Calculating standard deviation
Sample standard deviation describes spread around the mean. For at least two observations, square the deviations, divide their total by n − 1, then take the non-negative square root. Use the original measurement unit
- Find the mean Add all values and divide by the number of values \(\displaystyle \bar{x}=\frac{4+6+8+10+12}{5}=8\)
- Use signed deviations Subtract the mean from each value; values below the mean have negative deviations \(\displaystyle \begin{gathered}4-8=-4\\6-8=-2\\8-8=0\\10-8=2\\12-8=4\end{gathered}\)
- Square before adding Square every deviation so negative and positive deviations cannot cancel \(\displaystyle \begin{gathered}(-4)^2+(-2)^2=20\\0^2+2^2+4^2=20\\20+20=40\end{gathered}\)
- Divide and take the root Divide the sum of the squared deviations by one less than the number of observations, then take the non-negative square root \(\displaystyle s=\sqrt{\frac{40}{5-1}}=\sqrt{10}\approx3.2\)
Worked example
Five employees travel 8 km, 10 km, 12 km, 14 km and 16 km to work. Calculate the sample standard deviation to 1 decimal place
- \(\displaystyle \begin{gathered}8+10+12=30\\14+16=30\\T=30+30=60\\\bar{x}=60/5=12\end{gathered}\)Add the distances in two short groups. The total is 60 km; divide by five to find the mean
- \(\displaystyle \begin{gathered}8-12=-4\\10-12=-2\\12-12=0\\14-12=2\\16-12=4\end{gathered}\)Subtract the mean from each distance in the order given
- \(\displaystyle \begin{gathered}(-4)^2+(-2)^2=20\\0^2+2^2+4^2=20\\20+20=40\end{gathered}\)Square before adding, so negative and positive deviations cannot cancel
- \(\displaystyle \begin{gathered}s=\sqrt{40/4}=\sqrt{10}\\s\approx3.2\text{ km}\end{gathered}\)Divide by n − 1 = 4, then take the square root. Round only the final result
Common mistakes
Difficulty levels
Questions about this topic
Why does the sample formula divide by n - 1?+
The sample mean is estimated from the same data, so the sample standard deviation formula uses n - 1 rather than n as its divisor
Can a standard deviation be negative?+
No. It is the non-negative square root of a non-negative variance. For at least two observations, it is zero exactly when all values agree
Why are the deviations squared?+
Without squaring, negative and positive deviations would cancel to zero. Squaring also gives larger distances a larger contribution to the spread measure
Does adding a constant change standard deviation?+
No. The readings and their mean move together, so their deviations stay unchanged
Does multiplying all values change standard deviation?+
Yes. Multiply standard deviation by the absolute value of the scale factor. For example, multiplying every value by −3 multiplies standard deviation by 3.
Is standard deviation the average distance from the mean?+
Not the ordinary average of absolute distances. The sample calculation squares deviations, divides by n − 1 and takes a square root
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions