Formula sheet
Variance (ungrouped)
Both forms σ² = Σ(x−x̄)²/N = Σx²/N − x̄² are given on the SPM formula sheet. You must know that you need the mean x̄ first, that SPM divides by N (not N−1), and that the second form Σx²/N − x̄² is usually faster because it avoids one deviation column.
What the symbols mean
- σ² the variance, in the squared unit of the data
- x each individual data value
- x̄ the mean of the data
- N the number of data values
- Σ(x−x̄)² the sum of the squared deviations from the mean
- Σx² the sum of the squares of the data values
Given in the exam, or memorise?
Both forms σ² = Σ(x−x̄)²/N = Σx²/N − x̄² are given on the SPM formula sheet. You must know that you need the mean x̄ first, that SPM divides by N (not N−1), and that the second form Σx²/N − x̄² is usually faster because it avoids one deviation column.
Why it works
Variance measures how spread out the data is, as the average of the squared distances from the mean.
- Find each deviation x − x̄, the distance of a value from the mean.
- Square each deviation so that values below and above the mean do not cancel out.
- Average the squared deviations by dividing by N.
- Expanding the algebra turns Σ(x−x̄)²/N into Σx²/N − x̄², which needs only Σx² and the mean.
Worked example 1
Find the variance of the data 2, 4, 6, 8, 10.
- Find the mean: x̄ = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6.
- Deviations x − x̄: −4, −2, 0, 2, 4.
- Square and add: Σ(x−x̄)² = 16 + 4 + 0 + 4 + 16 = 40.
- Apply the formula: σ² = Σ(x−x̄)² / N = 40 / 5.
Worked example 2
Using the short form Σx²/N − x̄², find the variance of 5, 7, 3, 9, 6.
- Find the mean: x̄ = (5 + 7 + 3 + 9 + 6) / 5 = 30 / 5 = 6.
- Sum of squares: Σx² = 25 + 49 + 9 + 81 + 36 = 200.
- Divide by N: Σx²/N = 200 / 5 = 40.
- Apply the short form: σ² = Σx²/N − x̄² = 40 − 6² = 40 − 36.
Where students go wrong
- Forgetting to square the deviations, or squaring the sum instead of summing the squares, you need Σ(x−x̄)², not (Σ(x−x̄))².
- Dividing by N − 1 (the sample formula from other syllabuses), SPM 1449 divides by N.
- In the short form, subtracting x̄ instead of x̄², you must subtract the square of the mean.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Which form should I use, Σ(x−x̄)²/N or Σx²/N − x̄²?
Both give the same answer and both are on the formula sheet. The short form Σx²/N − x̄² is usually faster, since you only build an x² column and never a deviation column.
Use the first form when the mean is a whole number and deviations are easy to square by hand.
Can the variance be negative?
No. Variance is an average of squared quantities, so it is always zero or positive.
If your working gives a negative variance, you have made an error, most often subtracting x̄ instead of x̄², or mixing up Σx² with (Σx)². A variance of exactly 0 means every value is identical.
What unit does the variance have?
Variance is in the squared unit of the data, because every deviation is squared. If the data is in centimetres, the variance is in cm².
This is why the standard deviation, the square root of the variance, is often reported instead, it returns to the data's original unit.