Bell Curve Maker
Create and explore a bell curve using a mean and standard deviation or your own numerical data. Compare normal distributions, highlight probability regions, and see how the shape changes with the parameters.
Your bell curve
Inputs changed. Calculate to generate your updated graph.
Normal distribution
- Solid purple: Normal model
Highlighted interval: -1 to 1; probability 0.6827 (68.27%).
Graph window: −4 to 4. Areas use the complete normal distribution, not just the visible window.The mean shifts the center; standard deviation controls width. Curve height is density, not the probability of an exact value.
Displayed numbers are rounded; calculations retain full available precision. Central percentages describe probability intervals, not confidence intervals for a mean.Show calculation
- Selected model: X ~ N(0, 1).
- f(x) = exp(−(x−μ)²/(2σ²)) / (σ√(2π)); Normal model: μ = 0, σ = 1.
- Variance σ² = 1² = 1; peak f(μ) = 1 / (1√(2π)) = 0.3989423.
- zₐ = (-1 − 0) / 1 = -1; zᵦ = 1.
- P(-1 ≤ X ≤ 1) = Φ(zᵦ) − Φ(zₐ) = 0.6827 (68.27%). Stable tail differences are used internally.
Show histogram bins
Freedman–Diaconis bin width; Sturges fallback if IQR is zero; bounded to 2–40 bins. Last bin includes its upper bound. Density = count / (n × bin width).
How to use the Bell Curve Maker
- Choose whether to enter a mean and standard deviation, use your own dataset, or compare normal distributions.
- Enter the required values or select one of the example presets.
- Generate the graph and adjust the shading or other display options to explore the distribution.
- Review the curve properties, highlighted probabilities, and calculation details below the graph.
Worked example: creating a bell curve
Suppose a measurement is modeled by a normal distribution with a mean of 50 and a standard deviation of 5. We want to visualize the distribution and highlight values between 45 and 55.
- μ = 50.
- σ = 5.
- Variance = 25.
- Lower bound = 45.
- Upper bound = 55.
- z for 45 = −1.
- z for 55 = 1.
- Shaded probability ≈ 0.68268949.
- Shaded percentage ≈ 68.27%.
P(45 ≤ X ≤ 55) = Φ(1) − Φ(−1) ≈ 0.68268949
Under this normal model, approximately 68.27% of outcomes fall between 45 and 55.
This example uses sample values, not your current inputs.
Bell curve formula and properties
A bell curve represents the probability density of a normal distribution. The mean (μ) determines its center, while the standard deviation (σ) determines its spread. Every normal curve is symmetric about its mean, and its total area equals 1.
f(x) = exp(−(x−μ)²/(2σ²)) / (σ√(2π))
Here x is a value, μ is the mean, σ > 0 is the standard deviation, and f(x) is probability density. The standardized value is z = (x−μ)/σ; Φ(z) is the standard normal cumulative probability.
Increasing the standard deviation produces a wider, flatter curve. Decreasing it produces a narrower, taller curve. Moving the mean shifts the curve without changing its shape when the standard deviation remains fixed.
Formula & assumptions
- Standard deviation must be strictly positive.
- Fitted curves use sample SD with denominator n−1.
- A normal fit does not establish that observations are normally distributed.
- Histogram bar areas sum to 1; the y-axis shows probability density.
- Central probability intervals are not confidence intervals for an estimated mean.
- Parameters must have representable variance, density and graph bounds; rescale numerically unresolved inputs.
Reviewed: October 2026
The 68–95–99.7 rule
For a normal distribution, approximately 68.27% of outcomes lie within one standard deviation of the mean, 95.45% lie within two, and 99.73% lie within three. These percentages describe areas under the theoretical normal curve.
| Interval | Area |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
Bell curve vs. histogram
A bell curve is a smooth mathematical model. A histogram displays the distribution of observed values using bars. When the histogram is scaled to probability density, the two can be shown on the same axes and compared. A close visual match can support a normal-model approximation, but it does not prove that the data follow a normal distribution.
The fitted curve uses x̄ = Σx/n and s² = Σ(x−x̄)²/(n−1). Histogram bins use the Freedman–Diaconis width 2 × IQR / ∛n, with a Sturges fallback when IQR is zero and 2–40 bins. Repeated observations are retained. Each bar has density count / (n × bin width).
Comparing bell curves
Two normal distributions can have different centers, spreads, or both. Comparing them on the same axes makes these differences easier to understand. A larger mean moves the curve to the right, while a larger standard deviation makes it wider and lowers its peak.
For tail probabilities or values associated with percentiles, use the Normal Distribution Calculator. This bell curve creator and normal distribution graph generator focuses on constructing and comparing models.
Common questions
What is a bell curve?
A bell curve is the graph of a normal distribution. It is symmetric around the mean and becomes gradually lower toward both tails. The total area under the curve equals 1, representing 100% of the probability.
How do I create a bell curve?
Enter the mean and standard deviation of a normal distribution, then generate the graph. You can also paste numerical observations to fit a normal curve using their sample mean and sample standard deviation.
What is the difference between a bell curve and a normal distribution?
A normal distribution is a mathematical probability model. A bell curve is the familiar graphical representation of its probability density function.
What happens when I change the mean?
Changing the mean moves the center and peak of the normal curve along the horizontal axis. If the standard deviation stays the same, the curve's shape and height do not change.
What happens when I change the standard deviation?
A larger standard deviation makes the curve wider and flatter. A smaller standard deviation makes it narrower and taller. The total area under the curve stays equal to 1.
What is the 68–95–99.7 rule?
For a normal distribution, approximately 68.27% of outcomes fall within one standard deviation of the mean, 95.45% within two, and 99.73% within three.
Can I make a bell curve from my own data?
Yes. Enter or paste your numerical observations in From data mode. The calculator estimates the sample mean and standard deviation, displays a fitted normal curve, and can show a histogram of the original observations.
Does a bell curve fitted to my data mean the data are normally distributed?
No. A fitted curve is a model based on the dataset's center and spread. The observations may still be skewed, contain outliers, or follow a different distribution. Inspect the histogram and consider a validated normality assessment when the assumption matters.
Can I compare two or three bell curves?
Yes. Compare curves mode displays multiple normal distributions on shared axes. You can compare their mean positions, standard deviations, variances and peak density values.
What does the shaded area under a bell curve mean?
The shaded area represents the probability of a value falling within the selected interval under the specified normal distribution. The height of the curve at one point is a probability density, not the probability of that exact value.
Is a histogram the same as a bell curve?
No. A histogram shows observed data grouped into intervals, while a bell curve shows a theoretical normal probability density. They can be displayed together when the histogram uses the same density scale.
Can a bell curve have a negative mean?
Yes. The mean can be any finite real number, including a negative value. The standard deviation must be greater than zero because it describes the positive scale of the distribution.
What is the difference between this tool and a Normal Distribution Calculator?
This Bell Curve Maker focuses on creating and comparing graphs, including fitted curves and histograms. A Normal Distribution Calculator focuses on evaluating probabilities and finding values associated with particular percentiles.
Reference tables
Look up standard normal probabilities in the complete Z Table. Browse all statistical tables.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.