Measures of Dispersion for Grouped Data · Form 5
Measures of Dispersion for Grouped Data: Revision Notes
A tight revision summary of Measures of Dispersion for Grouped Data for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
When data is grouped into class intervals, you cannot use each value directly. This chapter shows how to find the mean, variance and standard deviation from a frequency table, and how to draw and read a histogram, frequency polygon and ogive (cumulative frequency curve).
Key ideas to revise
- Midpoints stand in for values. For grouped data you use the class midpoint as the value for the whole interval in every calculation.
- The ogive. A cumulative-frequency curve lets you read off the median and quartiles, and therefore the interquartile range.
- Standard deviation from a table. The given grouped formula uses Σfx and Σfx². A clear frequency table with these columns makes it almost mechanical.
One mini-example for each key idea
- Midpoints stand in for values: for the class 160–170, the midpoint is (160 + 170) ÷ 2 = 165. If 9 students fall in it, that class contributes fx = 9 × 165 = 1485 to the total, exactly as if all 9 sat at 165.
- Building the ogive: frequencies 4, 9, 12, 5 give cumulative frequencies 4, 13, 25, 30. Plot each against the upper boundary (160, 170, 180, 190), never the midpoint. The median sits at the n/2 = 30/2 = 15th value, read across from 15 on the cumulative-frequency axis.
- Standard deviation from the table: with Σf = 30, Σfx = 5130 and Σfx² = 879750, the mean is 5130 ÷ 30 = 171. Variance = 879750 ÷ 30 − 171² = 29325 − 29241 = 84, so the standard deviation = √84 = 9.17.
A pre-paper checklist for grouped dispersion
- Re-derive, don't just memorise: practise turning any class like 50–60 into its midpoint 55 and its boundaries 49.5 and 59.5 (or 50 and 60 for continuous data) until it is automatic, this one skill feeds every later calculation.
- Know what the paper gives you: the grouped variance formula σ² = Σfx²/Σf − x̄² and the median interpolation formula are printed on the formula sheet. You supply the table; the sheet supplies the formula.
- The habit that saves marks: build the midpoint (x), fx and fx² columns and write the three totals Σf, Σfx and Σfx² before touching the formula. Substituting from labelled totals prevents the commonest arithmetic slip.
- Before any ogive: choose a scale that fills the grid, plot the cumulative frequency against the upper boundaries, and join the points with one smooth curve, a cramped or jagged ogive quietly loses the median and quartile marks.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How should I set out the frequency table so the formula almost fills itself in?
Add three columns to the given table: the class midpoint x, then fx, then fx². Total each column to get Σf, Σfx and Σfx².
The mean is Σfx ÷ Σf and the variance is Σfx² ÷ Σf minus the mean squared. With the totals sitting under labelled columns, you just read them straight into the formula.
Do I have to memorise the grouped variance formula, or is it given?
The variance and standard-deviation formulas are printed on the SPM formula sheet, so you do not need to memorise them. What you must practise is the setup: forming midpoints, filling the fx and fx² columns, and substituting cleanly.
The formula is handed to you; the marks are in using it accurately.
Is the mean I calculate from grouped data the exact mean?
No, it is an estimate. Grouping throws away the individual values, so you replace each class with its midpoint and assume every member sits there.
The result is close to, but rarely identical to, the true mean of the raw data. SPM accepts this midpoint estimate as the full-mark answer.