Measures of Dispersion for Grouped Data · Form 5
What does the interquartile range measure?
The interquartile range is Q3 − Q1: it measures the spread of the middle 50% of the data, ignoring the highest and lowest values.
The spread of the middle half
The lower quartile Q1 is the value a quarter of the way through the ordered data, and the upper quartile Q3 is three-quarters of the way through. The interquartile range is the gap between them, Q3 − Q1, so it captures how widely the central half of the data is spread.
A small IQR means the middle values are packed tightly together; a large one means they are stretched out.
Why it beats the plain range
The ordinary range is largest minus smallest, so a single unusual value at either end can blow it up and give a misleading picture of the spread. The interquartile range trims away the top and bottom quarters, so those extreme values never touch it.
That makes the IQR a steadier measure of spread when the data has a few outliers.
It is Q3 − Q1, nothing else
Recognise an IQR question when it asks for the spread of the middle half, or gives you Q1 and Q3 to subtract; for grouped data these quartiles are usually read from an ogive. The common slips are subtracting the other way round (Q1 − Q3 gives a negative), reaching for the largest and smallest values instead of the quartiles, or dragging the median Q2 into the subtraction, the median marks the centre but plays no part in the interquartile range itself.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I actually find Q1 and Q3 for grouped data?
Build the cumulative frequency table, then use the N/4-th value's position for Q1 and the 3N/4-th value's position for Q3, either read them off an ogive or interpolate within the class they fall in. IQR = Q3 − Q1.
Exam questions often require the ogive to be drawn first.
Why use the IQR instead of just the range?
The range only compares the highest and lowest values, so one unusual score can make it misleading. The IQR looks only at the middle 50%, ignoring extreme values, so it gives a fairer sense of how spread out the typical data really is, this reasoning is often asked directly in comparison questions.
What's a common mistake when working out Q1 and Q3 here?
Students often confuse N/4 and 3N/4 with N/2 (used for the median), or read the wrong axis on the ogive. Another slip is forgetting the cumulative frequency must match the class boundaries, not the class midpoints, when interpolating, this small mix-up loses easy marks.