The Mean of Grouped Data: Quick Revision

A brief note on The Mean of Grouped Data. Read it before a test. Try each question at the end before you open its answer.

Class marks and the direct method

• The class mark is the middle of a class: $x = \dfrac{\text{lower end} + \text{upper end}}{2}$. For an inclusive class such as $1$ to $10$, it is $5.5$.
• Direct method: $\bar{x} = \dfrac{\sum fx}{\sum f}$.
• The mean of grouped data is an estimate, because every value in a class is replaced by the class mark.

The assumed mean method

• Choose an assumed mean $a$ near the middle. The deviation is $d = x - a$.
• Mean: $\bar{x} = a + \dfrac{\sum fd}{\sum f}$.
• Any $a$ gives the same mean. A negative $\sum fd$ means that the mean is below $a$.

The step deviation method

• For classes of equal width $h$, the step deviation is $u = \dfrac{x - a}{h}$.
• Mean: $\bar{x} = a + h \times \dfrac{\sum fu}{\sum f}$.
• Use it when the class marks are large and the classes have equal width.

A worked example

The weights of $30$ parcels, in kg, fall in the classes $5$ to $15$, $15$ to $25$, $25$ to $35$, $35$ to $45$ and $45$ to $55$ with frequencies $4, 9, 12, 3, 2$. Find the mean.

1. Class marks and $a$. The class marks are $10, 20, 30, 40, 50$. Take $a = 30$ and $h = 10$.
2. Step deviations. $u = -2, -1, 0, 1, 2$. The products $fu$ are $-8, -9, 0, 3, 4$, and $\sum fu = -10$.
3. The mean. $\bar{x} = 30 + 10 \times \dfrac{-10}{30} = 26.67$ kg.

Practice with new numbers

A program to try

The program finds the mean of the parcel table by the direct method and by the step deviation method.

Mistakes to avoid

• Dividing by the number of classes instead of by $\sum f$.
• Using an end of the class instead of the class mark.
• Leaving out $h$ at the end of the step deviation method.
• Using the step deviation method with classes of unequal width.

Check yourself

1. What is the class mark of the class $35$ to $55$?

2. For $N = 40$, $a = 50$ and $\sum fd = -80$, find the mean.

3. Why must the classes have equal width for the step deviation method?

4. Is the mean of grouped data exact?

Sources

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