Measures of Dispersion for Ungrouped Data
How to Find the standard deviation of ungrouped data
Use this to measure how spread out a list of values is around the mean.
Before you start
- Finding the mean of a list of numbers
- Squaring numbers, including decimals such as 7.5²
- Using the square-root key on a calculator
- Reading Σ (sigma) as 'add them all up'
When to use it
Use this to measure how spread out a list of values is around the mean.
The steps
- Find the mean by adding all values and dividing by how many there are (N).
- Build a table with a column for x and a column for x².
- Add up the x² column to get Σx².
- Use the given formula: variance = Σx²/N − (mean)².
- Take the square root of the variance to get the standard deviation.
Worked example
Find the standard deviation of the five numbers 4, 6, 8, 10, 12.
- Find the mean: (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
- Build a table of x and x²: x = 4, 6, 8, 10, 12 gives x² = 16, 36, 64, 100, 144.
- Add the x² column: Σx² = 16 + 36 + 64 + 100 + 144 = 360.
- Use the formula: variance = Σx²/N − (mean)² = 360 ÷ 5 − 8² = 72 − 64 = 8.
- Take the square root: standard deviation = √8 = 2.83 (2 d.p.).
A second example, with a twist
This time the mean is a decimal (7.5), so you must square 7.5 carefully, and the count N is even. The masses (in kg) of four boxes are 5, 7, 8 and 10.
Find the standard deviation.
- Find the mean: (5 + 7 + 8 + 10) ÷ 4 = 30 ÷ 4 = 7.5.
- Build a table of x and x²: x = 5, 7, 8, 10 gives x² = 25, 49, 64, 100.
- Add the x² column: Σx² = 25 + 49 + 64 + 100 = 238.
- Use the formula: variance = Σx²/N − (mean)² = 238 ÷ 4 − 7.5² = 59.5 − 56.25 = 3.25.
- Take the square root: standard deviation = √3.25 = 1.80 (2 d.p.).
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Do I divide by N or by N − 1?
For SPM 1449 use the population formula, variance = Σx²/N − (mean)², so you divide by N, the number of data values. Dividing by N − 1 is a later, university-level idea and is not what this syllabus asks for.
I got a negative number under the square root, what went wrong?
Variance can never be negative, so you have an arithmetic slip. Recheck Σx² and the mean; usually the mean was squared wrongly or a value was left out of Σx².
Fix that and the variance becomes positive.
Can I round the mean before finding the variance?
No, keep the mean exact and only round the final standard deviation. If you round the mean early, squaring it magnifies the error and your standard deviation will be off.
Round once, at the very end, to 2 decimal places unless told otherwise.