Summarising data: changes, averages and spread

In depth

Percentage change: three traps

Small bases. Lantern’s gift-box orders went from 9 in October to 88 in November, a rise of $(88 - 9) \div 9 \times 100 \approx 877.8\%$, then to 231 in December, a rise of 162.5%. December’s percentage is far smaller, but its absolute rise, 143 orders, is almost twice November’s 79. A percentage change from a small base is dramatic and often meaningless; this is a base effect. Report the counts beside it.

Reversals. Equal percentage rises and falls do not cancel, as the concept section showed. To undo a fall of 20% you need a rise of 25%, because the second change works on a smaller base.

Changes in rates. A change in a percentage has a points version and a relative version. Say which one you mean, every time.

Growth over several periods

Lantern’s total visits grew every month but one between July and December.

Over the five monthly steps visits grew by $16{,}200 \div 9{,}800 - 1 \approx 65.3\%$ in total. What was the average monthly growth? The tempting answer is the mean of the five monthly rates, about 11.0%. It is wrong: 9,800 visits growing at that average rate, 11.02% a month, for five months would reach about 16,500, not 16,200. Growth compounds, so the right average is the compound growth rate, the constant rate that turns the starting value into the final value over the same number of periods:

Check it, keeping more decimals than the rounded 10.6%: 9,800 × 1.10575⁵ ≈ 16,200. Over years, the same calculation gives the compound annual growth rate that F04 uses for money. Two cautions apply. A growth rate needs the number of steps, not the number of months: July to December is five steps. And an average growth rate describes the past; Lantern’s last two months were boosted by Christmas gift buying, and projecting 10.6% a month into 2027 would be a mistake that the next lesson explains.

The three averages in detail

For $n$ values $x_1, x_2, \dots, x_n$, the mean is $\bar{x} = (x_1 + x_2 + \dots + x_n) \div n$. The mean has a property the others lack: multiplied by the count, it gives the total. That is why average order value is a mean: AOV × orders = revenue. If you need totals, for a budget, a forecast or a stock order, use the mean.

The median of $n$ sorted values sits at position $(n + 1) \div 2$; when $n$ is even, it is halfway between the two middle values. Because it depends only on the order of the values, a few extreme values cannot drag it far. When data are skewed, the mean is pulled towards the long tail: above the median when there are a few very large values, as with order values, and below it when there are a few very small ones, as with Lantern’s ratings. For “what does a typical customer do?”, the median is usually the better answer.

The mode is the only average for nominal data (the most common product, region or channel) and is useful for counts with few possible values, such as packs per order. Some data have two modes, which often signals two groups mixed together: one-pack orders and gift boxes, say.

Measuring spread

The range is the simplest measure: branch A’s is $15 - 7 = 8$. It depends on just two values, the most extreme, so one unusual day changes it completely.

Quartiles split sorted data into four parts. The first quartile, Q1, has a quarter of the values below it and the third, Q3, three-quarters; the median is the second. There are several ways to calculate quartiles, and B02 explains the choice. This course uses quartiles by linear interpolation, as B02 does (and as Excel’s QUARTILE.INC does): the value at position $1 + (n - 1)p$ in the sorted list, where $p$ is 0.25 for Q1 and 0.75 for Q3, interpolating between neighbours when the position isn’t a whole number.

For branch A, $n = 9$, so Q1 is the 3rd value, 10, and Q3 the 7th, 12. The IQR is $12 - 10 = 2$: the middle half of branch A’s days lie within 2 orders of each other. For branch B the same positions give 8 and 14, an IQR of 6.

The standard deviation uses every value. Take each value’s distance from the mean, square it, add the squares, divide by $n - 1$ and take the square root. For branch A the distances from 11 are −4, −2, −1, −1, 0, 1, 1, 2 and 4; their squares add to 44; $44 \div 8 = 5.5$; and $\sqrt{5.5} \approx 2.35$ orders. For branch B the squares add to 220, and $\sqrt{220 \div 8} \approx 5.24$.

Squaring makes every distance positive and gives large distances extra weight. Dividing by $n - 1$ rather than $n$ is the convention for a sample, which B04 explains. For roughly symmetric, bell-shaped data, about 95% of values lie within two standard deviations of the mean (the empirical rule, also in B04).

Which measure to use follows the same logic as the averages. The standard deviation goes with the mean and suits roughly symmetric data. The IQR goes with the median and resists extreme values, which is why B02 builds its outlier fences on it. The range is quick, but describes only the two extremes.

Weighted averages

In a weighted average each value counts in proportion to a weight, $\bar{x}_w = \sum w_i x_i \div \sum w_i$. The most important example in this course is the overall conversion rate. In July Lantern’s website converted $205 \div 7{,}400 \approx 2.770\%$ of its visits and the app $89 \div 2{,}400 \approx 3.708\%$. The plain average of the two rates is about 3.24%, but the shop’s real conversion was $294 \div 9{,}800 = 3.00\%$. The overall rate is the channel rates weighted by each channel’s visits:

The website had three-quarters of the visits, so its rate counts three times as much. The plain average treats the two channels as equal and overstates the shop’s conversion. The same rule applies to any average of averages: to average monthly AOVs across several months, weight each month’s AOV by its orders; to average two branches’ ratings, weight by the number of ratings. The weighted result always equals the total of the numerators divided by the total of the denominators, which is the simplest way to compute it. It also means the overall rate can move when nothing but the mix of channels changes, a point the next lesson develops.

Rounding and precision

1. Round at the end. Rounding July’s channel rates to 2.8% and 3.7% before weighting gives $(7{,}400 \times 2.8 + 2{,}400 \times 3.7) \div 9{,}800 \approx 3.02\%$ instead of 3.00%. Keep full precision in the working; round the answer.
2. Match the precision to the data. July’s returning share is 38.0952…%; report 38.1%. Four decimal places suggest an accuracy that 294 orders can’t deliver.
3. Be consistent. Give every value in a column of a table the same number of decimal places, so that differences in digits don’t look like differences in size.
4. Rounded parts may not add up. Three shares of 33.3% total 99.9%. Say so in a note rather than fudging one figure.
5. Know when an input was rounded. The case’s average ratings are given to one decimal place, so a change from 4.4 to 4.5 could be anything from just above 0 to just under 0.2 stars.