Measures of Dispersion for Grouped Data · Form 5
Why is grouped data only an estimate?
Once data is sorted into class intervals, the exact values are lost, so every average and every measure of spread you calculate from it is a reasonable estimate, not an exact figure.
The raw numbers disappear
A grouped frequency table tells you how many values fall in each interval, but not what those values actually were. If 8 students score between 20 and 29, you know the count is 8, but not whether they scored 21, 25 or 29.
Because the individual numbers are gone, you cannot recover an exact mean or standard deviation from them.
The midpoint stands in for the whole class
To make progress, we assume the values in each interval are spread evenly and let the class midpoint represent them all, so 20–29 is treated as 24.5. This is a sensible approximation, not a mistake, but it is why the result is only as good as that assumption.
If the real values happen to cluster near one end of a class, the estimate drifts a little.
Estimate, not exact, and that is fine
In grouped-data questions the mean, mode, median and standard deviation are all estimates, and the mark scheme treats them that way. Recognise this whenever a table gives you intervals like 10–19, 20–29 rather than a list of separate numbers.
The common slip is to report the answer as if it were exact, or to think that using midpoints is somehow cheating, it is the accepted method for data you can no longer see one by one.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why can't I calculate an exact mean from grouped data?
Once raw values are packed into class intervals, you only know how many values fall in each class, not their exact figures. Calculations use the midpoint of each class to represent every value in it, so the mean, variance and standard deviation you get are all estimates, not exact answers.
How does this affect what I write in my exam answer?
Even though your working uses exact-looking formulas, remember the class midpoint assumption means your final answer is an approximation of the true mean or standard deviation, this is why some questions specifically ask you to explain or justify why the value is an estimate.
What's a common pitfall when choosing the midpoint for each class?
Students sometimes use the class boundary or class limit instead of the true midpoint, or make an arithmetic slip when finding it. The midpoint must be calculated as (lower limit + upper limit) ÷ 2 for each class before it's multiplied by frequency in any further calculation.