The mean (or average) is the sum of all data values divided by the number of values. It is represented by \(\bar{X}\). The mean can be calculated by two methods.
1. Direct Method
Formula for Continuous Data:
\(\begin{aligned}
& \bar{X} = \frac{\sum f m}{N}
\end{aligned}\)
where:
- \( m \) = Midpoint of each class interval
- \( f \) = Frequency of each class
- \( \sum f m \) = Sum of the product of midpoints and frequencies
- \( \sum f = N \) = Total frequency
2. Indirect Method
Formula for Continuous Data:
\(\begin{aligned}
\bar{X} = A + \frac{\sum fd}{N}
\end{aligned}\)
where:
- \( A \) = Assumed Mean
- \( d = m - A \) = Difference of Midpoint and Assumed mean
Example Questions
Q. Find the mean of the following data.
Marks
10-20
20-30
30-40
40-50
50-60
60-70
No. of Students
3
5
6
7
4
2
Solution:
Marks
Students (\( f \))
Mid-Point (\( m \))
\( f \times m \)
10-20
3
15
45
20-30
5
25
125
30-40
6
35
210
40-50
7
45
457
50-60
4
55
220
60-70
2
65
130
\(\sum f = N =27\)
\(\sum fm =1187 \)
Now, we have,
\(\bar{X} = \frac{\sum f m}{N}\)
\(or, \bar{X} = \frac{1187}{27}\)
\(\therefore \bar{X} = 43.96\) (Answer)
Median
The median is the middle value when the data is arranged in order. It is represented by \(M_d\).
- \( cf \) = Cumulative frequency before the median class
- \( f \) = Frequency of the median class
- \( h \) = Class width
Mode
The mode of a data set is the data with the highest frequency. For continuous data, the mode is the most frequently occurring class. It is represented by \(M_o\).
