Key takeaways
- The normal distribution is a symmetric bell curve set by its mean (μ) and standard deviation (σ).
- About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
- A z-score is z = (x − μ) ÷ σ: how many standard deviations a value is from the mean.
- Positive z means above the mean, negative z means below it.
In this guide
What is a probability distribution?
A probability distribution shows how likely each value is. It tells you where the data usually lands and how often you see unusual values.
The most famous one is the normal distribution, also called the bell curve. Many things follow it roughly: heights of adults, measurement errors, and scores on large exams.
Why does the normal distribution matter?
It lets you turn a single value into a meaningful comparison. A score of 85 is excellent if most people scored between 65 and 75, but average if most scored between 50 and 100.
It is also the base of much of modern statistics: quality control in factories, polls with a margin of error, and medical 'normal ranges' all build on it.
Knowing just two numbers, the mean and the standard deviation, tells you almost everything about a normal distribution.
The formula
The key tool is the z-score. It measures how many standard deviations a value sits from the mean.
| Symbol | What it means | Example |
|---|---|---|
| z | The z-score (standard score) | 1.5 |
| x | The value you are checking | 85 marks |
| μ | The mean of the distribution | 70 marks |
| σ | The standard deviation | 10 marks |
The 68-95-99.7 rule
| Range | Share of values inside |
|---|---|
| μ ± 1σ | about 68% |
| μ ± 2σ | about 95% |
| μ ± 3σ | about 99.7% |
So a value more than 3 standard deviations from the mean is very rare.
How to use z-scores step by step
- 1Find the mean and standard deviation of the distribution.
- 2Subtract the mean from your value:
x − μ. - 3Divide by the standard deviation to get z.
- 4Read the meaning. z = 0 is exactly average, z = 1 is one σ above, z = −2 is two σ below.
- 5Look up the probability in a z-table if you need a percentage.
Check the arithmetic with the z-score calculator. It shows each step.
Comparing two different tests
You got 85 in maths (mean 70, σ 10) and 80 in science (mean 60, σ 20). The maths z-score is 1.5. The science z-score is (80 − 60) ÷ 20 = 1. So your maths result is relatively better, even though the science average was lower.
Common mistakes to avoid
- Dividing by the variance instead of σ. Use the standard deviation, not σ².
- Losing the sign. A z-score of −1.5 means below average. Keep the minus sign.
- Assuming everything is normal. Incomes and waiting times are lopsided and do not follow a bell curve.
- Mixing up 'below' and 'above'. Most z-tables give the area to the left. For 'above', use
1 − table value.
Tips and tricks
- Density at a point: the normal distribution calculator gives the height of the curve f(x) for a value, mean and standard deviation.
- Other distributions: use the binomial probability calculator for yes/no outcomes, like the number of heads in 10 coin flips.
- Estimating a mean from a sample? The confidence interval calculator builds on the same z-values.
Where you'll use it in real life
- Exams: comparing results across different tests or years.
- Manufacturing: checking that bottles are filled close to the target amount.
- Health: growth charts that show which percentile a child is in.
- Finance: describing how much an investment's returns usually vary.
Quick summary
The normal distribution is a bell curve described by its mean and standard deviation. The 68-95-99.7 rule gives quick estimates, and the z-score z = (x − μ) ÷ σ lets you compare any value fairly. Use a z-table for exact percentages, and the free calculators linked above to check the steps.
Frequently asked questions
What is the 68-95-99.7 rule?
In a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
How do I interpret a z-score?
A z-score tells you how many standard deviations a value is from the mean. z = 0 is the mean, z = 1 is one standard deviation above, and z = −2 is two below.
Is the height of the bell curve a probability?
No. The height is the probability density. The probability of landing between two values is the area under the curve between them.
What z-score counts as unusual?
A common rule of thumb is that values with z below −2 or above 2 are unusual, because only about 5% of values lie that far from the mean.
