11.3 Averages and Dispersion: Mean, Median, Mode, Range, Variance, Standard Deviation and Coefficient of Variation

Why this topic is examined

Syllabus area D1(c) asks you to calculate summary measures of central tendency and dispersion for both grouped and ungrouped data: the arithmetic mean, median, mode, range, variance, standard deviation and coefficient of variation. In BA2 these measures support decisions under risk: the mean (or expected value) gives the typical outcome, and the standard deviation shows how far results may stray from it. Expect number-entry calculations and questions interpreting which option is riskier.

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Ungrouped data

Worked example

A shop records daily sales of a product over seven days: 12, 15, 11, 18, 15, 14, 13 units.

Ranked: 11, 12, 13, 14, 15, 15, 18.

For an even number of values, the median is the average of the two middle values.

Variance and standard deviation

The variance is the average squared deviation from the mean; the standard deviation is its square root, which puts it back into the original units:

$$\sigma^2 = \frac{\sum (x - \bar{x})^2}{n} \qquad \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$$ $$\sigma^2 = \frac{32}{7} = 4.571 \qquad \sigma = \sqrt{4.571} = 2.14 \text{ units}$$

The deviations always sum to zero, which is why they are squared before averaging. This guide divides by $n$, treating the data as the whole population, which is the usual approach in management accounting questions.

Coefficient of variation

$$\text{Coefficient of variation} = \frac{\sigma}{\bar{x}} = \frac{2.14}{14} = 0.153 \text{ (15.3\%)}$$

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Grouped data (frequency distributions)

When data are grouped into classes, each class is represented by its midpoint ($x$), weighted by its frequency ($f$).

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

Worked example: weekly overtime hours of 50 employees

Mean:

$$\bar{x} = \frac{456}{50} = 9.12 \text{ hours}$$

Standard deviation:

$$\sigma = \sqrt{\frac{5{,}064}{50} - 9.12^2} = \sqrt{101.28 - 83.17} = \sqrt{18.11} = 4.26 \text{ hours}$$

Coefficient of variation: 4.26 ÷ 9.12 = 0.467 (46.7%).

Median: the median is the value of the 25th employee (50 ÷ 2). The cumulative frequency reaches 20 at the end of the 4 to 8 class and 38 at the end of the 8 to 12 class, so the median lies in the 8 to under 12 class. Estimating by linear interpolation within the class:

$$\text{Median} = L + \frac{\frac{n}{2} - F}{f_m} \times c = 8 + \frac{25 - 20}{18} \times 4 = 8 + 1.11 = 9.11 \text{ hours}$$

where $L$ is the lower class limit, $F$ the cumulative frequency before the class, $f_m$ the frequency of the median class and $c$ the class width.

Mode: the modal class is the class with the highest frequency, 8 to under 12 (18 employees). If an estimate of the mode is required, one common interpolation is:

$$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times c = 8 + \frac{18 - 14}{36 - 14 - 9} \times 4 = 8 + 1.23 = 9.23 \text{ hours}$$

where $f_1$ is the modal class frequency and $f_0$ and $f_2$ are the frequencies of the classes either side.

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Choosing and interpreting the measures

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Using dispersion to compare risk

The standard deviation measures absolute spread. When two options have different means, the coefficient of variation measures risk relative to the expected return.

Project B has the higher standard deviation ($20,000) but also the higher expected NPV. Its coefficient of variation (0.25) shows it carries more risk per dollar of expected return than Project A (0.20). A risk-averse manager might prefer A; one focused on expected value would choose B. The measures inform the decision; they do not make it.