Home / A-Level Maths 9709 / Representation of data / Variance and standard deviation, including coded data
Cambridge International AS & A Level · Topic 5.1
A-Level Maths 9709 / Topic 5.1 / AS & A Level

The typical distance from the mean.

Standard deviation measures how far values typically sit from the mean. Paper 5 gives it to you raw, as totals \(\sum x\) and \(\sum x^2\), as coded totals, or split across two data sets, and expects the same two formulas every time.

Topic 5.1All representation of data lessons →
The key idea

\(\text{Variance} = \dfrac{\sum (x - \bar{x})^2}{n} = \dfrac{\sum x^2}{n} - \bar{x}^2\) and standard deviation \(= \sqrt{\text{variance}}\). Coding by subtracting \(a\) shifts the mean by \(a\) but leaves the standard deviation unchanged.

02468101214 mean = 8 x = 4: −4x = 7: −1x = 8: 0 x = 10: +2x = 11: +3
Fig. 1: deviations from the mean for 4, 7, 8, 10, 11. Squared and averaged: (16 + 1 + 0 + 4 + 9) ÷ 5 = 6, so the standard deviation is √6 ≈ 2.45
Section 01

Squared deviations.

Each value's deviation is \(x - \bar{x}\). The deviations always add up to zero, so square them first, average the squares (that is the variance) and then take the square root to get back to the original units. The second form of the formula, \(\dfrac{\sum x^2}{n} - \bar{x}^2\), is quicker and is the one Paper 5 totals are built for.

Worked example From raw data, both ways
Find the standard deviation of 4, 7, 8, 10, 11.
  1. \(\bar{x} = \dfrac{40}{5} = 8\).
  2. Definition: deviations \(-4, -1, 0, 2, 3\); squares sum to \(16 + 1 + 0 + 4 + 9 = 30\); variance \(= 30 \div 5 = 6\).
  3. Quick formula: \(\sum x^2 = 16 + 49 + 64 + 100 + 121 = 350\); variance \(= \dfrac{350}{5} - 8^2 = 70 - 64 = 6\).
  4. Standard deviation \(= \sqrt{6}\).
Answer: \(\sqrt{6} = 2.45\) (3 s.f.)
Worked example From totals
For 20 values, \(\sum x = 360\) and \(\sum x^2 = 6740\). Find the mean and standard deviation.
  1. \(\bar{x} = \dfrac{360}{20} = 18\).
  2. Variance \(= \dfrac{6740}{20} - 18^2 = 337 - 324 = 13\).
Answer: Mean 18, standard deviation \(\sqrt{13} = 3.61\) (3 s.f.)
Examiner trap

Writing \(\dfrac{\sum x^2}{n} - \bar{x}\) or \(\dfrac{(\sum x)^2}{n}\). It is the MEAN that is squared and subtracted: \(\dfrac{\sum x^2}{n} - \bar{x}^2\).

Section 02

Frequency tables and grouped data.

With frequencies, every sum gets an \(f\): \(\bar{x} = \dfrac{\sum fx}{\sum f}\) and variance \(= \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\). For grouped data use the class midpoints, and remember the answers are estimates.

Worked example Discrete frequency table
Values \(x = 1, 2, 3, 4, 5\) have frequencies 2, 5, 8, 4, 1. Find the mean and standard deviation.
  1. \(\sum f = 20\) and \(\sum fx = 2 + 10 + 24 + 16 + 5 = 57\), so \(\bar{x} = 2.85\).
  2. \(\sum fx^2 = 2 + 20 + 72 + 64 + 25 = 183\).
  3. Variance \(= \dfrac{183}{20} - 2.85^2 = 9.15 - 8.1225 = 1.0275\).
Answer: Mean 2.85, standard deviation 1.01 (3 s.f.)
Worked example Grouped data
Masses \(m\) kg: \(0 \le m < 10\): 4, \(10 \le m < 20\): 10, \(20 \le m < 40\): 6. Estimate the mean and standard deviation.
  1. Midpoints 5, 15, 30. \(\sum fm = 20 + 150 + 180 = 350\), so \(\bar{m} \approx 350 \div 20 = 17.5\).
  2. \(\sum fm^2 = 4(25) + 10(225) + 6(900) = 100 + 2250 + 5400 = 7750\).
  3. Variance \(\approx \dfrac{7750}{20} - 17.5^2 = 387.5 - 306.25 = 81.25\).
Answer: Mean about 17.5 kg, standard deviation about 9.01 kg
Examiner trap

Using \((\sum fx)^2\) or \(\sum (fx)^2\) in place of \(\sum fx^2\). Square the value \(x\) first, then multiply by its frequency.

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

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Find the standard deviation of 2, 4, 6.

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5 is added to every value in a data set. What happens to the standard deviation?

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For 10 values, \(\sum x = 50\) and \(\sum x^2 = 290\). Find the variance.

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Which is a correct formula for the variance?

Section 03

Coded totals.

To keep numbers small, data are often given as \(\sum (x - a)\) and \(\sum (x - a)^2\). Subtracting \(a\) slides every value along the number line: the mean moves by \(a\) but the spread does not change at all.

Worked example Mean and standard deviation from coded totals
For 12 values, \(\sum (x - 50) = -36\) and \(\sum (x - 50)^2 = 420\). Find the mean and standard deviation of \(x\).
  1. Mean of \(x - 50\) is \(-36 \div 12 = -3\), so \(\bar{x} = 50 - 3 = 47\).
  2. Variance \(= \dfrac{420}{12} - (-3)^2 = 35 - 9 = 26\), unchanged by the coding.
Answer: Mean 47, standard deviation \(\sqrt{26} = 5.10\) (3 s.f.)
Worked example Recovering \(\sum x\) and \(\sum x^2\)
For the same data, find \(\sum x\) and \(\sum x^2\).
  1. \(\sum x = \sum (x - 50) + 12 \times 50 = -36 + 600 = 564\).
  2. \(\sum (x - 50)^2 = \sum x^2 - 100 \sum x + 12 \times 2500\), so \(420 = \sum x^2 - 56\,400 + 30\,000\).
  3. \(\sum x^2 = 420 + 26\,400 = 26\,820\).
  4. Check: \(\dfrac{26\,820}{12} - 47^2 = 2235 - 2209 = 26\). Correct.
Answer: \(\sum x = 564\), \(\sum x^2 = 26\,820\)
Examiner trap

Thinking \(\sum x^2 = \sum (x - a)^2 + na^2\). The cross term \(-2a\sum x\) matters; expand the bracket properly or work via the variance.

Section 04

Two data sets together.

To combine two data sets, never average the means or the standard deviations. Turn each set back into \(n\), \(\sum x\) and \(\sum x^2\) (using \(\sum x^2 = n(\sigma^2 + \bar{x}^2)\)), add the totals, and use the formulas once on the combined totals.

Worked example Combined mean and standard deviation
Set A has 10 values with mean 15 and standard deviation 2. Set B has 15 values with mean 21 and standard deviation 3. Find the mean and standard deviation of all 25 values.
  1. A: \(\sum x = 150\), \(\sum x^2 = 10(2^2 + 15^2) = 2290\).
  2. B: \(\sum x = 315\), \(\sum x^2 = 15(3^2 + 21^2) = 6750\).
  3. Combined: \(n = 25\), \(\sum x = 465\), \(\sum x^2 = 9040\). Mean \(= 465 \div 25 = 18.6\).
  4. Variance \(= \dfrac{9040}{25} - 18.6^2 = 361.6 - 345.96 = 15.64\).
Answer: Mean 18.6, standard deviation \(\sqrt{15.64} = 3.95\) (3 s.f.)
Your turn
Ten values have mean 6 and standard deviation 2. Find \(\sum x\) and \(\sum x^2\).
Examiner trap

The combined standard deviation (3.95) is bigger than both separate ones (2 and 3), because the two means are far apart. Averaging the standard deviations misses this completely.

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For 25 values, \(\sum (x - 20) = 75\) and \(\sum (x - 20)^2 = 1350\). Find the standard deviation of \(x\).

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Set A: 8 values, mean 10, standard deviation 2. Set B: 12 values, mean 15, standard deviation 4. Find the standard deviation of the 20 values combined.

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Ten values have mean 6 and standard deviation 2. Find \(\sum x^2\).

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Temperatures have mean 20 °C and standard deviation 5 °C. They are converted to °F using \(F = 1.8C + 32\). Find the mean and standard deviation in °F.

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Six original Cambridge-style questions
Standard deviation from raw data, totals, frequency tables and coded totals, recovering \(\sum x^2\), and combining two data sets.