Cambridge IGCSE Statistics 5: Quartiles, percentiles and measures of dispersion

Cambridge IGCSE Statistics 0479 Topic 5 covers quartiles and percentiles, range and interquartile range, standard deviation for exact and grouped data, and justified selection of a dispersion measure. Dispersion describes how far observations vary, so it must be interpreted alongside a measure of centre.

1. Quartiles in exact data

Order the observations first. The lower quartile, Q1, marks approximately the 25th percentile; the median, Q2, marks the 50th; and the upper quartile, Q3, marks the 75th. Follow the positional convention stated in the question or used consistently by the course, because conventions can differ for small samples.

In an ungrouped frequency table, cumulative frequencies locate quartile positions without listing every repeated value. Always check the total frequency before selecting positions.

Quartiles divide ordered positions, not the numerical scale, into four parts. Q1 is not one quarter of the largest value.

2. Percentiles and cumulative frequency

The pth percentile is the value below which approximately p% of observations lie. On a cumulative frequency graph with total N, locate p ÷ 100 × N on the cumulative-frequency axis, move to the curve, then read the corresponding data value.

To estimate the percentile rank of a given value, read its cumulative frequency and calculate cumulative frequency ÷ N × 100. To estimate the number in an interval, subtract the cumulative readings at its endpoints. Above a value equals total frequency minus the cumulative frequency below it.

Graph readings are estimates and should reflect plotting accuracy. Do not report implausible precision.

3. Linear interpolation for grouped percentiles

Locate the class containing the target cumulative position. Then use:

estimate = lower boundary + (position into class ÷ class frequency) × class width

Position into class equals target cumulative position minus cumulative frequency before the class. This method assumes observations are spread approximately evenly within the class. It is more systematic than guessing from a graph but remains an estimate because the raw positions are unknown.

The same method can estimate Q1, the median, Q3 or any requested percentile.

4. Range and interquartile range

range = maximum - minimum

The range uses only the two extremes. It is quick and shows total span, but one unusual value can change it greatly.

interquartile range = Q3 - Q1

The interquartile range, or IQR, measures the spread of the middle 50%. It resists extreme values but ignores variation in the outer quarters. For grouped data, estimate both quartiles from the cumulative distribution and subtract.