Algebra
Data and statistics
Standard deviation and variance
20 practice questions
2 video lessons
Theory + worked examples
Theory
Standard deviation measures how spread out data are around the mean:
- Find the mean.
- Square each deviation \(x-\bar x\).
- Variance is the mean of those squares.
- Standard deviation \(\sigma\) is its square root.
Small SD: clustered; large SD: spread out.
Small vs large standard deviation.
Variance and standard deviation.
Variance and standard deviation:
\[\sigma^2=\dfrac{\sum(x-\bar x)^2}{n},\qquad \sigma=\sqrt{\sigma^2}\]
SD has the same units as the data.
How to find standard deviation
- Compute the mean.
- Square each deviation.
- Average them (variance).
- Take the square root.
Example 1 β Mean
Find the mean of \(2,4,6,8,10\).
Solution
Add and divide.
| \(\bar x\) | \(=\) | \(6\) |
Example 2 β Variance
Find the variance of \(2,4,6,8,10\).
Solution
Average the squared deviations from \(6\).
| \(\dfrac{16+4+0+4+16}{5}\) | \(=\) | \(8\) |
Example 3 β Standard deviation
Find the standard deviation.
Solution
Take the square root of the variance.
| \(\sigma\) | \(=\) | \(\sqrt8\approx2.83\) |
Example 4 β Compare spread
Which has a larger SD: tightly clustered or widely spread data?
Solution
Widely spread data have the larger standard deviation.
Common pitfalls
Square the deviations before averaging.
SD is the root of the variance.
SD keeps the data's units; variance is squared units.
Frequently asked questions
What is standard deviation?
A measure of spread around the mean.
What is variance?
The mean of the squared deviations.
How are they related?
SD is the square root of the variance.
What does a large SD mean?
The data are widely spread.
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