CBSE Class 10 Mathematics Statistics Notes Set 04

Mode

Mode is the value of the observation having the maximum frequency.

Mode of Grouped Data

In a grouped frequency distribution it is not possible to find the mode by looking at the frequencies. We can only locate the class having the maximum frequency, called the modal class. The mode is a value inside the modal class and is calculated by

where \( l \) = lower limit of the modal class, \( h \) = width of the modal class, \( f_1 \) = frequency of the modal class, \( f_0 \) = frequency of the class preceding the modal class, \( f_2 \) = frequency of the class succeeding the modal class.

Median

The middle-most value of the given data when it is arranged in ascending or descending order of magnitude is called the median. The median is also called a positional average.
1. If the total number of observations n is odd, Median = value of the \( \left( \frac{n+1}{2} \right)^{th} \) observation.
2. If n is even, Median \( = \frac{\text{value of the } \left( \frac{n}{2} \right)^{th} \text{ term} + \text{value of the } \left( \frac{n}{2} + 1 \right)^{th} \text{ term}}{2} \).

Example 4. Find the median of the following data: 25, 34, 31, 23, 22, 26, 35, 28, 20, 24, 32, 33.
Answer:
Arranging the data in ascending order: 20, 22, 23, 24, 25, 26, 28, 31, 32, 33, 34, 35.
Here the number of observations \( n = 12 \) (even).
Median \( = \frac{\text{value of the 6th observation} + \text{value of the 7th observation}}{2} = \frac{26 + 28}{2} = 27 \).

Example 5. Find the median of the following distribution: 37, 31, 42, 43, 46, 25, 39, 45, 32, 35, 49.
Answer:
Arranging the data in ascending order: 25, 31, 32, 35, 37, 39, 42, 43, 45, 46, 49.
Here the number of observations \( n = 11 \) (odd).
Median = value of the \( \left( \frac{11 + 1}{2} \right)^{th} \) observation = value of the 6th observation = 39.

Median of Grouped Data

In a grouped frequency distribution, the class corresponding to the cumulative frequency (c.f.) which is just greater than \( \frac{N}{2} \) is called the median class, and the median is obtained by the formula

where \( l \) = lower limit of the median class, \( c.f. \) = cumulative frequency of the class preceding the median class, \( f \) = frequency of the median class, \( h \) = class size and \( N = \sum f_i \).
Cumulative frequency is the running total of the frequencies. It is of two types: less than type, in which the frequencies of all the preceding classes are added to the frequency of the class, and more than type, in which the frequencies of all the succeeding classes are added to the frequency of the class.

Knowledge Hub

1. The sum of the absolute deviations taken from the median is less than the sum of the absolute deviations taken from any other observation in the data.
2. The median is not affected by extreme values (outliers).
3. Empirical relation: Mode = 3 Median − 2 Mean.
4. If the median class happens to be the first class interval, the c.f. of the preceding class is taken as zero.

Summary

1. Mean is the sum of all the observations divided by the total number of observations. For grouped data: direct method \( \bar{x} = \frac{\sum f_ix_i}{\sum f_i} \); assumed mean method \( \bar{x} = a + \frac{\sum f_id_i}{\sum f_i} \); step-deviation method \( \bar{x} = a + \frac{\sum f_iu_i}{\sum f_i} \times h \). Here \( \sum f_i \) is the total frequency, a is the assumed mean, \( d_i = x_i - a \), \( u_i = \frac{x_i - a}{h} \) and h is the class size.
2. Mode is the value of the observation having the maximum frequency. For grouped data, \( \text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \), where the class with the maximum frequency is the modal class.
3. Median is the middle-most value of the data when arranged in order. For n odd it is the \( \left( \frac{n+1}{2} \right)^{th} \) observation; for n even it is the mean of the \( \left( \frac{n}{2} \right)^{th} \) and \( \left( \frac{n}{2} + 1 \right)^{th} \) observations.
4. For grouped data, \( \text{Median} = l + \frac{\frac{N}{2} - c.f.}{f} \times h \), where the median class is the class whose cumulative frequency is just greater than \( \frac{N}{2} \).