The Normal Distribution
Understand The Normal Distribution for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. The normal distribution is a symmetric, bell-shaped distribution in which the mean, median and mode are equal.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
The normal distribution is a continuous distribution whose graph is a symmetrical bell-shaped curve. This is a Higher-only topic.
It is symmetrical about the mean, so the mean, median and mode are all equal and all lie at the centre. The total area under the curve is 1, representing all possible outcomes.
The key results concern proportions within standard deviations of the mean: approximately 68 per cent of values lie within one standard deviation, 95 per cent within two, and 99.7 per cent within three. These three figures are examined directly and should be learned precisely. Many natural measurements, such as heights, are approximately normally distributed.
Revision notes
The shape
A symmetrical bell-shaped curve centred on the mean.
The mean, median and mode are all equal and lie at the centre. The total area under the curve is 1, representing all possible outcomes.
The key proportions
About 68 per cent of values lie within one standard deviation of the mean.
About 95 per cent lie within two standard deviations. About 99.7 per cent lie within three. These are examined directly and must be learned.
Using the proportions
The symmetry means each half of an interval contains half its proportion.
So about 34 per cent lies between the mean and one standard deviation above it. Values more than two standard deviations from the mean make up about 5 per cent, split between the two tails.
Key points
- The normal distribution is bell-shaped.
- It is symmetrical about the mean.
- Mean, median and mode are equal.
- About 68 per cent lie within one standard deviation.
- About 95 per cent lie within two.
- About 99.7 per cent lie within three.
Worked examples
Example 1
A normal distribution has mean 50 and standard deviation 8. Work out the interval containing about 95 per cent of values. [3 marks]
Working
Example 2
State the proportion of values lying within one standard deviation of the mean. [1 mark]
Working
Example 3
Explain why the mean, median and mode are equal in a normal distribution. [2 marks]
Working
Common mistakes
Confusing the three proportions.
68, 95 and 99.7 per cent for one, two and three standard deviations.
Forgetting the distribution is symmetrical.
Symmetry lets you halve proportions for one-sided intervals.
Using the normal distribution for discrete data.
It is a continuous distribution.
Saying the area under the curve is 100.
It is 1, or 100 per cent.
Exam tips
- Learn 68, 95 and 99.7 per cent precisely.
- Use symmetry to find one-sided proportions.
- Multiply the standard deviation before adding and subtracting.
- Remember this is a Higher-only topic.
Key terms
- Normal distribution
- A symmetrical bell-shaped continuous distribution.
- Symmetrical
- Identical on both sides of the mean.
- Standard deviation
- The measure of spread defining the key intervals.
- Bell-shaped
- The characteristic curve of the normal distribution.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.