Cumulative Frequency Curves: Quick Revision

A brief note on Cumulative Frequency Curves. Read it before a test. Try each question at the end before you open its answer.

The two curves

• The less than total of a limit counts the values below it. Plot it at the upper limit. The curve rises from $0$ to $N$.
• The more than total of a limit counts the values above it. Plot it at the lower limit. The curve falls from $N$ to $0$.
• At every limit, less than plus more than equals $N$.
• The two curves cross at the height $\dfrac{N}{2}$. The horizontal reading of the crossing is the median.

Reading the curve

• Median: height $\dfrac{N}{2}$. Lower quartile: height $\dfrac{N}{4}$. Upper quartile: height $\dfrac{3N}{4}$. The $p$th percentile: height $\dfrac{p}{100}N$.
• Interquartile range: $Q_3 - Q_1$.
• The number of values between two limits is the difference of the readings of the curve.
• A reading at height $H$ is $l + \dfrac{H - cf}{f} \times h$ in the class where the running total reaches $H$.

Missing frequencies

• Call the missing frequency $x$. Write the mean, the median or the total as a formula in $x$. Solve, and check.
• For the median, decide the median class first, then check it.
• A frequency is a whole number that is not negative. Two missing frequencies need two equations.

A worked example

The frequencies $5, 11, 16, 8$ are in the classes $0$ to $20$, $20$ to $40$, $40$ to $60$ and $60$ to $80$. Find the quartiles and the interquartile range.

1. The running totals. $5, 16, 32, 40$, and $N = 40$.
2. $Q_1$. The height is $10$, in the class $20$ to $40$: $Q_1 = 20 + \dfrac{10 - 5}{11} \times 20 = 29.09$.
3. $Q_3$. The height is $30$, in the class $40$ to $60$: $Q_3 = 40 + \dfrac{30 - 16}{16} \times 20 = 57.5$.
4. The interquartile range. $57.5 - 29.09 = 28.41$.

Practice with new numbers

A program to try

The program reads the two quartiles and the median from the curve of the table.

Mistakes to avoid

• Plotting at the class mark instead of the upper limit.
• Mixing up the two types. Less than rises. More than falls.
• Using $\dfrac{N}{4}$ for the upper quartile. It is at $\dfrac{3N}{4}$.
• Forgetting that $N$ contains the missing frequency.

Check yourself

1. A table has $N = 60$. At what height do the two curves cross?

2. The quartiles of a set are $14$ and $31$. What is the interquartile range?

3. The curve is at height $9$ at $20$ and at height $24$ at $50$. How many values lie between $20$ and $50$?

4. The mean of a table with a missing frequency $x$ is given. What is the first thing to write?

Sources

• NCERT, Mathematics for Class 10, the chapter Statistics.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics.