Statistics From Zero · Lesson 3 of 22
Spread: Same Average, Very Different Night
In one sentence: The average tells you where the middle is; the spread tells you how far things wander from it, and two pubs with the same average temperature can be a cosy evening or a night of sweating and shivering.
The idea
In Lesson 2 we found the middle of the data. But the middle is only half the story. Two sets of numbers can have exactly the same average and be nothing alike.
The other half is the spread: how far the values wander away from the average. Small spread means everything sits close to the middle and the average is a good description. Big spread means the average is a place almost nobody actually is.
There are two common ways to measure it:
- The range: the biggest value minus the smallest. Quick, simple, and easily fooled.
- The standard deviation: roughly, how far a typical value sits from the average. Slower to calculate, much more useful.
Two pubs, one average
Uncle Gary has two locals. Aunt Linda, who keeps a thermometer in her handbag "for situations like this", measured the temperature in each one five times over an evening.
The Red Lion: 69, 70, 70, 71 and 70°F. Average: 70°F.
The Crown: 55, 85, 60, 80 and 70°F. Average: also 70°F.
On paper, identical. In real life, the Red Lion is a pleasant evening. The Crown has a roaring fireplace at one end and a door that never quite closes at the other, so you spend the night either sweating or wearing your coat. Gary sits by the fire. Linda sits by the door. They both describe the Crown as "average".
The average couldn't tell these pubs apart. The spread can. The Red Lion's range is 71 minus 69 = 2 degrees. The Crown's range is 85 minus 55 = 30 degrees. Same middle, completely different night.
Five temperature readings in each pub
Why the range isn't enough
The range only looks at two numbers: the biggest and the smallest. Everything in between is ignored. So a single outlier can blow it up. If someone opened the Red Lion's oven once and Linda happened to measure right next to it, the range would jump, even though the other four readings were perfectly calm.
The standard deviation fixes this by using every value, not just the two most extreme ones.
Standard deviation, step by step
It looks scary in textbooks. In words, it's four steps. Here they are for the Crown:
- Step 1. How far is each value from the average? 55 is 15 below 70. 85 is 15 above. 60 is 10 below, 80 is 10 above, and 70 is spot on. (Notice that if you simply added "15 below" and "15 above" together, they would cancel out to zero, as if there were no spread at all.)
- Step 2. Square each distance. 15 × 15 = 225, and so on: 225, 225, 100, 100 and 0. Squaring treats "below" and "above" the same, so nothing cancels, and it makes big distances count extra.
- Step 3. Average the squares. 225 + 225 + 100 + 100 + 0 = 650, divided by 5 = 130. This number is called the variance. It's in "squared degrees", which means nothing to anyone.
- Step 4. Take the square root to get back to normal degrees. The square root of 130 is about 11.4.
So the Crown's standard deviation is about 11.4°F: a typical reading is around 11 degrees away from the average. Do the same for the Red Lion and you get about 0.6°F. A typical reading there is less than one degree from the average, which is why nobody at the Red Lion has ever needed a coat indoors.
One handy rule for later: when data has the usual bell shape (Lesson 5), about two thirds of the values land within one standard deviation of the average, and about 95% within two.
Why anyone outside class cares
Spread is how statisticians talk about consistency and risk. Two delivery companies can both average two days, but if one sometimes takes ten, that's the one that ruins your birthday present. In finance, the standard deviation of returns is the usual measure of volatility: two investments can have the same average return while one of them swings much harder along the way. An average without a spread is only half an answer.
Try it yourself
Darts night. Over five rounds, Uncle Gary scored 40, 45, 50, 55 and 60. Cousin Kevin scored 10, 90, 20, 80 and 50.
Work out the average and the range for each of them. Then the hard question: the team needs a player who will score at least 40 in the final round. Who do you pick? And what if the team needs 85 to win?
Show the answer
Averages: Gary 250 / 5 = 50. Kevin 250 / 5 = 50. Identical.
Ranges: Gary 60 minus 40 = 20. Kevin 90 minus 10 = 80.
If you want the standard deviations too: about 7 for Gary and about 32 for Kevin.
Need at least 40? Pick Gary. He hasn't dropped below 40 all night. Kevin did it twice.
Need 85? Now pick Kevin. Gary's best was 60, and with that little spread he's very unlikely to suddenly hit 85. Kevin already hit 90 once.
Same average, opposite choices. Big spread means more chance of disaster and more chance of glory. Which one you want depends on what you need, and the average alone can't tell you.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Measures of spread (or dispersion, or variability) | Numbers that describe how spread out the data is |
| Range | Biggest value minus smallest value |
| Interquartile range (IQR) | The range of the middle half of the data, so outliers can't touch it |
| Deviation | How far one value is from the mean |
| Variance | The average of the squared deviations (step 3) |
| Standard deviation (SD, σ or s) | The square root of the variance: the typical distance from the mean (step 4) |
| Population vs sample standard deviation | For a sample, step 3 divides by one less than the number of values (n minus 1). It nudges the answer up a little, because a sample usually underestimates the spread of the whole pot. For the Crown, that gives about 12.7 instead of 11.4. |
| Volatility | In finance: the standard deviation of returns, or how wildly a price swings |
Three things to remember
- An average without a spread is half an answer. Always ask how much the values vary.
- The range is quick but one outlier can wreck it. The standard deviation uses every value.
- Standard deviation is roughly "how far a typical value is from the average". Small means consistent, big means anything can happen.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.