Algebra 2 / Unit 10: Statistics and Data Analysis
10.1 Standard Deviation and Spread
Notes
The standard deviation measures how far data values typically are from the mean. A bigger standard deviation means more spread.
- Population standard deviation: $\displaystyle \sigma = \sqrt{\frac{\sum (x - \mu)^2}{n}}$, where $\mu$ is the mean.
- Steps: find the mean, square each distance from the mean, average the squares (the variance), then take the square root.
- Adding a constant to every value shifts the mean but does not change the standard deviation.
- Multiplying every value by $c$ multiplies both the mean and the standard deviation by $|c|$.
Practice
The data set $2, 4, 4, 4, 5, 5, 7, 9$ has mean 5. What is its population standard deviation?
Squared distances sum to $9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$. $\displaystyle \frac{32}{8} = 4$ and $\displaystyle \sqrt{4} = 2$.
Which data set has the greatest standard deviation?
All have mean 10, but $1, 10, 19$ is the most spread out.
Every value in a data set is increased by 5. What happens?
Shifting every value moves the center but not the spread.
Ready for more? Take the Unit 10 test.