Standard deviationIB MYP Maths Extended: Revision notes

The mean tells you where the centre of a data set is. Two data sets can have the same mean but be very different, so you also need a measure of spread. The standard deviation (written $\sigma$) measures how far, on average, the values are from the mean.

• A small standard deviation means the values are close together and close to the mean (consistent).
• A large standard deviation means the values are spread out (less consistent).
• If all the values are equal, the standard deviation is $0$. It is measured in the same units as the data.

In MYP you find the standard deviation using technology (a graphic display calculator or spreadsheet). Enter the values into a list, then choose the one-variable statistics and read the value of $\sigma_x$, the population standard deviation. Example: $4,6,7,9,14$. The mean is $\frac{40}{5}=8$. The calculator gives $\sigma=3.41$ minutes (to 3 s.f.). What the calculator does: it finds the distance of each value from the mean ($-4,-2,-1,1,6$), squares them ($16,4,1,1,36$), finds the mean of the squares ($\frac{58}{5}=11.6$, the variance) and takes the square root.

For a frequency table, enter the values in one list and the frequencies in a second list, then ask for one-variable statistics using the frequencies. Example: 2 students read $0$ books, 5 read $1$, 6 read $2$, 4 read $3$ and 3 read $4$. The total is $n=20$. The mean is $\frac{\sum fx}{n}=\frac{0(2)+1(5)+2(6)+3(4)+4(3)}{20}=\frac{41}{20}=2.05$. Technology gives $\sigma=1.20$. Check by hand that the frequencies add to the number of students and that the mean lies within the range of the data.

Always link the number to the context. Say what the spread means for the situation.

• A bus company with a small standard deviation in its journey times is reliable: you can plan using the mean.
• A test with a large standard deviation means students' marks are very different from each other. You can also find the interval one standard deviation either side of the mean, $\bar{x}-\sigma$ to $\bar{x}+\sigma$. For the books example this is $0.85$ to $3.25$, so the values $1,2,3$ lie inside it (15 of the 20 students). Adding a value equal to the mean keeps the mean the same but reduces the standard deviation, because the new value has zero distance from the mean.

To compare two data sets, always make two comparisons: one using an average (the mean) and one using the spread (the standard deviation). Example: Company X and Company Y both have mean $28$ minutes. Company X has $\sigma=5.66$ and Company Y has $\sigma=1.41$.

• Same mean: on average the two take equally long.
• Smaller $\sigma$ for Y: its times are less spread out, so Y is more consistent and predictable. Different means give another comparison: for example, a higher mean score is better in a test, but a lower mean time is better in a race. Finish with a reasoned conclusion that uses both measures.