Standard Deviation
Standard deviation is a measure of spread that shows, on average, how far each data value lies from the mean; a small standard deviation means the data are clustered closely around the mean, while a large one means the data are more spread out. In GCSE Statistics it is calculated using the formula sqrt((sum of x^2)/n - mean^2) and is used to compare the consistency of two data sets.
Before you start
Make sure you're comfortable with these topics first:
Method
- Find the mean of the data set (for raw data), or the estimated mean using midpoints (for grouped data).
- Square every data value, or every midpoint for grouped data.
- Find the mean of the squared values: sum of x^2 divided by n, or sum of f x^2 divided by sum of f for a frequency table.
- Subtract the square of the mean from this value: variance = (mean of x^2) - (mean)^2.
- Take the square root of the variance to get the standard deviation.
- Compare standard deviations between two data sets: the smaller value shows more consistent, less spread out data.
Worked example
Find the standard deviation of the data set 4, 6, 7, 9, 9, giving your answer correct to 2 decimal places.
- Find the mean: (4+6+7+9+9)/5 = 35/5 = 7.
- Square each value: 16, 36, 49, 81, 81.
- Find the mean of the squares: (16+36+49+81+81)/5 = 263/5 = 52.6.
- Subtract the square of the mean: variance = 52.6 - 7^2 = 52.6 - 49 = 3.6.
- Take the square root: standard deviation = sqrt(3.6) = 1.897... which rounds to 1.90.
- Final answer: standard deviation = 1.90 (2 dp).
Practice questions
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Q1Find the standard deviation of the data set 1, 2, 3, 4, 5, giving your answer to 2 decimal places.Show answer
Answer: 1.41 (mean = 3, variance = 11 - 9 = 2, sd = sqrt(2)).
Q2A data set has a standard deviation of 0. What must be true about every value in the data set?Show answer
Answer: Every value is identical (equal to the mean).
Q3Find the standard deviation of 10, 12, 14, 16, 18, giving your answer to 2 decimal places.Show answer
Answer: 2.83 (mean = 14, variance = 204 - 196 = 8, sd = sqrt(8)).
Q4A shop records the number of items bought by 10 customers: 1 item (2 customers), 2 items (3 customers), 3 items (5 customers). Find the standard deviation, to 2 decimal places.Show answer
Answer: 0.78 (mean = 2.3, variance = 5.9 - 5.29 = 0.61, sd = sqrt(0.61)).
Q5Class A's test scores have mean 65 and standard deviation 4. Class B's test scores have mean 65 and standard deviation 9. Which class's scores were more consistent, and why?Show answer
Answer: Class A, because its smaller standard deviation shows the scores are more closely clustered around the mean.
Q6A gardener groups 10 plants by height: 10-20 cm (midpoint 15, 4 plants), 20-30 cm (midpoint 25, 6 plants). Estimate the standard deviation of the plant heights, to 2 decimal places.Show answer
Answer: 4.90 cm (mean = 21, variance = 465 - 441 = 24, sd = sqrt(24)).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
The heights, in cm, of 5 seedlings are: 8, 10, 12, 14, 16. Calculate the standard deviation of the heights, giving your answer to 2 decimal places.
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A sample of 8 values has a sum of x^2 equal to 340, and the mean of the sample is 6. (a) Find the variance of the sample. (b) Find the standard deviation, giving your answer to 2 decimal places.
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The table shows the number of hours 20 students spent revising in one week: 0-4 hours (midpoint 2, frequency 5), 4-8 hours (midpoint 6, frequency 9), 8-12 hours (midpoint 10, frequency 6). Calculate an estimate of the standard deviation of the hours spent revising, giving your answer to 2 decimal places.
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Free printable worksheet
Want more practice on paper? Download the standard deviation worksheet pack - 16 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 1 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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