Normal distribution calculator with a shaded curve

A normal distribution is the symmetric bell-shaped curve that many measurements follow, set by its mean μ (the centre) and standard deviation σ (the spread). The probability of a value in a range is the area under the curve over that range. For plant heights with μ = 50 cm and σ = 4 cm, the chance of a plant under 46 cm is 0.1587. This normal distribution calculator is part of our Probability Calculator: set μ and σ, choose below, above, between or outside, and it shades the curve and gives the probability and the z-scores.

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How to use the normal distribution calculator

1. Open the Probability Calculator. It opens on Normal, with the standard normal: μ = 0 and σ = 1.
2. Under Distribution, type the Mean μ and Std deviation σ.
3. Under Probability, choose a region in Shade: P(X < a), P(X > a), P(a < X < b) or P(X < a or X > b).
4. Type Bound a, and Bound b for a range.
5. Read the probability and, under it, the z-score of each bound. The curve shades the area you asked for.
6. To practise, press Random problem in the Practice section.

The calculator goes from values to probabilities. It doesn't go backwards from a probability to a value (inverse normal), but example 5 shows how to get one.

How normal probabilities work

The total area under the curve is 1. The curve is symmetric about the mean, so half the area is on each side. For a continuous distribution the chance of exactly one value is 0, so P(X < a) and P(X ≤ a) are the same. Every normal distribution becomes the standard normal when you convert a value to its z-score:

As a rough guide, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. The z score calculator guide goes into z-scores in more depth.

Worked examples you can check in the calculator

The plants in a greenhouse have heights that are normally distributed with mean 50 cm and standard deviation 4 cm. Set μ = 50 and σ = 4.

Example 1: below a value

What share of plants are under 46 cm? 46 is one standard deviation below the mean (z = −1). Choose P(X < a) with a = 46: P(X < 46) = 0.1587, about 16%.

Example 2: above a value

What share are over 57 cm? Here z = (57 − 50) ÷ 4 = 1.75. Choose P(X > a) with a = 57: P(X > 57) = 0.0401, about 4%.

Example 3: between two values

What share are between 45 and 55 cm? The z-scores are −1.25 and 1.25. Choose P(a < X < b) with a = 45 and b = 55: P(45 < X < 55) = 0.7887.

Example 4: outside a range

Plants under 42 cm or over 58 cm are more than two standard deviations from the mean. Choose P(X < a or X > b) with a = 42 and b = 58: the probability is 0.0455, matching the 95% rule, since 1 − 0.9545 = 0.0455.

Example 5: finding a value from a probability

How tall is a plant in the tallest 5%? Choose P(X > a) and move Bound a until the probability is as close to 0.05 as you can get: at a = 56.6 it reads 0.0495. With a table, the z with 5% above it is 1.645, so x = μ + zσ = 50 + 1.645 × 4 = 56.6 cm.

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

Quick check

Is my data normal?

The calculator assumes the data really is normal. Check that a histogram of your data is roughly symmetric and bell-shaped, with most values near the middle. Heights, measurement errors and many test scores are often close to normal. Waiting times, incomes and anything that can't go below zero but has a long tail usually aren't.

Common mistakes

• Using the variance instead of the standard deviation. If a question gives σ² = 16, then σ = 4.
• Picking the wrong tail. "More than" is the right-hand area; "less than" is the left.
• Subtracting the wrong way for a range. P(a < X < b) is the area left of b minus the area left of a.
• Forgetting symmetry. P(X > μ + k) equals P(X < μ − k), which can save work.
• Applying it to data that isn't bell-shaped. The probabilities are only as good as that assumption.

Read next

Maths · StatisticsNormal distribution: bell curve, z-scores and examplesThe normal distribution explained: the bell curve, the 68-95-99.7 rule, z-scores and the standard normal, probabilities and the normal approximation.