Statistics
Class 10: Mathematics
Mean, Median, Mode, Quartiles for continuous series data, Worked out example Question, See Questions with Answers
Mean (Arithmetic Mean)
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:
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:
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 \(M_d\).
Formula for Continuous data:
where:
- \( L \) = Lower boundary of the median class
- \( N \) = Total frequency
- \( cf \) = 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 \(M_o\).
Formula for Continuous Data:
where:
- \( L \) = Lower boundary of modal class
- \( f_1 \) = Frequency of modal class
- \( f_0 \) = Frequency of class before modal class
- \( f_2 \) = Frequency of class after modal class
- \( h \) = Class width
