LearnYMA01WST01 · Statistics 1Representation and summary of dataMean and standard deviation

Representation and summary of data · Subtopic 3 of 4

Mean and Standard Deviation

The mean is the balance point of the data; the standard deviation is a typical distance from it. One formula, "mean of the squares minus the square of the mean", gives the variance from almost any form of data.

Cambridge 9709 Paper 5Edexcel YMA01 WST01Edexcel 9MA0 Paper 3AQA 7357 Paper 3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Calculate the mean and standard deviation of raw data.
Use the totals $\Sigma x$ and $\Sigma x^2$ , including working backwards to find them.
Estimate the mean and standard deviation from a frequency table, using class midpoints for grouped data.
Choose a suitable average and measure of spread.

Pause and think

The values $4,5,6$ and $0,5,10$ both have mean $5$ . How could you measure, with one number, how spread out each set is?

Section 2 of 14

Core idea

Start hereSame average, different story

Two archers both average $6$ points. One always scores 5, 6 or 7; the other ranges from 1 to 11. Same mean, completely different reliability. A single number for spread, the standard deviation, captures the difference.

The big idea

Mean $\bar x=\frac{\sum x}n$ . Variance $\sigma^2=\frac{\sum(x-\bar x)^2}n=\frac{\sum x^2}n-\bar x^2$ ("mean of squares minus square of mean"). Standard deviation $\sigma=\sqrt{\text{variance}}$ .

1. Why squared deviations?

Two data sets with the same mean (6) but different spread

Both sets have mean

6

. Set A huddles round it; set B is scattered. The standard deviation puts a number on that difference.

Deviations always sum to zero

$\sum(x-\bar x)=\sum x-n\bar x$
Split the sum.
$=\sum x-n\cdot\frac{\sum x}n=0$
Definition of $\bar x$ .

Positive and negative deviations cancel, so we square them before averaging.

The computational formula

$\sigma^2=\frac1n\sum(x^2-2\bar xx+\bar x^2)$
Expand the square.
$=\frac{\sum x^2}n-2\bar x\frac{\sum x}n+\bar x^2$
Split; $\sum\bar x^2=n\bar x^2$ .
$=\frac{\sum x^2}n-2\bar x^2+\bar x^2=\frac{\sum x^2}n-\bar x^2$
$\frac{\sum x}n=\bar x$ .

Worked example $2,4,4,4,5,5,7,9$

Find the mean and standard deviation.

Goal: $\bar x$ , $\sigma$ .
Strategy: Computational formula.

$\bar x=\tfrac{40}8=5$
Mean.
$\sum x^2=4+16+16+16+25+25+49+81=232$
Squares.
$\sigma^2=\tfrac{232}8-25=4\Rightarrow\sigma=2$
Formula.

AnswerMean

5

, SD

2

✓ CheckDeviations

-3,-1,-1,-1,0,0,2,4

; squares sum

9+1+1+1+0+0+4+16=32

;

\frac{32}8=4

✓.

2. Grouped data and combining groups

\bar x=\frac{\sum fx}{\sum f},\qquad\sigma^2=\frac{\sum fx^2}{\sum f}-\bar x^2

Worked exampleCombining two groups

A: $n=10$ , $\sum x=50$ , $\sum x^2=290$ . B: $n=15$ , $\sum x=105$ , $\sum x^2=800$ . Find the combined mean and SD.

Goal: Combined stats.
Strategy: Add $n$ , $\sum x$ , $\sum x^2$ (never average the SDs).

$n=25,\ \sum x=155,\ \sum x^2=1090$
Add.
$\bar x=6.2,\ \sigma^2=\tfrac{1090}{25}-6.2^2=43.6-38.44=5.16$
Formulas.

AnswerMean

6.2

, SD

2.27

✓ CheckCombined mean between

5

and

7

✓.

Common misconceptionForgetting to square-root

\frac{\sum x^2}n-\bar x^2

is the variance; the SD is its square root.

Pause and think

Which is affected more by one extreme value, the mean or the median?

Summary

Variance = mean of squares − square of mean.
Grouped: use $f$ and midpoints.
Combine using totals.

Variance: Mean squared deviation.
Standard deviation: Square root of variance.

HistoryPearson again. Karl Pearson introduced the term "standard deviation" and the symbol

\sigma

in 1893–94. Before that, people used the "probable error" or the "mean error". The word "variance" came later, from Ronald Fisher in 1918.

Fun factMore legs than average. Because a few people have fewer than two legs and nobody has more, the mean number of legs is slightly below

2

. So almost everyone has more legs than average! The median (

2

) is the better average here. It is a classic example of a skewed distribution pulling the mean.

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