Reading a curve: median and quartiles

What you will be able to do

• read the median and the quartiles from the curve;
• find how many values lie between two limits;
• find the value that a given share of the data lies above or below.

Reading the curve

To read a value from the curve of the cycling students, choose a height on the vertical axis. Go across to the curve, then down to the horizontal axis.

• The height $\dfrac{N}{2} = 25$ gives the median, which is $24.71$ km.
• The height $\dfrac{N}{4} = 12.5$ gives the lower quartile $Q_1$. A quarter of the students are below it.
• The height $\dfrac{3N}{4} = 37.5$ gives the upper quartile $Q_3$. A quarter of the students are above it.

Because the curve is made of straight segments, each reading can also be found by a formula. It is the median formula with another height in place of $\dfrac{N}{2}$:

For $Q_1$: the height $12.5$ lies in the class $10$ to $20$, where the running total goes from $5$ to $17$. So $Q_1 = 10 + \dfrac{12.5 - 5}{12} \times 10 = 16.25$ km. For $Q_3$: the height $37.5$ lies in the class $30$ to $40$, where the running total goes from $34$ to $44$. So $Q_3 = 30 + \dfrac{37.5 - 34}{10} \times 10 = 33.5$ km.

The interquartile range

The interquartile range is $Q_3 - Q_1$. It is the width of the middle half of the data. For the cyclists:

Half of the students cycled a distance in a range that is $17.25$ km wide. Unlike the range, which uses the two extreme values, the interquartile range hardly changes when one ride is very long. IGCSE, GCSE, A-Level and IB use it, and also the percentiles: the $90$th percentile is the value that $90$ percent of the data lie below. It is read at the height $0.9N$.

How many lie between two limits?

To count the students who cycled between $15$ km and $35$ km, read the curve at both distances and subtract.

• At $15$ km: halfway through the class $10$ to $20$, so $5 + 0.5 \times 12 = 11$ students are below.
• At $35$ km: halfway through the class $30$ to $40$, so $34 + 0.5 \times 10 = 39$ students are below.

So $39 - 11 = 28$ students cycled between $15$ km and $35$ km.

Which value cuts off a share?

The top $20\%$. Which distance did only the top $20\%$ of the students, that is, $10$ students, exceed? There are $40$ students below it, so read the height $40$. It lies in the class $30$ to $40$, so the distance is $30 + \dfrac{40 - 34}{10} \times 10 = 36$ km.

The share below a value. To find what percent of the students cycled less than $25$ km: at $25$ km the curve is at $17 + 0.5 \times 17 = 25.5$ students, which is $\dfrac{25.5}{50} = 51\%$.

Practise with new numbers

Try it in Python

NumPy has a function that does this reading. The program uses it to read the median, the quartiles and the interquartile range from the curve.

Common mistakes

• Reading the horizontal axis first. The heights on the vertical axis are numbers of values. The quartiles are found by choosing a height.
• Using $\dfrac{N}{4}$ for $Q_3$. The upper quartile is at the height $\dfrac{3N}{4}$.
• Taking the number of values in a class as the difference of the class marks. Subtract two readings of the same curve.
• Mixing up the range and the interquartile range. The range uses the two ends. The interquartile range uses $Q_3 - Q_1$.

Remember these

• Read a median at the height $\dfrac{N}{2}$, the quartiles at $\dfrac{N}{4}$ and $\dfrac{3N}{4}$, and the $p$th percentile at $\dfrac{p}{100}N$.
• The interquartile range is $Q_3 - Q_1$.
• The number of values between two limits is the difference of the two readings on the curve.
• With straight segments, a reading equals the value from the formula $l + \dfrac{\text{height} - cf}{f} \times h$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A table has $N = 80$. At which heights do you read the quartiles?

2. The quartiles of a data set are $18$ and $34$. What is the interquartile range?

3. The curve at $20$ km is at height $12$ and at $40$ km it is at height $31$. How many values lie between $20$ km and $40$ km?

4. For $N = 50$, which height gives the $90$th percentile?

5. Why is the interquartile range often a better measure of spread than the range?

References

• NCERT, Mathematics for Class 10, the chapter Statistics. Quartiles and percentiles are not in NCERT. They are in Cambridge IGCSE and GCSE Mathematics, and A-Level Statistics.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics: quartiles and the interquartile range.