Median, Quartiles and Mode | Mathematics (Maths) Class 11 - Commerce PDF Download

Median

The median of a distribution is the value of the middle variable when the variables are arranged in ascending or descending order.

Median (M_d) is an average of position of the numbers.

1. Median for Simple Distribution

Firstly, arrange the terms in ascending or descending order and then find the number of terms n.

(a) If n is odd, then (n + 1 / 2)th term is the median.

(b) If n is even, then there are two middle terms namely (n / 2)th and (n / 2 + 1)th terms. Hence,

Median = Mean of (n / 2)th and (n / 2 + 1)th terms.

2. Median for Unclassified Frequency Distribution

(i) First find N / 2, where N = Σ f_i.

(ii) Find the cumulative frequency of each value of the variable and take value of the variable which is equal to or just greater than N / 2

(iii) This value of the variable is the median.

3. Median for Classified Data (Median Class)

If in a continuous distribution, the total frequency be N, then the class whose cumulative frequency is either equal to N / 2 or is just greater than N / 2 is called median class.

For a continuous distribution, median

M_d = l + ((N / 2 – C) / f) * h

where, l = lower limit of the median class

f = frequency of the median class

N = total frequency = Σ f

C = cumulative frequency of the class just before the median class

h = length of the median class

Quartiles

The median divides the distribution in two equal parts. The distribution can similarly be divided in more equal parts (four, five, six etc.). Quartiles for a continuous distribution is given by

Q₁ = l + ((N / 4 – C) / f) * h

Where, N = total frequency

l = lower limit of the first quartile class

f = frequency of the first quartile class

C = the cumulative frequency corresponding to the class just before the first quartile class

h = the length of the first quartile class

Similarly, Q₃ = l + ((3N / 4 – C) / f) * h

where symbols have the same meaning as above only taking third quartile in place of first quartile.

Mode

The mode (M_o) of a distribution is the value at the point about which the items tend to be most heavily concentrated. It is generally the value of the variable which appears to occur most frequently in the distribution.

1. Mode for a Raw Data

Mode from the following numbers of a variable 70, 80, 90, 96, 70, 96, 96, 90 is 96 as 96 occurs maximum number of times.

2 For Classified Distribution

The class having the maximum frequency is called the modal class and the middle point of the modal class is called the crude mode.

The class just before the modal class is called pre-modal class and the class after the modal class is called the post-modal class.

Mode for Classified Data (Continuous Distribution)

M_o = l + (f₀ – f₁ / 2 f₀ – f₁ – f₂) x h

Where, 1 = lower limit of the modal class

f₀ = frequency of the modal class

f₁ = frequency of the pre-modal class

f₂ = frequency of the post-modal class

h = length of the class interval

Relation between Mean, Median and Mode

(i) Mean — Mode = 3 (Mean — Median)

(ii) Mode = 3 Median — 2 Mean

Symmetrical and Skew distribution

A distribution is symmetric, if the same number of frequencies is found to be distributed at the same linear teance on either side of the mode. The frequency curve is bell shaped and A = M_d = M_o

In anti-symmetric or skew distribution, the variation does not have symmetry.

(i) If the frequencies increases sharply at beginning and decreases slowly after modal value, then it is called positive skewness and A > M_d > M_o.

(ii) If the frequencies increases slowly and decreases sharply after modal value, the skewness is said to be negative and A < M_d < M_o.