Management Accounting · Summarising and analysing data

Measures of Dispersion: Variance and Standard Deviation

Updated 11 October 2026 · Fact-checked

Measures of dispersion show how spread out data is around its average. The main ones are range, interquartile range, variance, standard deviation and coefficient of variation. To find standard deviation, work out the mean, square each deviation, average the squares to get variance, then take the square root.

Understand Measures of Dispersion

An average tells you where the middle of a data set is. It does not tell you how reliable that middle is. Two products can both average 100 units of monthly sales. One may sell between 98 and 102 each month. The other may swing between 40 and 160. A manager needs to know this difference, because the second product is much riskier to plan around. Dispersion measures that spread.

The simplest measure is the range: highest value minus lowest value. It is quick but uses only two values, so one extreme figure distorts it. The interquartile range (IQR) fixes this by looking at the middle half of the data. It is the upper quartile (Q3) minus the lower quartile (Q1), so extreme values are ignored.

The variance uses every value. You find how far each value is from the mean, square those gaps so negatives do not cancel positives, and take the average. The squaring means variance is in squared units, such as $ squared, which is hard to interpret.

The standard deviation is the square root of the variance. It returns to the original units, so it can be read directly. A standard deviation of $5 means values typically sit about $5 from the mean. A bigger standard deviation means more spread and more risk.

Standard deviation cannot compare data sets with different means or units on its own. The coefficient of variation (CV) solves this. It divides the standard deviation by the mean, giving relative spread, usually as a percentage. The lower the CV, the more consistent the data relative to its average.

Key formulas to remember

Range: Range = highest value − lowest value; Uses only two values, so it is affected by extreme figures.
Interquartile range: IQR = Q3 − Q1; Measures the spread of the middle 50% of the data. Q1 and Q3 are the lower and upper quartiles.
Variance (population, ungrouped): σ² = Σ(x − x̄)² ÷ n; Use n when the data is treated as the whole population, as is usual in MA questions. Check whether the question says sample.
Variance (shortcut form): σ² = (Σx² ÷ n) − x̄²; Often faster. Mean of the squares minus the square of the mean.
Standard deviation: σ = √variance; In the same units as the data.
Variance for a frequency distribution: σ² = (Σfx² ÷ Σf) − x̄²; Here x̄ = Σfx ÷ Σf. Σf is the total frequency.
Coefficient of variation: CV = (standard deviation ÷ mean) × 100%; Compares relative spread. Lower CV means more consistent data.

How to solve Measures of Dispersion questions

Use this method for any dispersion question. It works for lists of values and for frequency tables.

1Read the question for the exact measure needed: range, IQR, variance, standard deviation or CV. Note whether it asks for a percentage or a rounded figure.
2Check whether the data is a simple list or a frequency table. For a table, you will use f × x and f × x².
3Calculate the mean first: Σx ÷ n, or Σfx ÷ Σf.
4Calculate the mean of the squares: Σx² ÷ n, or Σfx² ÷ Σf.
5Find the variance: mean of squares minus the mean squared.
6Take the square root for the standard deviation. Do not forget this step if the question asks for standard deviation.
7For CV, divide standard deviation by the mean and multiply by 100.
8Sense-check: variance and standard deviation cannot be negative, and the standard deviation should be smaller than the range.

Quickest way: Shortcut variance on a calculator

When to use it: Use for number entry questions with a small data set or frequency table where you must give variance, standard deviation or CV.

Put your calculator in statistics mode and enter the data, adding frequencies if given.
Read off the mean (x̄) and the population standard deviation (σx or σn), not the sample one (sx or σn−1), unless told otherwise.
Square the standard deviation if the question asks for variance.
Divide by the mean for CV and multiply by 100.
If you do it by hand, use the shortcut form: mean of squares minus square of the mean.

Common mistakes in Measures of Dispersion

Giving the variance when the question asks for standard deviation, or the other way round.
Students stop after the variance step in a rush.
Fix: Underline the measure asked for. If it is standard deviation, finish by taking the square root.
Forgetting to square the mean in the shortcut formula.
Students subtract x̄ instead of x̄² from Σx² ÷ n.
Fix: Write the formula as (Σx² ÷ n) − (x̄)² and calculate x̄² as a separate line.
Using the sample standard deviation key on the calculator.
The calculator shows both values and they look similar.
Fix: For MA questions that treat the data as the full set, use the population value (divide by n). Do so unless the question says sample.
Using Σx² when the data has frequencies, ignoring f.
Students square the frequencies or forget to multiply by them.
Fix: Add columns for fx and fx². Square x first, then multiply by f.
Comparing standard deviations of data sets with very different means.
A bigger standard deviation looks like more risk.
Fix: Calculate the CV for each set. Compare relative spread, not absolute spread.
Treating a higher CV as better.
Students link a bigger number with a better result.
Fix: A higher CV means more variability relative to the mean, so less consistency and more risk.

Worked examples

Example 1

A machine fills bags with the following weights in kg: 4, 6, 8, 10, 12. Calculate the range, the variance and the standard deviation, treating this as the whole population. Give the standard deviation to two decimal places.

Show the solution

Range = 12 − 4 = 8.
Mean = (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
Σx² = 16 + 36 + 64 + 100 + 144 = 360. Mean of squares = 360 ÷ 5 = 72.
Variance = 72 − 8² = 72 − 64 = 8.
Standard deviation = √8 = 2.83 kg (to two decimal places).

Answer: Range = 8 kg, variance = 8 kg², standard deviation = 2.83 kg.

Example 2

Product A has monthly sales with a mean of $200 and a standard deviation of $30. Product B has monthly sales with a mean of $500 and a standard deviation of $60. Which product has the more consistent sales?

Show the solution

Product A: CV = 30 ÷ 200 × 100% = 15%.
Product B: CV = 60 ÷ 500 × 100% = 12%.
Product B's standard deviation is larger in dollars, but this is not a fair comparison because the means differ.
Product B has the lower CV, so its spread is smaller relative to its average.

Answer: Product B is more consistent: its CV is 12% compared with 15% for Product A.

Exam tips

Read whether the question asks for variance, standard deviation or CV. Many wrong answers come from stopping a step early.
In number entry questions, check the rounding instruction and the unit (percentage or decimal) before typing.
Use the calculator's statistics mode to save time, but select the population standard deviation unless told otherwise.
For comparison questions, go straight to the CV. Do not compare standard deviations when the means differ.
In multiple response questions, remember that a lower dispersion measure means more consistent data and lower risk.

Practice questions from Summarising and analysing data

Measures of Dispersion in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IMeasures of Dispersion and Risk

Measures of Dispersion: frequently asked questions

What is the difference between variance and standard deviation?

Variance is the average of the squared deviations from the mean, so it is in squared units. Standard deviation is the square root of variance, so it is in the same units as the data. Standard deviation is easier to interpret.

How do I calculate standard deviation in the ACCA MA exam?

Find the mean, then the mean of the squared values, and subtract the squared mean to get the variance. Take the square root. You can also use the statistics mode on the exam calculator, using the population standard deviation key.

What is the coefficient of variation used for?

It compares the relative spread of data sets that have different means or units. You divide the standard deviation by the mean and multiply by 100. The lower the result, the more consistent the data.

Why is the interquartile range better than the range?

The range depends only on the highest and lowest values, so one extreme figure can distort it. The interquartile range covers the middle half of the data and ignores extremes. It gives a more stable view of spread.