S2.2StatisticsCore

Other Measures of Location

Beyond the median, quartiles and percentiles pinpoint other positions in a data set. For grouped data you estimate them with linear interpolation — assuming values are spread evenly within each class.

25 min Measures of Location and Spread

Edexcel AS Level Maths: 2.2 Other Measures of Location (Linear Interpolation)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Find quartiles and percentiles
Locate a quartile’s position
Use linear interpolation on grouped data
Interpret these measures of location

Quartiles and percentiles

The $\textbf{lower quartile}$ $Q_1$ is one quarter of the way through the ordered data, the $\textbf{median}$ $Q_2$ is halfway, and the $\textbf{upper quartile}$ $Q_3$ is three quarters through. Percentiles divide the data into hundredths.

Q_1: \tfrac{n}{4}, \quad Q_2: \tfrac{n}{2}, \quad Q_3: \tfrac{3n}{4}

Positions for grouped/large data sets.

Linear interpolation

For grouped data, a quartile usually falls inside a class. Linear interpolation estimates it by assuming the data is spread evenly across that class.

Q = L + \dfrac{\left(\text{position} - \text{cf before}\right)}{f} \times w

L

= class lower bound,

f

= class frequency,

w

= class width.

Example 1Outline finding Q₁ by interpolation.

1Find the position

\tfrac{n}{4}

and the class it lands in.

2Use the lower bound, cumulative frequency before, class frequency and width in the formula.

Answerestimate within the class

Tip — Interpolation assumes even spread within a class — that’s why it’s an estimate, not exact.

Percentiles

A percentile is found the same way: the $p$ th percentile is at position $\tfrac{pn}{100}$ . The 25th percentile is $Q_1$ and the 75th is $Q_3$ .

Formula recap

Q_1: \tfrac{n}{4}, \ Q_3: \tfrac{3n}{4}

Quartile positions.

Q = L + \dfrac{(\text{pos} - \text{cf})}{f} \times w

Linear interpolation.

p\text{th percentile at } \tfrac{pn}{100}

Percentile position.

Common mistakes to avoid

Using the class midpoint instead of interpolating.

Interpolation places the quartile precisely within the class, not at the midpoint.

Mixing up the cumulative frequency before the class with the class frequency.

Subtract the cf BEFORE the class from the position; divide by THAT class’s frequency.

Key takeaways

Q₁, Q₂, Q₃ sit ¼, ½, ¾ through the data; percentiles divide into hundredths.
For grouped data, estimate with linear interpolation.
Q = L + ((position − cf)/f) × class width.

Test yourself

Answer a few questions, then review anything you missed.

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