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Standard deviation calculator (GCSE Statistics)

5 values

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Standard deviation

2.83

The values sit on average about 2.83 from their mean of 8.

Mean
8
Variance
8
Σf
5
Working table
xx − x̄(x − x̄)²
4−416
6−24
800
1024
12416
Total40

Working

  1. Mean x̄ = Σx ÷ n = 40 ÷ 5 = 8.
  2. Take the mean from each value and square it: (4 − 8)² = 16; (6 − 8)² = 4; (8 − 8)² = 0; (10 − 8)² = 4; (12 − 8)² = 16.
  3. Add them: Σ(x − x̄)² = 40. Divide by n: 40 ÷ 5 = 8 (the variance).
  4. Standard deviation = √8 = 2.8284.
  5. Check with the other formula: Σx² ÷ n − x̄² = 360 ÷ 5 − 8² = 8, and its square root is 2.8284.
  6. Outliers by the 3 standard deviation rule: below 8 − 3 × 2.8284 = −0.4853 or above 16.4853. No value is outside.

Your result

2.83. The values sit on average about 2.83 from their mean of 8. Roughly 95% of normally distributed data lies within 2 standard deviations of the mean: 2.34 to 13.66.

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GCSE Statistics divides by n: for 4, 6, 8, 10 and 12 the mean is 8, the squared distances add to 40, the variance is 40 ÷ 5 = 8 and the standard deviation is √8 = 2.83.

Questions people ask

What is the formula for standard deviation in GCSE Statistics?

Standard deviation = √(Σ(x − x̄)² ÷ n), or √(Σx² ÷ n − x̄²). For a frequency table multiply each term by its frequency and divide by Σf. Both forms give the same answer.

Why is my calculator's answer different?

Many calculators show two standard deviations: σx (or σn), which divides by n, and sx (or s), which divides by n − 1. GCSE Statistics uses σx. The sx answer is a little bigger, especially for small samples.

How do I find the standard deviation of grouped data?

Use the midpoint of each class as x, then work with frequencies as normal: Σfx, Σfx² and Σf. The result is an estimate because the midpoints stand in for the real values.

What does a bigger standard deviation mean?

The values are more spread out from the mean, so the data is less consistent. Compare standard deviations only for data measured in the same units.

Keep going

How we work this out

What it does

Find the standard deviation of a list or a frequency table the way GCSE Statistics does it, dividing by n, with every step.

Up to 60 values, or up to 60 values with frequencies. For grouped data, enter each class's midpoint and frequency.

Method

  1. Mean x̄ = Σfx ÷ Σf.
  2. Standard deviation = √(Σf(x − x̄)² ÷ Σf), or the equivalent √(Σfx² ÷ Σf − x̄²).
  3. Both GCSE Statistics specifications divide by n (Σf), not n − 1.
  4. Values more than 3 standard deviations from the mean count as outliers.