The Normal Distribution
The normal distribution is a symmetric, bell-shaped probability distribution in which data cluster around the mean, with the spread controlled by the standard deviation. In GCSE Statistics it is used to estimate the proportion of data lying within 1, 2 or 3 standard deviations of the mean, using the approximate percentages 68%, 95% and 99.7%.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check that the data is roughly symmetric and bell-shaped, with most values clustered near the mean.
- Identify the mean and standard deviation of the distribution.
- Use the rule that about 68% of data lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
- Work out which region of the curve the question is asking about, for example above a value, below a value, or between two values.
- Use symmetry to split percentages evenly either side of the mean when a question asks for one tail only.
- Convert the resulting proportion to a number of items by multiplying by the total sample or population size, rounding sensibly.
Worked example
The heights of 200 adult males are normally distributed with a mean of 175 cm and a standard deviation of 6 cm. Estimate the number of men with a height between 169 cm and 181 cm.
- Find how many standard deviations 169 and 181 are from the mean: 169 = 175 - 6 (1 sd below), 181 = 175 + 6 (1 sd above).
- Recognise that 169 cm to 181 cm is the range within 1 standard deviation of the mean.
- Use the rule that approximately 68% of data lie within 1 standard deviation of the mean.
- Calculate 68% of 200: 0.68 x 200 = 136.
- Final answer: approximately 136 men have a height between 169 cm and 181 cm.
Practice questions
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Q1A normal distribution has mean 50 and standard deviation 5. What range of values contains approximately 68% of the data?Show answer
Answer: 45 to 55 (mean plus or minus 1 standard deviation).
Q2A normal distribution has mean 100 and standard deviation 15. What range contains approximately 95% of the data?Show answer
Answer: 70 to 130 (mean plus or minus 2 standard deviations).
Q3The masses of 500 apples are normally distributed with mean 120 g and standard deviation 10 g. Estimate the number of apples with a mass between 110 g and 130 g.Show answer
Answer: 340 apples (68% of 500).
Q4A normally distributed data set has mean 40 and standard deviation 3. Estimate the percentage of values above 43.Show answer
Answer: 16% (50% - 34%, since 43 is 1 sd above the mean and 34% lie between the mean and 43).
Q5Test scores are normally distributed with mean 65 and standard deviation 8. Estimate the percentage of students scoring between 65 and 81.Show answer
Answer: 47.5% (half of the 95% within 2 sd, since 81 is 2 sd above the mean).
Q6IQ scores are normally distributed with mean 100 and standard deviation 15. Out of 1000 people, estimate how many have an IQ above 130.Show answer
Answer: 25 people (2.5%, since 130 is 2 sd above the mean and 2.5% lie beyond +2 sd).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
The lengths of 300 screws produced by a machine are normally distributed with a mean of 50 mm and a standard deviation of 2 mm. Estimate the number of screws with a length between 48 mm and 52 mm.
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A set of exam marks is normally distributed with mean 55 and standard deviation 10. Estimate the percentage of students who scored above 75.
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The diameters of 800 ball bearings produced by a factory are normally distributed with a mean of 12 mm and a standard deviation of 0.5 mm. Ball bearings with a diameter of less than 11 mm or more than 13 mm are rejected. (a) Show that 11 mm and 13 mm are each 2 standard deviations from the mean. (b) Estimate the number of ball bearings that are rejected.
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Free printable worksheet
Want more practice on paper? Download the the normal distribution worksheet pack - 14 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 31 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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