Generate Your Bell Curve
Shade area & find probability NEW
Labels & axis for reports NEW
Full Descriptive Statistics
Generate a curve above to see a full statistical summary here.
Multi-Curve Overlay
Plot up to 3 normal distributions on the same axes for direct comparison.
Normality Check
Available when you enter data. Uses skewness, excess kurtosis and empirical rule adherence.
Enter data in the Manual Data or CSV tab and click Generate Bell Curve to see normality indicators.
How to Make a Normal Distribution Graph
- Choose your input. Type a mean and standard deviation, pick a preset such as IQ scores, or paste raw data (the tool computes the sample mean and standard deviation for you).
- Shade the area you need. Open Shade area & find probability, select left tail, right tail, between or two tails and enter the bounds as x values or z-scores. The probability appears instantly above the chart.
- Label it. Open Labels & axis for reports and add a chart title and axis labels, and switch the x-axis ticks to μ ± kσ or z-scores, which is the format most textbooks use.
- Export or share. Save a high-resolution PNG for a report, copy the image straight into Word, Google Docs or PowerPoint, or copy a link that reopens exactly the same graph.
Area Under the Normal Curve: How the Shaded Probability Is Computed
Probability under a bell curve is an area, not a height. Every normal distribution is converted to the standard normal with the z-score, and the area to the left of z is given by the cumulative distribution function \(\Phi(z)\):
| Region | Formula | Standard normal example |
|---|---|---|
| Left tail | Φ(za) | P(Z < 1.645) = 0.9500 |
| Right tail | 1 − Φ(za) | P(Z > 1.96) = 0.0250 |
| Between | Φ(zb) − Φ(za) | P(−1 < Z < 1) = 0.6827 |
| Two tails | Φ(za) + 1 − Φ(zb) | P(|Z| > 1.96) = 0.0500 |
The inverse problem, finding the value below which a given percentage falls, uses the inverse CDF: \(x=\mu+z_p\,\sigma\) with \(z_p=\Phi^{-1}(p)\). The Percentile box in the generator does exactly this. For more probability calculations see the normal distribution calculator and the z-score calculator.
Worked Examples (Click to Load in the Generator)
1. What fraction of people have an IQ above 130?
IQ scores are scaled to μ = 100 and σ = 15. z = (130 − 100)/15 = 2.00, so P(X > 130) = 1 − Φ(2.00) = 0.0228 (2.28%).
2. What share of IQ scores lie between 85 and 115?
z = −1 and +1, so P(85 < X < 115) = Φ(1) − Φ(−1) = 0.6827 (68.27%): the first band of the empirical rule.
3. Exam scores N(70, 10): what score is needed for the top 10%?
The 90th percentile has z = 1.2816, so x = 70 + 1.2816 × 10 = 82.8. Enter 90 in the Percentile box to see it marked.
4. Quality control: parts must be 50 ± 0.5 mm with σ = 0.2 mm
The limits are ±2.5σ, so the reject rate is P(|Z| > 2.5) = 2 × 0.00621 = 1.24%, about 12 parts per 1,000.
Bell Curve Theory & Formulas
The normal distribution (Gaussian distribution) is the most widely used probability distribution in statistics. It appears naturally due to the Central Limit Theorem wherever many small independent effects add together.
1. Probability Density Function
The PDF gives the height of the bell curve at each x. Probabilities are areas, not heights. The peak is at \(x=\mu\) with height \(\frac{1}{\sigma\sqrt{2\pi}}\).
2. Sample Statistics (for data-driven curves)
Using \(n-1\) (Bessel's correction) gives an unbiased estimate of population variance from a sample.
3. Skewness and Excess Kurtosis
A perfect normal has skewness = 0 and excess kurtosis = 0. Both metrics are used in the normality check above.
4. Empirical Rule
For any normal distribution: \(\approx68.27\%\) of values fall within \(\pm1\sigma\), \(\approx95.45\%\) within \(\pm2\sigma\), and \(\approx99.73\%\) within \(\pm3\sigma\) of the mean. The coloured bands on the chart visualise these regions directly.
Frequently Asked Questions
1. What is a bell curve?
A bell curve is the graph of a normal distribution. It is symmetric about its mean, peaks at the mean, and tapers off equally on both sides. The total area under the curve is always 1, representing 100% of all probability.
2. How do I generate a bell curve from my own data?
Switch to the Manual Data tab, paste your numbers separated by commas or line breaks, then click Generate Bell Curve. The tool computes the sample mean and standard deviation and plots the fitted normal curve with an optional histogram overlay.
3. Why does my bell curve look flat and wide?
A large standard deviation means data is spread out, producing a wide, shorter curve. This is mathematically correct. A larger σ always produces a flatter, wider bell. The area under the curve remains exactly 1 regardless of how it looks.
4. Can I compare multiple distributions?
Yes. Use the Multi-Curve Overlay section to plot up to 3 normal distributions with different parameters simultaneously. A comparison table shows mean, std dev, variance and peak PDF side by side.
5. What are the empirical bands on the chart?
The shaded bands mark ±1σ (68.27%), ±2σ (95.45%), and ±3σ (99.73%) from the mean. Select which bands to show or choose 'All three' to show all simultaneously. They give immediate visual context for data spread.
6. How does the histogram overlay work?
When you enter data manually or via CSV, a normalised frequency histogram is overlaid on the bell curve. The histogram is scaled to probability density (count divided by n times bin width) so it sits on the same y-axis as the PDF curve.
7. What does the normality check tell me?
The normality check reports skewness and excess kurtosis and classifies the result as approximately normal, mild departure or significant departure. It also checks whether your data respects the empirical rule proportions. For formal testing use Shapiro-Wilk (n ≤ 50) or Kolmogorov-Smirnov (n > 50).
8. What CSV format is accepted?
Upload a CSV or TXT file with numeric values in the first column. The header row is skipped automatically. Rows with non-numeric values are ignored. Only one column is needed.
9. Can I export the chart?
Yes. Click Save PNG under any chart to download it as a PNG file. The exported image includes all visible datasets, bands and labels exactly as displayed.
10. What is the difference between this tool and the Normal Distribution Calculator?
This tool focuses on visualisation: plotting curves, comparing distributions, showing empirical bands, computing descriptive statistics and normality checks. The calculator computes specific probabilities (CDF, tail areas, inverse CDF) and shows shaded probability regions.
11. Why use sample standard deviation (n-1) instead of population (n)?
Dividing by n-1 (Bessel's correction) gives an unbiased estimate of the population variance from a sample. Dividing by n systematically underestimates the true spread. For directly entered parameters, your exact σ value is used unchanged.
12. What is the Central Limit Theorem and why does it make the bell curve so important?
The CLT states that the distribution of the sample mean of any independent, identically distributed random variables converges to a normal distribution as sample size increases, regardless of the original distribution. This is why the bell curve appears so frequently across science, engineering, finance and social research.
13. How do I shade the area under a normal curve?
In the generator, open Shade area & find probability, choose left tail, right tail, between or outside, and enter the bound(s) as x values or z-scores. The tool shades that region on the bell curve and shows the probability as a decimal and a percentage, together with the z-scores and the complementary area.
14. How do I find the value at a given percentile?
Enter the percentile (for example 95) in the Percentile box. The tool computes x = μ + zσ using the inverse normal distribution and marks that value on the graph. For the standard normal, the 95th percentile is z = 1.645.
15. Can I share or save the exact graph I made?
Yes. Click Copy share link: the link stores the mean, standard deviation, shading, colours and labels, so anyone who opens it sees the same graph. Save PNG exports a high-resolution image with a solid background that is ready for reports and slides.