Measures of Spread (AA HL)
Two data sets can share the same mean and still look completely different once you plot them - measures of spread are what capture that difference. This page covers variance and standard deviation, including how to compute them from a frequency table, and what happens to them when you shift or scale a whole data set. It's part of the broader Descriptive Statistics topic.
11 questions on this sub-topic.
Variance and standard deviation
Covered under IB syllabus reference SL4.3: measures of dispersion (IQR, standard deviation, variance), and the effect of constant changes - adding a constant or scaling - on the mean and standard deviation. The variance formula below is in the formula booklet, though you're still expected to use your GDC's statistics mode to evaluate it in practice.
Variance
\(\sigma^2 = \dfrac{\sum(x-\bar x)^2}{n} = \dfrac{\sum x^2}{n} - \bar x^2\)
The second form is usually quicker by hand: find \(\sum x^2\) and \(\bar x\) first, then subtract.
Standard deviation
\(\sigma = \sqrt{\sigma^2}\)
Adding a constant to every value leaves \(\sigma\) unchanged; multiplying every value by \(k\) scales \(\sigma\) by \(|k|\).
Worked examples
The number of goals scored in 40 matches is shown.
| Goals \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Freq \(f\) | 6 | 14 | 11 | 6 | 3 |
(a) Find the mean number of goals.
(b) Find the standard deviation (3 significant figures).
Worked solution
(a) Mean. \(\sum f = 40,\ \sum fx = 0+14+22+18+12 = 66.\) \(\bar x = \dfrac{66}{40}\) M1
\(= 1.65.\) A1
(b) Standard deviation. \(\sum fx^2 = 160\); \(\sigma^2 = \dfrac{160}{40} - 1.65^2\) M1
\(= 1.2775\) A1
\(\sigma \approx 1.13.\) A1
A data set has mean \(20\) and standard deviation \(4.\) Each value is increased by \(5.\) State the new mean and standard deviation.
Worked solution
Adding a constant shifts every value: the mean increases by 5 to \(25.\) M1
\(25.\) A1
The standard deviation is unchanged: \(4.\) M1
\(4.\) A1
Common mistakes
- Forgetting to subtract \(\bar x^2\). Variance is \(\dfrac{\sum x^2}{n} - \bar x^2\), not just \(\dfrac{\sum x^2}{n}\) - dropping the second term gives a much too large answer.
- Applying a shift and a scale the same way. Adding a constant to every value leaves the standard deviation alone, but multiplying every value by \(k\) scales it by \(|k|\) - these two effects are easy to mix up under pressure.
- Skipping the frequency weighting. With grouped data, both \(\sum fx\) and \(\sum fx^2\) need the frequency \(f\) built in - using raw \(x\) values without their weights silently corrupts the mean and variance.
Ready to practise properly?
11 measures-of-spread questions, marked instantly like the real exam. Not sure how your GDC handles frequency data? Check the parent topic's GDC guidance.
Quick answers
What is the formula for variance and standard deviation?
Variance is \(\sigma^2 = \dfrac{\sum(x-\bar x)^2}{n}\), equivalent to \(\dfrac{\sum x^2}{n} - \bar x^2\). Standard deviation is \(\sigma = \sqrt{\sigma^2}\).
What happens to the standard deviation if a constant is added to every value?
Adding a constant shifts the mean by that amount but leaves the standard deviation unchanged, since the spread of the data hasn't changed. Multiplying every value by a constant \(k\) scales the standard deviation by \(|k|\).