Bell Curve Generator
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Bell Curve Examples
Browse normal distribution examples or generate your own above
What Is a Bell Curve?
A bell curve, also known as a normal distribution or Gaussian distribution, is a symmetric probability distribution that forms a characteristic bell shape when graphed. The highest point of the curve represents the mean (average), and the data spreads symmetrically on both sides. Bell curves are fundamental to statistics and appear naturally in many real-world phenomena, from test scores and height distributions to measurement errors and quality control processes.
The shape is defined by two parameters: the mean (μ) which determines the center, and the standard deviation (σ) which determines the spread. A larger σ produces a flatter, wider curve; a smaller σ produces a taller, narrower curve. The total area under any normal distribution curve always equals exactly 1 (or 100%), making it directly interpretable as probability.
The 68–95–99.7 Rule (Empirical Rule)
| Range | Probability | In plain English | Applications |
|---|---|---|---|
| μ ± 1σ | 68.27% | About 2 in 3 observations fall within one SD of the mean | Typical range, one standard deviation interval |
| μ ± 2σ | 95.45% | About 1 in 20 falls outside this range | 95% confidence intervals, control limits |
| μ ± 3σ | 99.73% | Only 1 in 370 falls outside — statistically rare | Six Sigma quality control, outlier detection |
| μ ± 1.96σ | 95.00% | Exactly 95% (used in hypothesis testing) | Z-test critical value, confidence intervals |
| μ ± 2.576σ | 99.00% | Exactly 99% of observations | 99% confidence intervals, strict quality control |
Applications of Bell Curves in Research
Bell Curve vs Other Distributions
Not all data follows a normal distribution. Skewed distributions have longer tails on one side — common in income data (right-skewed) or age at death (left-skewed). Bimodal distributions have two peaks, seen in mixed populations. Uniform distributions have equal probability across all values. Using bell curve assumptions on non-normal data leads to incorrect statistical conclusions. Common normality tests include the Shapiro-Wilk test, Kolmogorov-Smirnov test, and Q-Q plot inspection.
Related Topics
Sources & further reading:
- NIST Engineering Statistics Handbook — Normal Distribution
- Gauss, C.F. (1809). Theoria motus corporum coelestium. [Origin of the Gaussian distribution]
- Khan Academy — Normal Distributions Review
Frequently Asked Questions
This tool is fully free, browser-based, and requires no sign-up. It supports parameter-based curves (set μ and σ), data analysis mode (paste raw data to fit a normal distribution), and two-distribution comparison with Cohen's d effect size. You can shade regions, display confidence intervals, show the 68-95-99.7 rule, and download publication-ready SVG files.
In the Parameters tab, set the mean to your class average (e.g. μ = 75) and standard deviation to the SD of your scores (e.g. σ = 10). Select "Display style: 68–95–99.7 rule" to visualize grade boundaries. Alternatively, use the Data Analysis tab to paste your actual scores — it will fit the normal curve, calculate mean and SD from your data, and check how closely your grade distribution matches normality.
The empirical rule states that in any normal distribution: 68% of observations fall within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. This rule is useful for quickly estimating probabilities and identifying outliers. Values beyond ±3σ are considered statistically rare (only 0.3% of all observations). Select "Display style: 68–95–99.7 rule" in the Parameters tab to see this visualized with color-coded shading.
Use the "Compare Two" tab. Enter the mean and standard deviation for each distribution, add labels (e.g. "Control" and "Treatment"), and click Generate. The tool draws both curves with transparent fill, highlights the overlap region, and calculates Cohen's d effect size, the overlap coefficient, and the probability of superiority — all standard metrics for comparing two normal distributions in research.
Yes. Use the "Z-score boundaries" display style to show critical values on the bell curve. Enter your test's z-score as the boundary value. For a two-tailed test at α=0.05, enter z=1.96. The tool will shade the rejection regions (tails) and acceptance region (center). For a dedicated rejection region diagram with left/right/two-tailed options, see our Rejection Region Diagram Maker.
A histogram shows the actual frequency distribution of your data using bars. A bell curve is the theoretical normal distribution that your data is assumed to follow. The Data Analysis tab overlays both — it draws the empirical histogram from your actual data alongside the fitted normal curve. Comparing them visually is one of the simplest ways to assess normality. If your bars closely follow the curve shape, your data is approximately normal.