Normal Distribution Calculator

What is the standard normal distribution Z-score formula?

The Z-score formula standardises any value x from a normal distribution with mean μ and standard deviation σ: z = (x − μ) / σ. The resulting z is the number of standard deviations x lies above or below the mean, and the standard normal cumulative distribution Φ(z) gives the probability of a value below x.

Normal probabilities in three steps

A normal distribution is fixed by two numbers, its mean μ and its standard deviation σ, and every normal curve is the same bell, shifted and stretched. That is why one function, Φ, answers every question about all of them, in three steps:

1. Standardise. Turn each bound x into a z-score, z = (x − μ) / σ, the number of standard deviations it lies from the mean.
2. Look up Φ(z), the area under the standard bell curve to the left of z, from a table or a calculator.
3. Combine. Subtract two areas for a range, P(a < X < b) = Φ(z_b) − Φ(z_a), and take an area from 1 for the part above a bound.

The whole area under the curve is 1, so every area is a probability: the chance that one value drawn from the distribution lands there. Going the other way, from an area back to a value, is the inverse normal, and the calculator prints its working too.

Using the calculator

Pick a question: Below, Above, Between, Outside or Percentile. Type the mean and the standard deviation, then drag along the axis or type the bounds, and the shading, its probability, both z-scores and the working follow. With the curve selected, the arrow keys move a bound by 0.01σ, or by 0.1σ with Shift, and Tab moves to the other bound. The 68-95-99.7 button marks the rule beneath the axis, z-table shows the row of a printed table for each bound, and Grade bands splits the curve into five. Save image gives a credited PNG of the curve as it stands, and the link keeps every setting.

Worked example: IQ-style scores, N(100, 15²)

Modern IQ tests, the Wechsler scales among them, are normed so that their scores have a mean of 100 and a standard deviation of 15, which is the curve the calculator opens with. That is a property of how the tests are scaled, and these are the areas it gives.

• Between 85 and 115. z = (85 − 100) / 15 = −1 and z = (115 − 100) / 15 = 1, so P(85 < X < 115) = Φ(1) − Φ(−1) = 0.841345 − 0.158655 = 0.68269, about 68.27 percent of scores.
• Above 130. z = (130 − 100) / 15 = 2 and P(X > 130) = 1 − Φ(2) = 1 − 0.97725 = 0.02275, about 1 score in 44.
• The 90th percentile. z = Φ⁻¹(0.9) = 1.2816, so x = 100 + 1.2816 × 15 = 119.2.

Reading a z-table

A printed z-table gives Φ(z) to four decimals for z to two. Find the row for z to one decimal, then the column for its second decimal: for z = 1.28, row 1.2 and column 0.08 give 0.8997. For a negative z, use the symmetry Φ(−z) = 1 − Φ(z), so Φ(−1.00) = 1 − 0.8413 = 0.1587, or a table with negative rows. The calculator’s z-table button shows the row for each bound with its cell ringed, and the working uses the exact z rather than z rounded to two decimals.

Percentiles and the inverse normal

The inverse runs the steps backwards: from an area to a z with z = Φ⁻¹(p), then back to the scale with x = μ + zσ. Percentile mode takes the area to the left, to the right, in the middle or in both tails, typed as a decimal or a percent, or set by dragging the edge of the area. The central 95 percent of any normal curve lies within 1.95996 standard deviations of the mean, so for N(100, 15²) it runs from 70.6 to 129.4, and the same z leaves 2.5 percent in each tail. Φ⁻¹ is Wichura’s Algorithm AS 241 (1988), good to about one part in 10¹⁶.

Fitting a bell curve to your data

To make a bell curve from your own data, paste a column of numbers into the second tab, straight from a spreadsheet or typed with spaces or commas. The calculator finds their mean and standard deviation, the sample one by default, as the standard deviation calculator does, and draws a histogram on the density scale, where each bar’s area is its share of the data. The bars then add up to an area of 1, the same as the fitted curve, so the curve is drawn to the same scale as the bars and the two can be compared directly; the frequency density and histogram calculator explains the density scale. The number of bars follows the Freedman-Diaconis rule, bars about 2 × IQR × n^(−1/3) wide, and a slider changes it.

A histogram can look bell-shaped without being normal, so the tab adds a Q-Q plot, which puts each sorted value against the z a normal distribution would give it. Normal data lie along the straight line through the quartiles. The example’s 100 made-up test scores have a mean of 65.93 and a sample standard deviation of 13.75, and their lowest score, 27, sits about 6 below the line while their highest, 94, sits about 4 below it: a longer lower tail and a shorter upper one than a bell curve has, a slight left skew. Use this fit carries the mean and standard deviation to the probability tab.

Grading on a curve

Grading on a curve sets grade boundaries by position in the distribution rather than at fixed marks. With cut-offs at μ ± 0.5σ and μ ± 1.5σ, a normal curve splits into shares of 6.68, 24.17, 38.29, 24.17 and 6.68 percent, from E up to A. The Grade bands button draws those bands under the curve, lets you move each cut-off as a z or as a percentile, and in the data tab counts your own scores in each band beside what the fitted curve expects: for the example, 6, 23, 39, 24 and 8 against 6.68, 24.17, 38.29, 24.17 and 6.68.

Whether to grade on a curve at all is a policy decision for whoever sets the marks, and the calculator makes no recommendation. Real marks are often skewed, so a fixed share for each grade can mean very different marks from one test to the next, which a Q-Q plot of the marks themselves will show.

Common mistakes

• Using the variance for σ. A variance of 225 means σ = 15. Dividing by 225 instead gives z-scores fifteen times too small.
• Asking for P(X = a). One value has no width, so it has no area: P(X = 115) is 0, and P(X < 115) and P(X ≤ 115) are the same number, 0.84134.
• Forgetting to standardise. Φ(130) is not P(X < 130). A table is read at z = (130 − 100) / 15 = 2.
• Mixing up the sample and population standard deviations. For data that are a sample of something larger, divide by n − 1. For the example scores that gives 13.75, against 13.68 dividing by n.
• Taking a far tail by subtraction. 1 − Φ(4) needs Φ(4) to nine figures to keep five in the answer, 3.1671 × 10⁻⁵. The calculator works out each tail in its own tail, which is how it gives P(5 < Z < 6) = 2.8566 × 10⁻⁷ to every figure shown, where Φ(6) and Φ(5) agree to six decimal places.

Further tools

Two normal curves and a cut-off between them make the sensitivity and specificity visualiser, where the tails beyond the cut-off are the false positives and the false negatives. The chi-square calculator compares counts with the counts a ratio predicts, and the error propagation calculator combines standard deviations through a formula.