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Your exam score is 75. The class average is 60. Is that good? It depends on how spread out the marks are. If everyone scored between 55 and 65, a 75 is outstanding. If marks ranged from 20 to 100, it is merely above average. The bell curve gives you a way to answer this question with a single number, and it works the same for marks, heights, blood pressure, delivery times and machine tolerances.
What the bell shape means
Imagine measuring the height of 10,000 adults and stacking the results into bars. Most people are close to the average. Very tall and very short people are rare, and the bars get lower the further you go from the middle. Draw a smooth line over the tops of those bars and you get the famous bell.
The bell appears whenever a result is the sum of many small, independent influences, such as genes, diet and sleep for height, or tiny random errors in a measurement. This is why it shows up everywhere, and why statisticians treat it as the default model for many everyday measurements. It is not universal, though. Incomes, website visits and waiting times are usually skewed, and we will come to that.
The curve is highest at the mean and falls away equally on both sides. The values are the density of the standard normal curve multiplied by 100 so they fit on the axis. At ±3 standard deviations the curve is almost flat on the floor.
The two numbers that describe it
Only two values are needed to draw a normal curve. The mean (written μ) says where the centre is. The standard deviation (written σ) says how wide the bell is. A small σ gives a tall narrow bell. A large σ gives a low wide one. Same shape, different scale.
- Values cluster tightly
- A big gap from the mean is unusual
- Typical of precise machines
- Values are scattered
- A big gap from the mean is common
- Typical of mixed groups
Two classes can have the same average mark of 60 and very different stories. The standard deviation is what tells them apart.
The 68-95-99.7 rule
This rule, also called the empirical rule, is the part worth memorising. Walk out from the mean in steps of one standard deviation and count how much of the data you have captured.
68.27% lies within one σ. Another 27.18% lies between one and two σ (so 95.45% within two). A further 4.28% lies between two and three σ (99.73% within three). Only 0.27% is beyond three σ.
- Within 1σ of the mean68
- Between 1σ and 2σ27
- Further than 2σ5
The exact figures are 68.3, 27.2 and 4.6, rounded to whole people.
Many process checks, from factory quality control to blood-test reference ranges, are built on exactly these bands.
Z-scores: how unusual is this value?
A raw number such as 75 means little without context. A z-score turns it into a universal unit: how many standard deviations the value is above or below the mean. After that you can compare things that have nothing in common, like a maths mark and a running time.
- 1Subtract the mean75 − 60 = 15 marks above average.
- 2Divide by the standard deviation15 ÷ 10 = 1.5. The score is one and a half spreads above the centre.
- 3Look up the percentileA z of 1.5 has about 93.3% of the curve below it.
- 4Say it in wordsThe score is better than about 93 of every 100 students, or in the top 7%.
| z-score | Percentile | Share above | Roughly 1 in… |
|---|---|---|---|
| 0 | 50.0% | 50.0% | 2 |
| 0.5 | 69.1% | 30.9% | 3 |
| 1.0 | 84.1% | 15.9% | 6 |
| 1.5 | 93.3% | 6.7% | 15 |
| 2.0 | 97.7% | 2.3% | 44 |
| 2.5 | 99.4% | 0.6% | 161 |
| 3.0 | 99.9% | 0.13% | 741 |
The ringed row is z = 2: about 97.7% of values are lower, so only about 2.3% (1 in 44) are higher. Shading is just to show that rows get rarer as z grows. The last column is for one side only.
Anything inside −1 to +1 is ordinary. Beyond ±2 is unusual enough that you would notice it. These are common rules of thumb, not strict laws.
Worked examples
1. An exam
Marks have a mean of 60 and a standard deviation of 10. A score of 80 is two standard deviations above the mean, so z = 2 and about 2.3% of students did better. A score of 45 has z = −1.5, so it is better than only about 6.7% of the class.
2. IQ scores
IQ tests are designed to give a mean of 100 and a standard deviation of 15. A score of 130 is z = 2, so roughly 2.3% of people score higher. About 95% of people fall between 70 and 130, which is exactly the two-σ band. This is a design choice of the test, not a law of nature.
3. A delivery promise
A courier’s average delivery time is 40 minutes with a standard deviation of 8 minutes. If you promise 56 minutes (two σ above the mean), about 97.7% of orders arrive in time and 2.3% are late. Promise 64 minutes (three σ) and only about 0.13% are late. Each extra σ of cushion costs time but cuts late deliveries sharply.
Promising the average time means half of the orders are late. Adding a little cushion gives enormous gains at first, then smaller ones.
How rare is “rare”?
The further out you go, the faster the odds collapse. People talk about “five sigma” events in physics and “six sigma” in manufacturing. Here is what those numbers mean for the chance of landing beyond that many standard deviations from the mean on either side.
- 1σAbout 1 in 331.7% of values. Hardly rare at all.
- 2σAbout 1 in 224.6% of values. You would see this in a class of thirty.
- 3σAbout 1 in 3700.27% of values. Rare enough to check for a cause.
- 4σAbout 1 in 15,8000.0063% of values. A real surprise.
- 5σAbout 1 in 1.7 millionThe usual bar in particle physics for claiming a discovery.
- 6σAbout 1 in 500 millionThe pure normal-curve figure. Quality programmes quote 3.4 defects per million because they allow the mean to drift by 1.5σ.
Real-world data often has heavier tails than the bell predicts, so extreme events can happen more often than these numbers suggest.
When the bell curve does not fit
The normal curve is a model. It is excellent for many natural measurements and poor for others. If the data has a hard floor at zero and a long tail on the right, such as salaries, house prices, city sizes or the time before a customer calls support, the mean is dragged upward by a few giants and the 68-95-99.7 rule fails.
- Check the shape first. A quick histogram will show if the data is symmetric.
- Be careful with small samples. Ten values rarely show a clean bell, and the estimated standard deviation is itself uncertain.
- Beware of outliers. One mistaken entry can inflate the standard deviation and hide real surprises. The mean, median and mode guide shows how a single value can change everything.
- Do not confuse “normal” with “healthy” or “good”. In statistics, normal only describes the shape.
A short checklist
- Find the mean and the standard deviation of your data.
- Compute z = (value − mean) ÷ σ for any value you care about.
- Read the percentile from a table or the calculator above.
- Check that the data is roughly symmetric before trusting the answer.
- Report it in plain words: “better than 93%”, “1 in 22”, “within the usual range”.
Rare events sound alarming and ordinary ones sound dull, but the z-score gives both the same fair measure. Pair it with the odds ideas in can two UUIDs ever be the same, and you have a solid feel for how likely, or unlikely, things really are.
Questions people ask
What is the 68-95-99.7 rule?
In a normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
What is a z-score?
A z-score is the number of standard deviations a value is from the mean: (value − mean) ÷ standard deviation. Positive means above the mean, negative means below.
What does a z-score of 2 mean?
The value is two standard deviations above the mean. About 97.7% of values are lower and about 2.3% are higher.
How do I turn a z-score into a percentile?
Use a standard normal table or a calculator. For example, z = 1 is about the 84th percentile and z = 1.5 is about the 93rd.
Is all data normally distributed?
No. Heights and measurement errors are close to normal, but incomes, prices and waiting times are usually skewed with long right tails.
What does six sigma mean?
In pure normal-curve terms it means the nearest limit is six standard deviations away. Six Sigma programmes quote 3.4 defects per million because they allow for a 1.5σ drift in the process mean.