Key takeaways
- The normal distribution is a symmetric bell curve set by its mean μ and standard deviation σ.
- f(x) is the height of the curve at x, not a probability.
- About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.
- σ must be greater than 0. It is the standard deviation, not the variance.
In this guide
What is the normal distribution?
Picture measuring the height of 1,000 adults and drawing a bar chart. Most people are close to the average height, a few are very short and a few are very tall. The chart looks like a smooth hill, highest in the middle and falling evenly on both sides. That shape is the normal distribution, also called the bell curve or Gaussian distribution.
Just two numbers describe every bell curve: where its centre is, and how wide it is.
Why does the normal distribution matter?
Many things in nature and in factories come out roughly normal, because they are the result of lots of small random effects added together. Once you know the mean and standard deviation, the bell curve tells you what is common and what is rare.
That is why it sits behind exam grading, quality control, medical growth charts and opinion polls.
About 95% of values in a normal distribution lie within two standard deviations of the mean. A value outside that band is unusual, and worth a second look.
The formula
| Symbol | What it means | Example |
|---|---|---|
| f(x) | Height of the curve (density) at x | 0.0242 |
| x | The value you are interested in | 180 cm |
| μ (mu) | The mean, the centre of the curve | 170 cm |
| σ (sigma) | The standard deviation, must be > 0 | 10 cm |
| e | A fixed number, about 2.71828 | 2.71828 |
| π (pi) | A fixed number, about 3.14159 | 3.14159 |
The part (x − μ) ÷ σ is the z-score: how many standard deviations x is from the mean. The curve is highest when x equals μ, and it drops quickly as x moves away.
How to calculate the density step by step
- 1Find the distance from the mean. Work out
x − μ. - 2Divide by σ. This gives the z-score.
- 3Square it and multiply by −½. This is the power of e.
- 4Raise e to that power. Use the eˣ button on a calculator.
- 5Multiply by 1 ÷ (σ√(2π)). The result is f(x).
Try it with your own mean, standard deviation and value in the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using the variance instead of σ. σ is the square root of the variance. A variance of 100 means σ = 10. The variance calculator shows the link.
- Entering σ = 0 or a negative σ. The spread must be positive, or the formula makes no sense.
- Forgetting the minus sign in the power. Without it the curve would shoot upward instead of falling away.
- Assuming all data is normal. Incomes and waiting times are often lopsided, not bell shaped.
Tips and tricks
| Distance from the mean | Share of values inside |
|---|---|
| Within 1σ | about 68% |
| Within 2σ | about 95% |
| Within 3σ | about 99.7% |
- The peak height is always
1 ÷ (σ√(2π)), so double σ and the peak drops to half. - The curve is symmetric: f(μ + 5) is exactly the same as f(μ − 5).
Where you'll use it in real life
- Education: grading on a curve and comparing scores from different exams.
- Manufacturing: checking that bottle volumes or bolt sizes stay close to target.
- Health: reading growth charts for babies and children.
- Surveys and science: building a confidence interval around an average.
Quick summary
The normal distribution is a bell curve set by its mean and standard deviation. The formula gives the height of the curve at a value x, and probabilities come from areas under the curve. Use the calculator above to get f(x) with every step shown.
Worked example
Adult heights in a town follow a normal distribution with a mean of 170 cm and a standard deviation of 10 cm. How tall is the bell curve at a height of 180 cm?
- 01f(180) = (1/10√(2π)) × e^(-½((180-170)/10)²)
- 02= (0.04) × e^(-0.5)
- 03= 0.04 × 0.607
- 04= 0.024
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the standard normal distribution?
It is the normal distribution with mean 0 and standard deviation 1. Its peak height at x = 0 is 1 ÷ √(2π) ≈ 0.399.
Why is the normal distribution called a bell curve?
Because its graph is high and round in the middle and falls away evenly on both sides, like the outline of a bell.
How do I find a probability from the normal distribution?
Convert your value to a z-score with z = (x − μ) ÷ σ, then look up the area under the curve in a z-table. The density f(x) alone does not give a probability.
Can the normal density be greater than 1?
Yes. The peak height is 1 ÷ (σ√(2π)), so when σ is small, for example 0.1, the peak is about 3.99. Only areas under the curve must stay between 0 and 1.
