Revision: Statistics >> Measures of Central Tendency (Mean, Median, Quartiles and Mode) Maths (English Medium) ICSE Class 10 CISCE

The mean is the value that is derived by summing all the values and dividing it by the number of observations.

`bar"x" = "Sum of observations"/"No. of observations"`

The arithmetic mean (or, simply, mean) of a set of numbers is obtained by dividing the sum of the numbers in the set by the number of numbers.

\[\mathbf{Mean}=\frac{\left(x_1+x_2+x_3+\ldots+x_n\right)}{n}=\frac{\Sigma x_i}{n}\] 

Median is the value of the middle-most observation(s). The median is a measure of central tendency which gives the value of the middle-most observation in the data.

The observations which divide the whole set of observations into four equal parts are known as quartiles.

Before finding quartiles, the given data must always be arranged in ascending order of magnitude.

The difference between the largest and smallest values in a data set is called the range.

Range = Largest value − Smallest value

The mode is the value of the observation that occurs most frequently; i.e., the observation with the maximum frequency is called the mode.

Direct Method:

\[\bar{x}=\frac{\sum f_ix_i}{\sum f_i}\]

where x_i​ = class mark, f_i​ = frequency

Short-cut (Assumed Mean) Method:

\[\bar{x} = A+\frac{\sum f_id_i}{\sum f_i}\]

where d_i = x_i - A
A is the assumed mean

Step-deviation Method:

\[\bar{x}=a+h\frac{\sum f_iu_i}{\sum f_i}\]

where \[u_i=\frac{x_i-a}{h}\]

h is the class width / common factor

If the number of data points (n) is odd, the median is,

Median = `((n+1)/2)^(th)` term

If n is even, the median is the average of the values at positions

Median = Average of  `(n/2)^(th)` and `(n/2+1)^(th)` values

Inter-quartile range:

The difference between the upper quartile (Q₃) and the lower quartile (Q₁) is called the inter-quartile range.

Inter-quartile range = Q₃ − Q₁

• It is always positive, since Q₃ > Q₁.

Semi-interquartile range:

Half of the inter-quartile range is called the semi-interquartile range.

Semi-interquartile range = `1/2` (Q₃ − Q₁)

Case I: When n is ODD

• Lower Quartile, Q₁ = (n + 1) / 4 th term

• Upper Quartile, Q₃ = 3(n + 1) / 4 th term

Case II: When n is EVEN

• Lower Quartile, Q₁ = n / 4 th term

• Upper Quartile, Q₃ = 3n / 4 th term

Types of Quartiles

1. Lower Quartile (Q₁)
   The observation lies midway between the lowest value and the median.

2. Middle Quartile (Q₂)
   The median of the data.

3. Upper Quartile (Q₃)
   The observation lies midway between the median and the highest value.