Heights, test scores and measurement errors often follow a normal (bell-shaped) distribution. Enter the mean, standard deviation and two values to get the probability below, above and between them.
Enter the mean and standard deviation.
Enter the lower and upper values.
Read the probabilities.
Standardise: z = (x − μ) ÷ σ
Between: P(a < X < b) = Φ(z_b) − Φ(z_a)
The normal distribution is the familiar bell-shaped curve. Heights of adults, measurement errors, weights of packets filled by a machine and scores on large exams often follow it approximately. This calculator takes the mean and standard deviation of such a distribution plus two values, a and b, and returns three probabilities: of a value falling below a, above b, and between them.
These are the questions people actually ask. What share of students will clear a cut-off? What fraction of cement bags will weigh under 49.5 kg? How many customers will fall inside the size range a garment maker stocks? Before software, answering needed a z-table and careful subtraction. The calculator does the standardising and the area look-ups exactly.
1. Write down the mean μ and the standard deviation σ.
2. Standardise the lower value: z_a = (a − μ) ÷ σ.
3. Standardise the upper value: z_b = (b − μ) ÷ σ.
4. Look up Φ(z_a) and Φ(z_b), the areas to the left of each z under the standard normal curve.
5. Below a: P(X < a) = Φ(z_a). Above b: P(X > b) = 1 − Φ(z_b).
6. Between: P(a < X < b) = Φ(z_b) − Φ(z_a).
7. Multiply by 100 for percentages, or by the group size for an expected count.
The normal curve has the formula f(x) = (1 ÷ (σ√(2π))) × e^(−(x − μ)² ÷ (2σ²)). The height of the curve is not itself a probability. Probability is the area under the curve between two points, and the total area is exactly 1. The curve is symmetric about μ and its width is set by σ: the points of inflection, where the bell changes from curving down to curving up, sit exactly at μ ± σ. Changing μ slides the bell; changing σ stretches or squeezes it while keeping the area at 1.
Any normal variable X can be converted into the standard normal Z = (X − μ) ÷ σ, which has mean 0 and SD 1. Areas are preserved by this change, so one table of Φ(z) serves every normal distribution. That is also why the empirical rule works for all of them: about 68.27% of values lie within one SD of the mean, 95.45% within two and 99.73% within three. For a continuous distribution, P(X = a) is zero, so it makes no difference whether you write < or ≤.
The central limit theorem says that a quantity made by adding many small independent effects tends towards a normal shape, whatever the individual effects look like. Height depends on many genes and environmental factors; measurement error comes from many small disturbances. But incomes, waiting times and house prices are skewed, and some data has heavier tails than the normal curve predicts. For those, normal probabilities in the tails can be badly wrong. Also note the calculator returns zero for the between probability if a is larger than b.
A school uniform supplier in Lucknow assumes the heights of senior students follow a normal distribution with mean 165 cm and SD 7 cm. It stocks blazers for heights from 158 cm to 180 cm and wants to know what share of students fall inside, below and above that range.
Standardise both values: z_a = (158 − 165) ÷ 7 = -1 z_b = (180 − 165) ÷ 7 = 2.1429
Cumulative probabilities: Φ(z_a) = 0.1587, Φ(z_b) = 0.9839
P(a < X < b): 0.9839 − 0.1587 = 0.8253
Answer: P(158 < X < 180) 82.528%; P(X < 158) 15.866%; P(X > 180) 1.606%
Using the variance in place of the standard deviation when standardising.
Reading Φ(z) as the area above z when a table gives the area below, or vice versa; check the diagram at the top of the table.
Subtracting in the wrong order for the between probability and getting a negative number.
Assuming data is normal without looking at it, especially for skewed quantities like income.
Forgetting a continuity correction when using the normal curve to approximate counts such as binomial data.
Setting size ranges for garments, helmets or furniture to cover most of a population.
Estimating the share of candidates above a cut-off in a large exam.
Quality control of filled packets, bolts or cement bags against weight limits.
Estimating the chance a measurement error exceeds a tolerance in lab and field work.
Risk estimates in finance, with care about tails that are fatter than normal.
What is P(μ − σ < X < μ + σ)?
About 68.27% for any normal distribution.
Are these one-tailed or two-tailed?
P(X < a) and P(X > b) are one-tailed; the between value covers the middle region.