Less than and more than curves

What you will be able to do

• build the less than and the more than cumulative frequency tables;
• draw the two curves from the tables;
• explain why the curves cross at the median.

From a running total to a picture

A school asks $50$ students how far they cycle in a week. The table groups the answers.

A running total tells us how many students cycled less than a given distance. A graph of the running totals shows this for every distance at once, and we can read answers off it. Such a graph is called an ogive, which is said "oh-jive". Books also call it a cumulative frequency curve or cumulative frequency diagram.

The less than curve

The less than cumulative frequency of a limit is the number of values that are below it. The first class holds $5$ students who cycled less than $10$ km. Add the next class: $5 + 12 = 17$ students cycled less than $20$ km. Go on adding.

Plot each pair as a point: the upper limit on the horizontal axis and the cumulative frequency on the vertical axis. We use the upper limit, because all the values of a class are below it, but only some are below the class mark. Join the points. The curve starts at the lowest lower limit, at height $0$, and ends at height $N = 50$. It never goes down.

The more than curve

The more than cumulative frequency of a limit is the number of values that are above it. Everyone cycled more than $0$ km, so the first entry is $50$. Take away the first class: $50 - 5 = 45$ students cycled more than $10$ km. Take away the next class: $45 - 12 = 33$ cycled more than $20$ km.

This time use the lower limit of each class, because the values counted are above it. The curve starts at height $N = 50$ and falls to $0$. For any limit, the more than total equals $N$ minus the less than total.

The two curves cross at the median

Draw both curves on one graph. At any distance, the number of students below it and the number above it add up to $50$. At the point where the curves cross, the two numbers are equal. Each must be $\dfrac{50}{2} = 25$. So the crossing is at height $\dfrac{N}{2}$, and its horizontal position is the median: half of the students are below it and half are above it.

In the graph the crossing is at $24.71$ km. The median formula gives the same number: the median class is $20$ to $30$, and $20 + \dfrac{25 - 17}{17} \times 10 = 24.71$ km.

Straight lines or a smooth curve

In this book we join the points with straight lines. Then the reading at any height is the same number that the median formula gives, because the formula also assumes that values are spread evenly through each class. NCERT and IGCSE ask for a smooth free-hand curve. A smooth curve gives almost the same readings, but a small difference is to be expected, so a reading from a hand drawn curve is correct if it is close.

Inclusive classes. For classes such as $1$ to $10$ and $11$ to $20$, plot at the boundaries: less than $10.5$, less than $20.5$ and so on. The first point is at $0.5$ with height $0$.

Other names. Cambridge IGCSE, GCSE and A-Level books mostly use the less than type and call it the cumulative frequency graph or diagram. The more than type is mostly met in Indian books.

Practise with new numbers

Try it in Python

This program builds both cumulative frequency tables for the cycling data and draws the two curves.

Common mistakes

• Plotting at the class mark. For a less than curve use the upper limit. For a more than curve use the lower limit.
• Leaving out the starting point. The less than curve starts at height $0$, at the lowest lower limit.
• Mixing the two types. A less than total goes up as the distance grows. A more than total goes down.
• Adding the wrong way. Each more than total is $N$ minus the less than total at the same limit.

Remember these

• The less than curve plots the upper limits against the running totals. It rises from $0$ to $N$.
• The more than curve plots the lower limits against $N$ minus the running totals. It falls from $N$ to $0$.
• At every limit, less than plus more than equals $N$.
• The two curves cross at the height $\dfrac{N}{2}$, and the horizontal position of the crossing is the median.

Check yourself

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

1. The frequencies of a table with classes $0$ to $10$, $10$ to $20$, $20$ to $30$ are $4$, $9$ and $7$. Write the less than cumulative frequencies at $10$, $20$ and $30$.

2. For the table of question 1, write the more than cumulative frequencies at $0$, $10$ and $20$.

3. Why is the less than curve drawn with the upper limits?

4. A table has $N = 64$. At what height do the two curves cross?

5. What is the number that the horizontal reading of the crossing gives?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the graphical representation of cumulative frequency, the ogive. Some boards have removed it from the syllabus, so check your own.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics. Cambridge IGCSE Mathematics uses the cumulative frequency diagram.