Bell Curve Generator & Maker
Normal distribution curves from a mean and SD: the 68-95-99.7 bands, a shaded area with its exact probability, z-scores, percentile cut-offs or two overlaid curves, for statistics lessons, exams and papers.
Bell curve styles
Four styles, each on a different bell curve: exam grade bands A to F in textbook colour, the share of women between 155 and 176 cm hatched in black and white for a test paper, a curve fitted to 40 fruit fly wing lengths as a journal panel with its density axis and rug, and the top 10% of daily step counts on a dark slide.
Bell curve examples
Four real renders, each drawn from the prompt on its card: IQ scores with the empirical rule's 68%, 95% and 99.7% brackets, newborn birth weight with the tail below 2500 g shaded at P = 0.0359, the two 0.0250 rejection regions of a z test at α = 0.05, and placebo and drug blood pressure curves sharing 66.82% of their area. Click a card to load its prompt.
What is a bell curve?
A bell curve is the graph of a normal distribution: a symmetric, single-peaked curve centred on the mean, whose width is set by the standard deviation. The area under it between two values is the probability of a value landing between them, so shading an area answers a question such as what share of IQ scores fall below 85 (15.87%). The curve never touches the axis; one standard deviation either side of the mean it has dropped to 61% of its peak height, and that is exactly where it changes from bending down to bending out.
About this bell curve generator
- Type the parameters the way your course writes them: mean 100, SD 15; μ = 0, σ = 1; 72 ± 8; N(100, 15²). Paste a list of raw values instead and the curve is fitted with the sample mean x̄ and the sample SD s (n − 1).
- Shaded areas are computed from the normal CDF before drawing and printed to four decimals: P(X < 85) = 0.1587 and P(85 < X < 115) = 0.6827 for mean 100, SD 15.
- Percent cut-offs run backwards through the inverse CDF: the top 5% of IQ scores starts at 124.67, and the 90th percentile of the standard normal sits at z = 1.282.
- The x axis spans μ − 4σ to μ + 4σ with a tick at every standard deviation (or at round numbers when a stated boundary falls between them); up to three curves can share it, each named in a legend.
- Only the normal distribution is drawn, and no bars: for measured data with a normal curve on top, the histogram maker bins the sample on a density scale.
Bell curve graphs you can generate
Seven normal distribution figures people come here for, what each one shows, and the words that get it.
Empirical rule (68-95-99.7) bell curve
The curve cut into strips at each standard deviation: 34.13% either side of the mean, 13.59% from one to two SD, 2.14% from two to three, and 0.13% in each tail. Write "empirical rule" or "68-95-99.7" and the labels follow the rule as it is taught (68%, 95% and 99.7% on nested brackets, 34%, 13.5%, 2.35% and 0.15% in the strips); write "standard deviation bands" for the exact two-decimal areas.
Bell curve with standard deviations
A normal curve whose axis is marked at μ − 3σ through μ + 3σ, the figure behind every question about how far a value is from the mean. IQ with mean 100 and SD 15 gives ticks at 55, 70, 85, 100, 115, 130 and 145; ask for σ labels or z-scores and a second row appears under the numbers.
Shaded area under the normal curve
One region below, above, between or outside stated values, with its probability printed on it. Mean 3400 g, SD 500 g, shaded below 2500 g prints P(X < 2500) = 0.0359; women's heights with mean 162 cm and SD 7 cm, shaded from 155 to 176 cm, print 0.8186. Each boundary gets its own vertical line up to the curve.
Z-score bell curve
The standard normal curve, mean 0 and SD 1, drawn on a z axis from −4 to 4. Mark a z-score, shade the area to its left for a z-table figure, or shade both rejection regions of a two-tailed test: α = 0.05 puts the critical values at ±1.96 with 0.0250 in each tail, α = 0.01 at ±2.576.
Grading curve with letter grade cut-offs
An exam's score distribution with each grade band shaded in its own fill and the share of students printed in it. Mean 72 and SD 8 with A above 88, B from 80 to 88, C from 64 to 80, D from 56 to 64 and F below 56 gives 2.28%, 13.59%, 68.27%, 13.59% and 2.28%. Grading by percentile works the other way: "the top 10% get an A" returns the cut-off score.
Two overlaid normal distributions
Treatment against control, or one method against another, on one shared axis with a legend. Ask for the overlap and the shared area is computed: placebo at 138 mmHg and drug at 126 mmHg, both with SD 14, overlap by 66.82% with Cohen's d = 0.86. Curves with different SDs cross at two points, and both are located.
Normal curve fitted to a sample
Paste the raw measurements and the curve is drawn from their mean and sample standard deviation, with n, x̄ and s printed beside it and a rug of the values on request. If you want the binned data drawn under the curve as bars, that figure is a histogram with a normal overlay.
The empirical rule (68-95-99.7), exact values
The rounded rule is what gets memorised; the exact areas are what a figure should print. Every row is computed from Φ, the standard normal CDF, and shown on the IQ scale for scale.
| Interval | Area inside | Area outside | IQ scale (mean 100, SD 15) |
|---|---|---|---|
| μ ± 1σ | 68.27% | 31.73% | 85 to 115 |
| μ ± 1.645σ | 90.00% | 10.00% | 75.3 to 124.7 |
| μ ± 1.96σ | 95.00% | 5.00% | 70.6 to 129.4 |
| μ ± 2σ | 95.45% | 4.55% | 70 to 130 |
| μ ± 2.576σ | 99.00% | 1.00% | 61.4 to 138.6 |
| μ ± 3σ | 99.73% | 0.27% | 55 to 145 |
95% of a normal distribution lies within 1.96 standard deviations of the mean, not 2. Two SD hold 95.45%; the 1.96 is the multiplier behind a 95% confidence interval and the critical value of a two-tailed test at α = 0.05.
The normal curve formula and the z-score
The bell curve is the normal density f(x), and every probability read off it is an area under f, found by standardising with z = (x − μ)/σ. A z-score counts how many standard deviations a value sits from the mean, so one table of Φ answers questions on any normal scale, from IQ points to grams.
f(x) = e^(−(x − μ)² / 2σ²) / (σ√(2π)) z = (x − μ) / σ P(X < x) = Φ(z)
In words
Height falls with the squared distance from μ; the area left of x is Φ(z).
Φ has no closed form. The generator evaluates it from the error function to double precision, Φ(z) = ½[1 + erf(z/√2)], so a printed 0.1587 or 0.9750 is right in all four decimals, and runs the inverse, Φ⁻¹, to turn a stated percentile into its cut-off.
Where the normal curve is used
Bell curves appear wherever a measurement is summarised by a mean and a standard deviation and a reader needs to see what that spread means.
Reference ranges in laboratory medicine
Many reference intervals are the central 95% of a healthy population, the mean ± 1.96 SD when the analyte is normally distributed; a shaded middle region shows which results get flagged as high or low.
Grading on a curve
Cut-offs set at fixed SD distances or at percentiles; the figure shows a class where each grade boundary sits and what share of students each band holds.
Hypothesis tests and confidence intervals
Rejection regions of a z test, a p value as the tail area beyond the observed z, a 95% interval as the middle 95%: the standard figures of an introductory statistics course.
Standardised test scores
IQ (mean 100, SD 15) and T-scores (mean 50, SD 10) are defined on a normal scale, so a percentile rank reads straight off the curve: an IQ of 130 is two SD up, above 97.72% of scores.
Normal distribution FAQ
How do I make a bell curve?
Give a mean and a standard deviation, say which area to shade, and generate. For example: "IQ scores, mean 100, SD 15, shade above 130 and print the probability." The axis at each standard deviation, the curve's heights and P(X > 130) = 0.0228 are worked out before anything is drawn. In a spreadsheet the same figure means a column of x values, NORM.DIST for each, and a smoothed line chart.
What does the empirical rule (68-95-99.7) say?
About 68% of a normal distribution lies within one standard deviation of the mean, 95% within two and 99.7% within three. The exact areas are 68.27%, 95.45% and 99.73%. The rule holds only for normal data; for any distribution at all, Chebyshev's inequality guarantees just 75% within two standard deviations.
How do I find the probability of a shaded area under a bell curve?
Turn each boundary into a z-score, z = (x − μ)/σ, then take Φ(z) for the area below, 1 − Φ(z) for the area above, and the difference of two Φ values for the area between. With mean 100 and SD 15: P(X < 85) = Φ(−1) = 0.1587, P(X > 115) = 0.1587, P(85 < X < 115) = 0.6827. This page is a figure maker rather than a normal distribution calculator, but it runs exactly that arithmetic and prints the result on the shaded region.
What is a z-score on a bell curve?
The number of standard deviations a value lies from the mean, z = (x − μ)/σ. A height of 176 cm, with mean 162 cm and SD 7 cm, has z = 2.00, and 97.72% of the distribution lies below it. On a standard normal curve the x axis is the z scale itself, so the critical values ±1.96 can be marked directly.
How do I shade the top 5% or mark a percentile?
Ask for it as a percent and the cut-off is found with the inverse normal CDF. The top 5% of IQ scores starts at 124.67 (z = 1.645), the bottom 10% ends at 80.78 (z = −1.282), and the 90th percentile of daily steps with mean 8000 and SD 2000 is 10,563.
Should I use a bell curve or a histogram?
A histogram shows data you measured; a bell curve shows a model, the normal distribution with a stated mean and SD. To check whether a sample looks normal, draw its histogram with a normal curve overlay on a density scale. Draw a bare bell curve to teach a distribution, mark cut-offs or show a probability.
What should a figure legend for a bell curve say?
Name the distribution, its parameters and what the shading means, for example: "Normal distribution with mean 100 and SD 15; the shaded area, P(X < 85) = 0.1587, is the proportion of scores below 85." For a fitted curve, give n, x̄ and s and say the curve is a normal fit rather than the data. When μ and σ come from a published population, cite it.
Can two normal curves be compared on one graph?
Yes: up to three curves share one axis, each in its own colour and named once in the legend. The point where two curves with equal SDs cross is halfway between their means; with unequal SDs they cross twice, once on each side of the narrower curve's peak. Ask for the overlap, or for Cohen's d, and those are printed too.
Does the AI change my mean, SD or probabilities?
No. Every tick, the curve's height at seventeen points, each boundary's z-score and every shaded probability are computed from your parameters first, and the image model is told to place them, not to work them out. A generic request with no parameters is drawn as the standard normal (mean 0, SD 1); IQ and T-score figures use the scales' defined values.
Is my data normal enough to draw as a bell curve?
A normal curve suits a variable that is roughly symmetric around one peak; check with a histogram or a normal Q-Q plot first. Right-skewed measurements such as reaction times, incomes or antibody titres misrepresent badly under a normal curve and are better shown as they are, or after a log transform.
Sources
- 1.3.6.6.1. Normal Distribution — NIST/SEMATECH, e-Handbook of Statistical Methods
- 6.1 The Standard Normal Distribution — OpenStax, Introductory Statistics 2e
Keep exploring
Our other diagram tools, and the guides behind them.
Tools
Generate a bell curve with the probability already worked out
Type a mean and an SD, name the area to shade, and get a normal distribution graph ready for a lesson, an exam or a paper.
Make a bell curve