Coefficient of variation and statistical investigations

25 min

Which province's districts are more alike?

The 2022 census gives the population density (people per km²) of every district. In the Southern Province the eight districts range from 409 (Nyaruguru) to 683 (Kamonyi); in the Eastern Province the seven range from 338 (Kayonza) to 740 (Rwamagana) (NISR, Table 1). The Eastern districts have a larger standard deviation, but they also have a different mean. To compare spread fairly when means differ, or when units differ, statisticians use a relative measure: the coefficient of variation.

Definition

Coefficient of variation (CV): the standard deviation as a percentage of the mean,

\text{CV} = \frac{\sigma}{\bar{x}} \times 100\% .

It has no units, so it compares the relative spread of data sets with different means or different units. The smaller the CV, the more consistent (less variable) the data. It is meaningful only for data measured from a true zero and with a positive mean, such as masses, yields, incomes or densities.

The investigation cycle

Every statistical study, from the national census to a class survey, follows the same cycle. Skipping a step is the usual cause of a wrong conclusion.

Step	What to decide	Watch out for
1. Pose a question	What exactly will be measured, for whom, and when	Vague questions ("Is transport bad?") that no data can answer
2. Plan and collect	Census or sample; how to choose the sample; questionnaire, observation, measurement or secondary data	Bias: a sample of only those easiest to reach; leading questions
3. Organise	Frequency tables; classes; checking for errors	Overlapping classes; impossible values
4. Present	The graph that answers the question	Misleading scales; pie charts with too many sectors
5. Analyse	Averages, quartiles, standard deviation, CV	Using the mean for strongly skewed data
6. Interpret and conclude	Answer the question in words, with figures; state limits	Claiming more than the data show; ignoring outliers

Sampling methods. In a simple random sample every member has an equal chance of selection (names drawn by lot, or random numbers). A systematic sample takes every $k$ th member from a list. A stratified sample divides the population into groups (strata), such as S4, S5 and S6, and samples each in proportion to its size. A convenience sample (whoever is nearest) is quick but usually biased.

Tip

Same units and similar means? Compare standard deviations. Different units or very different means? Compare CVs. And always end an investigation with a sentence that answers the original question, quotes the key figures and admits one limitation.

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Worked example 1: densities in two provinces

Southern Province densities: 545, 584, 409, 657, 441, 573, 555, 683. Eastern Province: 740, 373, 435, 338, 398, 498, 450 (people per km², NISR 2022). Compare their means and relative spread.

Southern ( $n = 8$ ): $\sum x = 4447$ , $\sum x^2 = 2\,534\,335$ .

\bar{x} = \frac{4447}{8} \approx 555.9, \qquad \sigma^2 = \frac{2\,534\,335}{8} - 555.875^2 \approx 7794.9, \qquad \sigma \approx 88.3

\text{CV} = \frac{88.3}{555.9} \times 100\% \approx 15.9\%

Eastern ( $n = 7$ ): $\sum x = 3232$ , $\sum x^2 = 1\,599\,106$ .

\bar{x} = \frac{3232}{7} \approx 461.7, \qquad \sigma^2 = \frac{1\,599\,106}{7} - 461.714^2 \approx 15\,263.6, \qquad \sigma \approx 123.5

\text{CV} = \frac{123.5}{461.7} \times 100\% \approx 26.8\%

Conclusion: Eastern districts are, on average, less densely populated (about 462 against 556 people per km²) and much less alike: their relative spread is about $\frac{26.8}{15.9} \approx 1.7$ times as large. Much of it comes from Rwamagana (740), close to Kigali, against the wide, less crowded districts near Akagera.

Worked example 2: return and risk

A cooperative compares two ways of investing its savings. Over five years, investment A returned 12%, 8%, 10%, 14% and 6%; investment B returned 25%, −5%, 20%, 30% and 5%. (Illustrative returns.)

\bar{x}_A = \frac{50}{5} = 10\%, \quad \sigma_A = \sqrt{\frac{540}{5} - 10^2} = \sqrt{8} \approx 2.83, \quad \text{CV}_A \approx 28.3\%

\bar{x}_B = \frac{75}{5} = 15\%, \quad \sigma_B = \sqrt{\frac{1975}{5} - 15^2} = \sqrt{170} \approx 13.04, \quad \text{CV}_B \approx 86.9\%

Interpretation: B pays more on average, but its risk per unit of return is about three times A's, and it lost money in one year. A cooperative that cannot afford a bad year should prefer A; one that can wait out losses might accept B. Statistics informs the decision; it does not make it.

Worked example 3: planning a class investigation

Question: "How long do pupils at our school take to travel to school in the morning, and do day pupils in S4, S5 and S6 differ?"

Plan: the school has 300 day pupils (S4: 120, S5: 100, S6: 80). A stratified sample of 30 takes $\frac{120}{300} \times 30 = 12$ from S4, 10 from S5 and 8 from S6, chosen by drawing names from each class list.
Collect: each pupil records the time in minutes from leaving home to arriving at the gate on three mornings; use the mean of the three to reduce day-to-day variation. A short questionnaire also asks the means of transport.
Organise and present: a grouped frequency table with classes such as $0 \le t \lt 10$ , …; a histogram for the whole sample; box plots for S4, S5 and S6 side by side.
Analyse: median and IQR for each class (times are likely skewed by a few long journeys); mean, standard deviation and CV for comparison.
Interpret: answer the question in words with the figures, and note the limits: a sample of 30, one season, possible errors in self-reported times.

Warning

Common mistakes. Comparing standard deviations of data with very different means or units (people per km² against Frw). Forgetting to multiply by 100% (CV $= 0.159$ means 15.9%). Using a CV when the mean is near zero or negative, where it means nothing. Concluding about a whole population from a biased sample, for example surveying only pupils who arrive early.

The Organic Chemistry Tutor, "Standard Deviation and Coefficient of Variation". The video works with a sample, dividing by n − 1; in this unit we divide by n. Notice that the CV step is the same either way: standard deviation ÷ mean × 100%.

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Practice

Try each one before opening the answers. They get harder as you go.

A data set has mean 40 and standard deviation 6. Find its coefficient of variation.
Maize variety P gave yields (tonnes per hectare) of 3.2, 3.6, 3.4, 3.0, 3.8 on five plots; variety Q gave 4.5, 3.1, 5.2, 3.6, 4.6 (illustrative). Find the mean, standard deviation and CV of each. Which is more consistent? Which would you recommend, and why?
The mean mass of pupils in a class is 52 kg ( $\sigma = 6.5$ kg) and their mean height is 160 cm ( $\sigma = 8$ cm). Which varies more, relative to its mean?
A school of 600 pupils has 250 in S4, 200 in S5 and 150 in S6. How many from each should a stratified sample of 48 include?
Name the sampling method: (a) every 10th name on a register; (b) the first 20 customers to leave a shop; (c) 30 names drawn by lot from all members.
Write a clear, measurable investigation question about mobile-money use among traders in your sector, and describe how you would collect the data.
A student surveyed 20 friends at a football match and concluded that "90% of young people in Rwanda prefer football to any other sport." Evaluate this conclusion.

Show answersHide answers

$\text{CV} = \frac{6}{40} \times 100\% = 15\%$ .
P: mean 3.4, $\sigma \approx 0.283$ , CV $\approx 8.3\%$ . Q: mean 4.2, $\sigma \approx 0.751$ , CV $\approx 17.9\%$ . P is more consistent; Q yields more on average. A farmer who needs a reliable harvest might choose P; one aiming for the largest average yield, Q, after checking why its yields vary (soil, rain).
Mass: $\frac{6.5}{52} \times 100\% = 12.5\%$ ; height: $\frac{8}{160} \times 100\% = 5\%$ . Mass varies more relative to its mean.
S4: $\frac{250}{600} \times 48 = 20$ ; S5: 16; S6: 12.
(a) systematic; (b) convenience; (c) simple random.
Example: "What percentage of market traders in our sector received at least one mobile-money payment last week, and what was the median number of such payments?" Collect by a short questionnaire given to a random or systematic sample of traders from the market's list of stalls.
Not justified. The sample is tiny and biased (friends at a football match are likely to prefer football), it is not random, and it cannot represent all young people in Rwanda. A fair study needs a clear question, a larger random or stratified sample from the whole population of interest, and neutral questions.

Quick check

Key points

$\text{CV} = \frac{\sigma}{\bar{x}} \times 100\%$ measures relative spread; it has no units and compares data sets with different means or units.
Rwanda 2022: Eastern districts' densities (CV ≈ 26.8%) vary more than Southern districts' (CV ≈ 15.9%).
An investigation runs: question → collect → organise → present → analyse → interpret.
Random, systematic and stratified samples reduce bias; convenience samples usually introduce it.
A conclusion answers the question with figures and states its limits.

1Which province's districts are more alike?

2The investigation cycle

Which province's districts are more alike?

The investigation cycle