Right on average, wrong most days: Every night at Mill Lane, Priti bakes 400 loaves for Jonas's Park Road shop. That is the shop's average, its mean (6.7.1).
In one week of September 2026 Park Road sold exactly 400 loaves a day on average. Yet on six of the seven days Priti baked the wrong number.
| Loaves sold, one week in September | High Street (Aisha) | Park Road (Jonas) |
|---|---|---|
| Monday | 410 | 350 |
| Tuesday | 425 | 330 |
| Wednesday | 430 | 345 |
| Thursday | 415 | 360 |
| Friday | 440 | 400 |
| Saturday | 405 | 510 |
| Sunday | 415 | 505 |
| Mean | 420 | 400 |
Baking 400 every night, Park Road had 215 loaves left over on the weekdays and ran 215 loaves short at the weekend. Baking 420 for High Street left just 35 over and 35 short across the whole week.
Both averages were right. What differs is how far each day wanders from the average. That is the spread of the figures (dispersion).
The range
- Highest figure minus lowest figure
- High Street: 440 − 405 = 35 loaves
- Park Road: 510 − 330 = 180 loaves
The interquartile range
- The spread of the middle half of the figures
- Leaves out the very high and very low days
The standard deviation
- How far a typical figure sits from the mean
- Uses every figure in the set
The range is the quickest, but it rests on just two figures, the highest and the lowest. One unusual day decides it. The interquartile range and the standard deviation look past the two ends.
The average is half the story: An average says where the middle is. The spread says how far a single day can be from it.
For Lena's Bakery that means how many loaves to bake, how many staff to put on, and how many loaves end up unsold.
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Station Road opened in September 2026, hoping to sell 250 loaves a day. Its first eleven trading days sold 390 on the opening day, with free coffee for every customer, then 240, 225, 210, 250, 230, 260, 245, 265, 230 and 260.
The range is 390 − 210 = 180 loaves. But almost all of that comes from one day that will never be repeated. Priti needs the spread of a normal day.
Finding the quartiles
Put the figures in order
210, 225, 230, 230, 240, 245, 250, 260, 260, 265, 390.
Find the median
Eleven figures, so the middle one is the 6th: 245 (6.7.1).
Split into two halves
Leave the median out. Lower half: 210, 225, 230, 230, 240. Upper half: 250, 260, 260, 265, 390.
Find the quartiles
The lower quartile (Q1) is the middle of the lower half: 230. The upper quartile (Q3) is the middle of the upper half: 260.
Take one from the other
Interquartile range (IQR) = Q3 − Q1 = 260 − 230 = 30 loaves.
The method used in these notes: Put the figures in order and find the median. With an odd number of figures, leave the median out of both halves; with an even number, split the list into two equal halves.
Q1 is the median of the lower half and Q3 the median of the upper half. The interquartile range is Q3 − Q1.
The quartiles cut the ordered days into four quarters. At Station Road, a quarter of the days sold 230 loaves or fewer, and a quarter sold 260 or more.
So the middle half of the days sold between 230 and 260 loaves: a spread of just 30. The opening day sits far outside it and hardly matters. Take it out and the range falls from 180 to 55, while the quartiles barely move.
The range
- Two figures only. The highest and the lowest day.
- Pulled by one odd day. The opening day makes it 180 loaves.
- Quick. One subtraction, no ordering needed.
The interquartile range
- The middle half. Q3 − Q1, from the ordered figures.
- Not pulled by odd days. 30 loaves, with or without the opening day.
- Leaves things out. It ignores the top and bottom quarters.
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The interquartile range looks at the middle half of the figures. The standard deviation uses every one of them: it measures how far a typical figure sits from the mean.
A small standard deviation means the figures bunch close to the average. A large one means they are spread out.
High Street's standard deviation, step by step
The mean
2,940 loaves ÷ 7 days = 420 a day.
Each day's distance from the mean
−10, +5, +10, −5, +20, −15, −5. They always add up to zero, so they cannot simply be averaged.
Square them, then add
100 + 25 + 100 + 25 + 400 + 225 + 25 = 900. Squaring turns every distance into a positive number.
Divide by one less than the count
Seven days, so divide by 6: 900 ÷ 6 = 150.
Take the square root
√150 = 12.2 loaves. A typical High Street day is about 12 loaves from the mean.
The formula, and which version: Standard deviation = √(sum of (each figure − mean)² ÷ (n − 1)), where n is the number of figures.
Dividing by n − 1 is the version a spreadsheet's STDEV function gives. Dividing by n gives a slightly smaller figure, 11.3 here, and is read in the same way. In a case the standard deviation is usually given to you; the task is to say what it shows.
| High Street | Park Road | |
|---|---|---|
| Mean | 420 loaves | 400 loaves |
| Standard deviation | 12.2 loaves | 76.5 loaves |
| Compared with the mean | About 3%: small | About 19%: large |
| What it shows | Days bunch close to 420 | Days swing far from 400 |
When most figures bunch near the mean and thin out evenly on both sides, in a hill shape, a rule of thumb applies. About two-thirds (68%) of the figures lie within one standard deviation of the mean, about 95% within two, and almost all within three.
For High Street that means between 408 and 432 loaves on about two days in three, and between 396 and 444 on about 95 days in 100. In the September week, 5 of the 7 days fell in the first band and all 7 in the second.
When the rule of thumb does not fit: Park Road's week is not a hill. It is two groups: weekdays from 330 to 400, weekends above 500.
A standard deviation of 77 warns that the average hides something; the days themselves show what. Priti should bake by the day of the week, not to the mean.
How this comes up: A case gives a mean and a standard deviation, sometimes in a table of figures, and asks you to comment on the standard deviation or on the data, for two marks. The standard deviation is given; you are not asked to work it out.
One mark is for showing what a standard deviation measures, the other for using the case's own figures to say what it means for that business.
The two-mark pattern
- Say what it measures. How far the figures typically sit from the mean.
- Judge its size against the mean. Small: the figures bunch together. Large: they are spread out.
- Put numbers on it. Mean ± one standard deviation for about two-thirds of the figures, ± two for about 95%.
- Say what it means here. Steady or unpredictable sales, and what that means for stock, staff or planning.
The trap: a definition with no case: Defining standard deviation and stopping there shows you know the tool, but it never touches the business.
The opposite trap: copying the figures from the case without saying what they show about spread. A comment needs both.
Using the standard deviation, comment on the daily loaf sales at Lena's Market Square shop.
Model answer plan
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