Formula sheet
Mean (grouped)
x̄ = Σfx / Σf is given on the SPM formula sheet. You still must know how to build the fx column, and, crucially, that for a class interval x is the class midpoint (lower boundary + upper boundary, then ÷ 2), not the upper or lower limit.
What the symbols mean
- x̄ the mean of the grouped data
- f the frequency of each value or class
- x each value, or the midpoint of each class interval
- Σfx the sum of every frequency × value
- Σf the total frequency (the total number of data)
Given in the exam, or memorise?
x̄ = Σfx / Σf is given on the SPM formula sheet. You still must know how to build the fx column, and, crucially, that for a class interval x is the class midpoint (lower boundary + upper boundary, then ÷ 2), not the upper or lower limit.
Why it works
Each value x occurs f times, so multiplying x by f is a shortcut for adding that value f times over.
- For each row, multiply the value (or class midpoint) x by its frequency f to get fx.
- Add all the fx to get the grand total Σfx, this is the same as Σx would be for a full raw list.
- Add all the frequencies to get Σf, the total count.
- Divide: x̄ = Σfx / Σf gives the mean.
Worked example 1
The number of goals scored in 20 matches is recorded. Goals x = 0, 1, 2, 3, 4 with frequencies f = 3, 5, 6, 4, 2.
Find the mean number of goals per match.
- Total frequency: Σf = 3 + 5 + 6 + 4 + 2 = 20.
- Multiply each value by its frequency: fx = 0, 5, 12, 12, 8.
- Add them: Σfx = 0 + 5 + 12 + 12 + 8 = 37.
- Apply the formula: x̄ = Σfx / Σf = 37 / 20.
Worked example 2
The masses of 20 parcels are grouped: class 1–5 (midpoint 3) f = 4, class 6–10 (midpoint 8) f = 6, class 11–15 (midpoint 13) f = 7, class 16–20 (midpoint 18) f = 3. Find the mean mass.
- Use the class midpoints as x: 3, 8, 13, 18.
- Total frequency: Σf = 4 + 6 + 7 + 3 = 20.
- Compute fx: (3×4), (8×6), (13×7), (18×3) = 12, 48, 91, 54.
- Add: Σfx = 12 + 48 + 91 + 54 = 205, then x̄ = Σfx / Σf = 205 / 20.
Where students go wrong
- Dividing by the number of classes instead of by Σf (the total frequency), the denominator is always the total count of data.
- For class intervals, using the upper or lower limit instead of the midpoint x when there is a range like 6–10.
- Computing Σf × Σx or (Σf)(Σx) separately, you need Σ(fx), the sum of each row's f × x.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What value of x do I use for a class interval like 21–30?
Use the class midpoint. Add the lower and upper limits and divide by two: for 21–30 that is (21 + 30) ÷ 2 = 25.5.
Enter this midpoint as x for that row, multiply by the frequency f, and continue. Never use the limits themselves as x.
Why is my grouped mean slightly different from the raw-data mean?
Because grouping replaces every value in a class with the one midpoint, some information is lost. The grouped mean is therefore an estimate, usually very close but not identical to the exact mean of the original list.
That is expected and acceptable in SPM answers.
Is this the same formula as the ungrouped mean?
Yes, it is the same idea. Σfx is just Σx written efficiently, because f tells you how many times each value repeats, and Σf equals N.
Use x̄ = Σfx / Σf whenever the data comes as a frequency table or grouped classes rather than a plain list.