 ≈ 68.4.

Grading on a curve

Some courses assign grades by percentile. If the top 10% receive an A on an exam with mean 72 and SD 9, the A cut-off is 72 + 1.2816 × 9 ≈ 83.5. The next 20% (the 70th to 90th percentiles) might receive a B: the B cut-off is 72 + 0.5244 × 9 ≈ 76.7. Inverse mode gives each cut-off directly.

Two-tailed probabilities and p-values

In hypothesis testing, the p-value for a z statistic is the area in the tail beyond it. For a two-sided test, double the one-tail area: a z of 2.1 has a one-tail area of about 0.0179, so the two-sided p-value is about 0.036 – below the common 0.05 threshold. The calculator's 'outside' mode gives the two-tailed area for any pair of values symmetric around the mean.

Step by step with a z-table

1. Write down μ, σ and the value(s): μ = 75, σ = 10, between 70 and 90.
2. Convert to z-scores: (70 − 75) ÷ 10 = −0.5 and (90 − 75) ÷ 10 = 1.5.
3. Look up the areas to the left: Φ(−0.5) = 0.3085 and Φ(1.5) = 0.9332.
4. Subtract: 0.9332 − 0.3085 = 0.6247, so about 62.5% of scores lie between 70 and 90.

Common mistakes

• Using the variance instead of the standard deviation – σ is the square root of the variance.
• Forgetting to subtract from 1 for 'greater than' questions.
• Applying the normal model to strongly skewed data.
• Mixing up percentiles and probabilities: the 90th percentile is the value with 0.90 of the area below it.
• Using a one-tailed critical value (1.645) when a two-tailed one (1.96) is needed, or the reverse.

Sources

1. NIST/SEMATECH e-Handbook of Statistical Methods – Normal distribution — itl.nist.gov/div898/handbook