Statistics › 📊 Describing Data

Range, Variance, and Standard Deviation

Two data sets can share the same average and still tell completely different stories — spread is what sets them apart.

Intro StatisticsAP Statistics level
💡
The big idea: The center of a data set — its mean — only tells half the story. Spread tells the other half: how far the values sit from that center. The range gives a quick reach from smallest to largest, the variance averages the squared distances from the mean, and the standard deviation — the square root of the variance — brings that spread back into the original units as the typical distance a value sits from the mean.
🎯 By the end, you'll be able to
  • Explain why the mean alone can hide big differences between data sets
  • Compute the range of a data set and say why it is fragile
  • Define variance as the average squared deviation from the mean
  • Compute a standard deviation and read it as a typical distance from the mean
📎 You should already know
  • The mean (the average of a data set)
  • Reading a dot plot or number line

Same average, different stories

Two archery students each average a score of 5. One of them lands every arrow right around 5 — steady and predictable. The other fires wild shots at 1 and 9 that just happen to average out to 5. Same center, completely different consistency — and the mean cannot tell them apart.

To capture that difference we need a second kind of number: a measure of spread, describing how far the values stray from the center. This lesson builds up the three you will meet most often — the range, the variance, and the standard deviation.

Range: quickest, but fragile

The simplest measure of spread is the range: the largest value minus the smallest.

For the data set 2, 4, 4, 4, 5, 5, 7, 9 the range is 9 − 2 = 7. Easy to compute — but notice its weakness. The range looks at only two numbers and ignores everything in between, so a single unusually large or small value (an outlier) can blow it up and give a misleading picture of the typical spread. We want a measure that listens to all the data.

🔑 Start with deviations from the mean
Every measure of spread that uses all the data begins the same way — with each value's deviation: how far it sits from the mean, written \( x_i - \mu \). Values far from the mean have large deviations; values near it have small ones. But you cannot simply average the raw deviations: the positives and negatives always cancel to exactly zero. The standard fix is to square each deviation first.

Variance: the average squared deviation

Squaring each deviation does two useful things at once: it removes the minus signs (so nothing cancels), and it punishes far-away points more than nearby ones. Add up all those squared deviations and divide by how many there are, and you get the variance — the average squared deviation from the mean.

\[ \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}\left(x_i-\mu\right)^2 \qquad \sigma = \sqrt{\sigma^2} \]
Variance σ² is the mean of the squared deviations from the mean μ; the standard deviation σ is its square root, back in the original units. (Dividing by N describes a whole population; for a sample, statisticians divide by n − 1 instead.)

Standard deviation: spread in the original units

There is one catch with variance: squaring the deviations also squares the units. If the data are in dollars, the variance comes out in dollars-squared — a quantity nobody can picture. Taking the square root undoes that. The result, the standard deviation, lands back in the same units as the data and reads naturally as the typical distance of a value from the mean.

The tool below makes that reading concrete. Each point shows its deviation from the mean as a line; drag the spread control and watch how stretching those lines changes the standard deviation.

🎮 Interactive: watch the spread grow the SD LIVE
Points sit on a number line with their deviations from the mean drawn as lines. Increase the spread and every deviation stretches — squared, they stretch even more — so the standard deviation climbs. Pull the points toward the mean and the SD shrinks toward zero.
✨ Spread the points out, and the SD grows
This is the heart of the idea. Move the values away from the mean and every deviation gets longer; because variance squares them, it grows even faster, and its square root — the standard deviation — climbs with it. Pull the values toward the mean and the SD shrinks. If every value is identical, all deviations are zero and the standard deviation is exactly 0: no spread at all.
⚠️ Variance lives in squared units
Variance is convenient for the math, but awkward to interpret because its units are squared — min², dollars². That is precisely why reports almost always quote the standard deviation instead: it carries the same information but in the data's own, readable units.
📝 Worked example: A data set is 2, 4, 4, 4, 5, 5, 7, 9. Find its range, variance, and standard deviation (treat it as the whole population).
  1. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5.
  2. Range: max − min = 9 − 2 = 7.
  3. Deviations from the mean (each value minus 5): −3, −1, −1, −1, 0, 0, 2, 4.
  4. Square each deviation: 9, 1, 1, 1, 0, 0, 4, 16 — these add up to 32.
  5. Variance = average squared deviation = 32 / 8 = 4 (in squared units).
  6. Standard deviation = √4 = 2, back in the original units.
✓ The range is 7, the variance is 4, and the standard deviation is 2 — so a typical value sits about 2 units away from the mean of 5, even though the two extremes reach 7 apart.

Check your understanding

1. Which measure of spread depends on only two data values — the largest and the smallest?
The range is max − min, so it ignores everything in between and is easily distorted by a single outlier.
2. The standard deviation is best described as…
Standard deviation = √variance. (The sum of the deviations from the mean is always 0, which is exactly why variance squares them first.)
3. You move the data points farther away from the mean without changing the mean. The standard deviation will…
Larger deviations give larger squared deviations, so the variance and its square root — the SD — both grow. Spreading points out increases the spread.
4. The values 1, 3, 5, 7 have a mean of 4. Treating them as a population, what is the variance (the average squared deviation)?
Deviations are −3, −1, 1, 3; squared they are 9, 1, 1, 9, summing to 20; the average is 20 / 4 = 5. (20 is the sum, not the average, and 2.24 is the standard deviation √5.)
✅ Key takeaways
  • The mean describes the center of a data set but says nothing about how spread out the values are.
  • Range = max − min is the quickest measure of spread, but it uses only two values and is easily distorted by an outlier.
  • Variance is the average of the squared deviations from the mean; squaring keeps positive and negative deviations from cancelling.
  • Standard deviation = √variance puts spread back into the data's original units and reads as the typical distance of a value from the mean.
  • Spreading values away from the mean increases the standard deviation; an SD of zero means every value is identical.