Class marks and the direct method

What you will be able to do

• read a grouped frequency table and find the class width and the class marks;
• change classes such as $1$ to $10$ and $11$ to $20$ into continuous classes;
• find the mean of grouped data by the direct method.

Why we group data

A teacher asks $40$ students how many minutes they take to reach school. The answers are $40$ different numbers, and nobody can read them aloud and remember them. So she makes groups. One group holds the students who take from $0$ to $10$ minutes, the next group holds those who take from $10$ to $20$ minutes, and so on. A group is called a class. The number of students in it is the frequency of the class. The difference between the two ends of a class is the class width. Here the width is $10$ minutes.

A value that falls on the end of a class, such as exactly $10$ minutes, is put in the upper class, here $10$ to $20$. This is the usual rule. The table has $N = 6 + 11 + 14 + 6 + 3 = 40$ students.

The class mark

In the table we no longer know each journey time. We only know that $14$ students took between $20$ and $30$ minutes. To go on, we make one assumption: every value in a class is equal to the middle of the class. The middle is the class mark.

For the class $20$ to $30$ the class mark is $\dfrac{20 + 30}{2} = 25$ minutes. Other books call it the mid-point or the mid-interval value. IGCSE and IB books say mid-point.

The direct method

For a short list of values, the mean is the sum divided by the number of values. In a grouped table, a class with frequency $f$ and class mark $x$ stands for $f$ values that are each $x$. Their total is $f \times x$. So we multiply each class mark by its frequency, add all the products, and divide by the total frequency:

The symbol $\sum$ is the Greek capital sigma and means "add up". The symbol $\bar{x}$ is read "x bar". We write $N$ for $\sum f$, the total number of values. Let us use a table with a column for the class mark $x$ and a column for the product $fx$.

The mean journey time is about $22$ minutes. Notice that it is not the middle class mark. It lies below $25$ minutes because the table has more students on the left of the middle class than on the right.

Why is the answer only an estimate? We pretended that all $14$ students in the class $20$ to $30$ took exactly $25$ minutes. Some took $21$ minutes and some took $29$ minutes. If we had the 40 original numbers, their true mean would be a little different. The difference is small because the errors above and below the class mark partly cancel. IGCSE and GCSE questions say "estimate the mean" for this reason.

When the classes have gaps

Marks are whole numbers, so a table of marks out of $50$ is often written $1$ to $10$, $11$ to $20$, $21$ to $30$ and so on. The class $1$ to $10$ holds the marks $1, 2, \dots, 10$, and the next class begins at $11$. There is a small gap between the classes. Such classes are called inclusive, because each includes both of its ends.

Two ways give the same class mark:

• take the class mark directly: $\dfrac{1 + 10}{2} = 5.5$;
• or close the gaps first. Move each lower end down by $0.5$ and each upper end up by $0.5$. The class becomes $0.5$ to $10.5$, and its middle is $\dfrac{0.5 + 10.5}{2} = 5.5$.

The numbers $0.5$ and $10.5$ are the class boundaries. A-Level and IB books use boundaries for the median and for histograms, and the next course needs them. For the mean, the class marks are all we use.

Watch out. Suppose a student uses $5$ as the first class mark, because the class starts at $1$ and ends at $10$. Then every class mark is wrong by the same $0.5$. The mean is wrong by $0.5$ too.

What other courses say

• IGCSE and GCSE ask you to estimate the mean from a grouped table. They use the mid-point of each class and the direct method.
• A-Level and IB also use the mid-point. A-Level calls the shortcut methods of the next two lessons coding.
• SAT Math and the Common Core data standards give lists, frequency tables and histograms. They ask for the mean and what it says about the data. The idea of $\dfrac{\sum fx}{\sum f}$ for a frequency table is the same.

Practise with new numbers

Try it in Python

A loop can do the multiplying and adding for us. This program goes through the five classes of the journey table. For each class it finds the class mark, multiplies it by the frequency, and adds the product to a running total.

Common mistakes

• Using the end of the class instead of the class mark. Every value in the class is taken to be in the middle, not at an end.
• Dividing by the number of classes. Divide by $\sum f$, the number of values. With five classes and $40$ values, the divisor is $40$.
• Forgetting to multiply by the frequency. Adding the class marks $5 + 15 + 25 + 35 + 45$ treats each class as holding one value.
• Taking $5$ as the class mark of $1$ to $10$. For inclusive classes, add the two ends and halve: $5.5$.

Remember these

• The class mark is the middle of the class: $\dfrac{\text{lower end} + \text{upper end}}{2}$.
• The direct method is $\bar{x} = \dfrac{\sum f x}{\sum f}$.
• The result for grouped data is an estimate, because each value was replaced by its class mark.
• For inclusive classes such as $1$ to $10$, the class mark is $5.5$, the same as for the boundaries $0.5$ and $10.5$.

Check yourself

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

1. A class runs from $15$ to $25$ minutes. What is its class mark?

2. Two classes have class marks $10$ and $30$ minutes, with frequencies $3$ and $1$. Find the mean.

3. What is the class mark of the inclusive class $51$ to $60$?

4. Why is the mean found from a grouped table only an estimate of the mean of the original values?

5. In a table, a class with class mark $25$ minutes has frequency $14$. What does the product $14 \times 25 = 350$ stand for?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the mean of grouped data by the direct method.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics: the mean of a frequency table and of grouped data.