Descriptive Statistics
Statistics
Descriptive Statistics
Syllabus tag: KASNEB CPA | Foundation Level | CA15 Quantitative Analysis | Topic 5 Descriptive Statistics
Lesson objectives
By the end of this topic, you will be able to:
- Compute the mean, median and mode and say when each is appropriate
- Compute the range, quartiles and interquartile range
- Compute variance and standard deviation, and distinguish the population from the sample formula
- Compute and interpret the coefficient of variation
- Describe the shape of a distribution
Why this matters
A set of figures tells you nothing until it is summarised. Every summary discards information, so the skill is knowing which measure to use and what it hides.
One dataset throughout
Ten monthly sales figures, in KES thousands:
12, 15, 18, 18, 21, 24, 27, 30, 33, 42
Measures of location
Mean = sum / n = 240 / 10 = 24.0
Median — the middle value. With ten items, the average of the 5th and 6th: (21 + 24) / 2 = 22.5
Mode — the most frequent value = 18
Which to use. The mean uses every figure and is therefore pulled by extreme values — the 42 here drags it above the median. The median ignores extremes entirely, which is why it is preferred for skewed data such as incomes or house prices. The mode is the only measure available for categorical data, where "average colour" is meaningless.
Note what the mean and median tell you together. Mean above median signals a distribution with a long right tail, as here.
Measures of dispersion
Range = 42 − 12 = 30. Simple, and determined entirely by the two most extreme values.
Quartiles. Q1 = 18, Q3 = 30. Interquartile range = 30 − 18 = 12, covering the middle half of the data and ignoring the tails.
Variance and standard deviation.
| Formula denominator | Result | |
|---|---|---|
| Population variance | n = 10 | 75.60 |
| Population standard deviation | 8.6948 | |
| Sample variance | n − 1 = 9 | 84.00 |
| Sample standard deviation | 9.1652 |
The n − 1 matters and is examined. Where the data is a sample used to estimate the spread of a wider population, dividing by n understates it, because the sample mean is closer to the sample's own values than the true population mean would be. Dividing by n − 1 corrects for that.
An examiner will state whether the figures are a population or a sample. If the wording is ambiguous, say which you assumed.
Coefficient of variation
CV = Standard deviation / Mean, usually expressed as a percentage.
= 8.6948 / 24.0 = 36.23%
The CV allows datasets of different scales to be compared. A standard deviation of 8.69 means nothing until you know the mean it sits against — a spread of 8.69 around a mean of 24 is wide, and around a mean of 2,400 is negligible.
:::checkpoint Branch A has mean sales of 24,000 with a standard deviation of 8,695. Branch B has mean sales of 180,000 with a standard deviation of 30,000. Compute both coefficients of variation and say which branch is more variable. :::
Skewness
Symmetrical — mean = median = mode. Positively skewed (long right tail) — mean > median > mode. Our data. Negatively skewed (long left tail) — mean < median < mode.
Pearson's coefficient of skewness = 3(Mean − Median) / Standard deviation
= 3(24.0 − 22.5) / 8.6948 = 0.5175, confirming a modest positive skew.
Income distributions are almost always positively skewed, which is why median income is the figure usually reported. Quoting the mean would overstate what a typical person earns.
Grouped data
Where data arrives in classes, use the class midpoint as though every item in the class took that value.
Mean = Σfx / Σf, where x is the midpoint and f the frequency.
The result is an approximation, because the assumption that items are evenly spread within each class is rarely exactly true. Narrower classes reduce the error.
Index numbers
An index expresses a value relative to a base period set at 100.
Simple price index = (Current price / Base price) × 100
A price rising from KES 80 to KES 92 gives (92 / 80) × 100 = 115, an increase of 15%.
Laspeyres uses base-period quantities as weights; Paasche uses current-period quantities. Laspeyres is easier to compute over time because the weights are fixed, and it tends to overstate inflation, since it assumes consumers do not switch away from goods that have become expensive.
:::checkpoint A dataset has a mean of 50 and a median of 62. Describe the shape of the distribution, state which measure of location you would report, and explain why. :::