Algebra 2
Statistics and probability
Standard deviation and variance
20 practice questions
0 video lessons
Theory + worked examples
Theory
Standard deviation measures how spread out data are around the mean:
- Find the mean \(\bar x\).
- Find each deviation \(x-\bar x\) and square it.
- The variance is the mean of those squares.
- The standard deviation \(\sigma\) is its square root.
Small SD: data clustered near the mean; large SD: spread out.
Standard deviation is small for tight data, large for spread data.
Variance and standard deviation.
Variance and standard deviation:
\[\sigma^2=\dfrac{\sum(x-\bar x)^2}{n},\qquad \sigma=\sqrt{\sigma^2}\]
The standard deviation has the same units as the data.
How to find the standard deviation
- Compute the mean.
- Subtract the mean from each value and square.
- Average the squares (variance).
- Take the square root.
Example 1 β Find the mean
Find the mean of \(2,4,6,8,10\).
Solution
Add and divide by \(5\).
| \(\bar x\) | \(=\) | \(\dfrac{2+4+6+8+10}{5}=6\) |
Example 2 β Variance and SD
Find the standard deviation of \(2,4,6,8,10\).
Solution
Square the deviations from \(6\), average, then root.
| \(\text{deviations}\) | \(:\) | \(-4,-2,0,2,4\) |
| \(\text{variance}\) | \(=\) | \(\dfrac{16+4+0+4+16}{5}=8\) |
| \(\sigma\) | \(=\) | \(\sqrt8\approx2.83\) |
Example 3 β Compare spread
Which has a larger SD: \(5,5,5,5\) or \(1,5,5,9\)?
Solution
The first has no spread; the second varies.
| \(1,5,5,9\) | \(\Rightarrow\) | \(\text{larger SD}\) |
Example 4 β Units
If data are in meters, what unit is the standard deviation?
Solution
Standard deviation shares the data's units.
| \(\sigma\) | \(\text{in}\) | \(\text{meters}\) |
Common pitfalls
Square the deviations before averaging.
The SD is the root of the variance, not the variance itself.
SD keeps the data's units; variance uses squared units.
Frequently asked questions
What is standard deviation?
A measure of how spread out data are around the mean.
What is variance?
The mean of the squared deviations from the mean.
How are they related?
The standard deviation is the square root of the variance.
What does a large standard deviation mean?
The data are widely spread from the mean.
β Previous subtopic
This is the first subtopic
Next subtopic β
Normal distribution and fitting data with mean & SD
More in Statistics and probability