Standard deviation and variance
Standard Deviation and Variance
Standard Deviation and Variance is the opening topic of Statistics & Probability in the Common Core State Standards. It is aligned to Standard S-ID.2, which requires students to use the mean and standard deviation of a data set to compare and interpret spread.
Variance is the mean of the squared deviations from the mean; the standard deviation is its square root and measures spread.
Theory
- 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.
Variance and standard deviation:
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.
Add and divide by \(5\).
| \(\bar x\) | \(=\) | \(\dfrac{2+4+6+8+10}{5}=6\) |
Square the deviations from \(6\), average, then root.
| \(\text{deviations}\) | \(:\) | \(-4,-2,0,2,4\) |
| \(\text{variance}\) | \(=\) | \(\dfrac{16+4+0+4+16}{5}=8\) |
| \(\sigma\) | \(=\) | \(\sqrt8\approx2.83\) |
The first has no spread; the second varies.
| \(1,5,5,9\) | \(\Rightarrow\) | \(\text{larger SD}\) |
Standard deviation shares the data's units.
| \(\sigma\) | \(\text{in}\) | \(\text{meters}\) |
Common pitfalls
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.