Standard Deviation vs Variance Explained
Standard deviation and variance both measure the same thing: how spread out a set of numbers is. But they express that spread in different ways, and knowing which one to use and when is an important skill for anyone working with data.
Why Spread Matters
The mean gives you the centre of a data set. But the mean alone can be misleading — two very different data sets can have the same mean.
Data set A: 48, 49, 50, 51, 52 — Mean = 50. Tightly clustered.
Data set B: 10, 30, 50, 70, 90 — Mean = 50. Widely spread.
Variance and standard deviation reveal this difference.
What Is Variance?
Variance measures how far, on average, each value is from the mean. It squares each deviation so that negative differences do not cancel out positive ones.
Population variance: Sum of (each value minus mean)², divided by the number of values.
Sample variance: Sum of (each value minus mean)², divided by (n − 1).
Example using Data set A (48, 49, 50, 51, 52):
Mean = 50. Deviations: −2, −1, 0, 1, 2. Squared: 4, 1, 0, 1, 4. Sum = 10. Population variance = 10/5 = 2. Sample variance = 10/4 = 2.5.
The unit of variance is the square of the original unit. If your data is in metres, variance is in metres squared — making it harder to interpret directly.
What Is Standard Deviation?
Standard deviation is simply the square root of variance. Taking the square root converts the unit back to the original unit, making the result much easier to interpret.
From the example above: Population standard deviation = √2 ≈ 1.41. Sample standard deviation = √2.5 ≈ 1.58.
Population vs Sample: When Does It Matter?
| Type | When to use | Denominator |
|---|---|---|
| Population | You have data for every member of the group. | Divide by n |
| Sample | You have data for a subset of the full group. | Divide by n − 1 (Bessel's correction) |
Dividing by n − 1 in the sample formula compensates for the fact that a sample tends to underestimate the true spread of the population.
Variance vs Standard Deviation: Key Differences
| Variance | Standard Deviation | |
|---|---|---|
| Unit | Squared units (e.g. m²) | Same unit as the data (e.g. m) |
| Interpretation | Harder to interpret directly | Easy to interpret on the same scale as data |
| Mathematical use | Easier to work with in formulas (variances can be added) | Used in most everyday statistics and reporting |
The 68-95-99.7 Rule
In a normal (bell-shaped) distribution, standard deviation has a special relationship with the percentage of data it covers.
- 68 percent of values fall within 1 standard deviation of the mean
- 95 percent of values fall within 2 standard deviations
- 99.7 percent of values fall within 3 standard deviations
Example: Exam scores have a mean of 70 and SD of 10. 68% of students scored between 60 and 80. 95% between 50 and 90. 99.7% between 40 and 100.
Worked Example: Full Calculation
Data set: 4, 7, 13, 2, 1 — Mean = 27 / 5 = 5.4
Squared deviations:
(4 − 5.4)² = 1.96
(7 − 5.4)² = 2.56
(13 − 5.4)² = 57.76
(2 − 5.4)² = 11.56
(1 − 5.4)² = 19.36
Sum of squared deviations = 93.2
Population variance = 93.2 / 5 = 18.64 → Population SD ≈ 4.32
Sample variance = 93.2 / 4 = 23.3 → Sample SD ≈ 4.83
Use the Standard Deviation Calculator to do all of this in one step. The Statistics Calculator gives you mean, median, mode and range at the same time.
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Frequently Asked Questions
Is a higher standard deviation always bad?
Not necessarily. A high standard deviation means data is more spread out, which is bad if you want consistency (e.g. manufacturing tolerances) but expected and normal in other contexts (e.g. income distribution). Whether it is good or bad depends entirely on what you are measuring.
When should I use population vs sample formulas?
Use the population formula when your data set contains every member of the group you are studying. Use the sample formula when your data is a subset drawn from a larger population — which is the case in most real-world research and surveys.
Can variance be negative?
No. Because variance squares every deviation, all values are zero or positive. A variance of zero means every value in the data set is identical. Any data set with at least two different values will have a positive variance.
What does a standard deviation of zero mean?
It means every value in the data set is exactly the same. There is no spread at all. For example, if every student in a class scored 75, the standard deviation is zero.