The score in Maths of 50 students of a class is given in the following table. Calculate the mean score.
Interval Frequencies
0-10 2
10-20 11
20-30 28
30-40 6
40-50 3

Question

The score in Maths of 50 students of a class is given in the following table. Calculate the mean score.IntervalFrequencies0-10210-201120-302830-40640-503

Prepp Team · Accepted Answer

Calculating Mean Score from Grouped Frequency Data

The problem asks us to calculate the mean score from a grouped frequency distribution table representing the Maths scores of 50 students. To find the mean for grouped data, we use the following formula:

\text{Mean} = \frac{\sum f \times x}{\sum f}

Where:

\(\sum f\) is the sum of frequencies (total number of students).
\(\sum f \times x\) is the sum of the product of each frequency (\(f\)) and its corresponding class midpoint (\(x\)).

First, we need to find the midpoint (\(x\)) for each class interval. The midpoint is the average of the lower and upper limits of the interval. Then, we multiply each frequency (\(f\)) by its corresponding midpoint (\(x\)) to get \(f \times x\).

Interval	Frequency (\(f\))	Midpoint (\(x\))	\(f \times x\)
0-10	2	\(\frac{0+10}{2} = 5\)	\(2 \times 5 = 10\)
10-20	11	\(\frac{10+20}{2} = 15\)	\(11 \times 15 = 165\)
20-30	28	\(\frac{20+30}{2} = 25\)	\(28 \times 25 = 700\)
30-40	6	\(\frac{30+40}{2} = 35\)	\(6 \times 35 = 210\)
40-50	3	\(\frac{40+50}{2} = 45\)	\(3 \times 45 = 135\)

Now, we sum the frequencies (\(\sum f\)) and the products (\(\sum f \times x\)):

\(\sum f = 2 + 11 + 28 + 6 + 3 = 50\) (This matches the given total number of students).
\(\sum f \times x = 10 + 165 + 700 + 210 + 135 = 1220\)

Finally, we calculate the mean using the formula:

\text{Mean} = \frac{\sum f \times x}{\sum f} = \frac{1220}{50}

To simplify the calculation:

\text{Mean} = \frac{122}{5} = 24.4

The calculated mean score is 24.4.

Steps to Calculate Mean from Frequency Table

Determine the midpoint (\(x\)) for each class interval.
Multiply the frequency (\(f\)) of each class by its midpoint (\(x\)) to get \(f \times x\).
Sum all the frequencies (\(\sum f\)).
Sum all the \(f \times x\) values (\(\sum f \times x\)).
Divide the sum of \(f \times x\) by the sum of \(f\).

Revision Table: Mean Calculation Formulas

Data Type	Formula for Mean	Notes
Ungrouped Data	\(\bar{x} = \frac{\sum x}{n}\)	Sum of all observations divided by the number of observations.
Discrete Frequency Distribution	\(\bar{x} = \frac{\sum f \times x}{\sum f}\)	Sum of (frequency × value) divided by sum of frequencies.
Grouped Frequency Distribution	\(\bar{x} = \frac{\sum f \times x}{\sum f}\)	Sum of (frequency × midpoint) divided by sum of frequencies. Midpoint represents the class value.

Additional Information: Measures of Central Tendency

The mean is one of the key measures of central tendency. For grouped data, other important measures include the median and the mode.

Median: The middle value when the data is arranged in order. For grouped data, it is calculated using a specific formula involving cumulative frequencies and the median class.
Mode: The value that appears most frequently. For grouped data, it is the midpoint of the modal class (the class with the highest frequency) or calculated using a specific formula involving the modal class and frequencies of adjacent classes.

Understanding how to calculate the mean score from a frequency distribution is fundamental in statistics for summarizing data.