Measures of Dispersion for Ungrouped Data · Form 4
Measures of Dispersion for Ungrouped Data: Common Mistakes
The mistakes that quietly cost marks in Measures of Dispersion for Ungrouped Data, and how to avoid each one in the SPM exam.
In our experience teaching Measures of Dispersion for Ungrouped Data, most lost marks come from a small set of repeated slips, not from a lack of understanding. Here they are, with the fix for each.
Mistakes to avoid
- Forgetting to take the square root for standard deviation
- Dividing by the wrong N
- Rounding too early and carrying an error through the calculation
Six more slips that cost marks in this chapter
- What students write: Σx² = (Σx)², so for 2, 4, 6 they write (12)² = 144. → Why it loses marks: Σx² means square each value then add, giving 4 + 16 + 36 = 56, not add then square; the whole variance is now wrong. → Correct working: build an x² column, sum it to 56, then use Σx²/N − x̄².
- What students write: variance = Σx²/N − x̄, subtracting the mean itself. → Why it loses marks: the formula subtracts x̄ squared, not x̄, so the units and the value are both wrong. → Correct working: for 2, 4, 6, Σx²/N − x̄² = 56/3 − 4² = 18.667 − 16 = 2.667.
- What students write: quartiles read straight off the unordered list 12, 4, 9, 6, 15. → Why it loses marks: quartiles are meaningless until the data is in order, so Q1 and Q3 come out wrong. → Correct working: order to 4, 6, 9, 12, 15 first, then locate Q1 and Q3.
- What students write: 'standard deviation = 12.25' after computing the variance. → Why it loses marks: that figure is the variance; the standard deviation still needs the square root. → Correct working: σ = √12.25 = 3.5.
- What students write: 'variance = 12.25 kg' for masses in kilograms. → Why it loses marks: variance is in squared units (kg²), and only the standard deviation shares the data's units (kg). → Correct working: quote variance as 12.25 kg² and standard deviation as 3.5 kg.
- What students write: they centre the spread on the median, computing Σ(x − median)². → Why it loses marks: variance and standard deviation are measured about the mean, not the median, so every deviation is off. → Correct working: find x̄ first, then use deviations from x̄.
- What students write: 'range = 5' for the five values 3, 5, 8, 12, 20, counting how many numbers there are. → Why it loses marks: range is a spread, not a count; it is highest minus lowest. → Correct working: range = 20 − 3 = 17.
The costliest slip to guard against
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
I got a negative variance, where did it go wrong?
Variance can never be negative, so a minus sign means a slip in Σx²/N − x̄². Usually Σx² was computed as (Σx)² and came out too small, or the mean was rounded up before squaring.
Recompute Σx² value by value and keep x̄ exact; the subtraction will then give a positive number.
Do I lose marks if I round the mean to 2 decimal places along the way?
Often yes, because the rounded mean is squared and the error grows. Markers accept a rounded final answer but penalise a final value that drifts because of premature rounding.
Carry the exact mean through the variance, and only round the standard deviation at the very end to the accuracy the question asks for.
The question says 'measure of dispersion' but not which one, how do I avoid picking wrong?
Look at the following clause. If it points at the whole spread or the effect of an extreme value, range fits; if it mentions the middle half or being resistant to outliers, use the IQR; if it asks how values sit around the mean, it wants the standard deviation.
Underline that phrase before choosing, so the wording, not a guess, decides.