Bell Curve — Core Maths Compendium

Build it up, step by step

Understanding bell curve

Click each step below to reveal it — work through them in order the first time round.

Step 1 · What the normal distribution is

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The normal distribution is a symmetric, bell-shaped probability distribution that describes many real-world continuous variables (e.g. heights, exam scores, measurement errors). It's defined by two parameters: the mean, $\mu$, and the standard deviation, $\sigma$.

Step 2 · Key properties of the bell curve

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The curve is perfectly symmetric about the mean; mean = median = mode, all at the centre. Most values cluster near the mean, with fewer values further away in either direction (the ‘tails’). The total area under the curve equals 1 (representing 100% probability).

Step 3 · The 68-95-99.7 rule

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For any normal distribution: about 68% of values lie within 1 standard deviation of the mean; about 95% lie within 2 standard deviations; about 99.7% lie within 3 standard deviations. This gives a quick way to estimate probabilities without a calculator.

Step 4 · Recognising when to use a normal model

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The exam will often tell you a variable ‘can be modelled by a normal distribution’ — this means you can use the mean, standard deviation and the properties above (or a calculator/table) to find probabilities. Not every data set is normal — very skewed data may not be well modelled this way.

Worked example

The heights of adult women in a population are normally distributed with mean 165 cm and standard deviation 7 cm. Use the 68-95-99.7 rule to estimate the percentage of women with height between 158 cm and 172 cm.

158 cm = 165−7 (1 s.d. below mean), 172 cm = 165+7 (1 s.d. above mean). This range is within 1 standard deviation of the mean either side — about 68% of women have a height in this range.

Test yourself

Past-paper style question

The time taken for a machine to complete a task is normally distributed with mean 40 seconds and standard deviation 3 seconds.

(a) Using the 68-95-99.7 rule, estimate the percentage of tasks completed in between 34 and 46 seconds.
(b) Explain what feature of the bell curve means the mean, median and mode are all equal for this distribution. [4 marks]

Show the answer

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(a) 34 = 40−2(3), 46 = 40+2(3), so this range is within 2 standard deviations of the mean — about 95% of tasks are completed in this time range.

(b) The normal distribution is perfectly symmetric about its mean, so the middle value (median) and the most common value (mode) both coincide exactly with the centre of symmetry, which is the mean.

Practice

Bell Curve worksheet

Five short questions on bell curve. Work through them, then reveal the mark scheme to check.

State two key properties of a normal distribution's bell-shaped curve.
A normal distribution has mean 50 and standard deviation 4. Using the 68-95-99.7 rule, estimate the percentage of values between 46 and 54.
Using the same distribution as Q2, estimate the percentage of values between 42 and 58.
Explain why the total area under a normal distribution curve must equal 1.
Give one example of a real-world variable that might reasonably be modelled by a normal distribution, and one that would NOT be well modelled this way, with a reason.

Mark scheme

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Any two of: symmetric about the mean; bell-shaped; mean=median=mode; total area under the curve = 1; most values close to the mean, fewer in the tails.
46 = 50−4, 54 = 50+4, within 1 s.d. — about 68%.
42 = 50−2(4), 58 = 50+2(4), within 2 s.d. — about 95%.
The area under the curve represents total probability, and since one of all possible outcomes must occur, the total probability (area) must equal 1 (100%).
Example that fits well: adult heights (clusters around an average, roughly symmetric). Example that doesn't fit well: household income (typically strongly skewed, with a long tail of very high earners, not symmetric).

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