Revision Notes: Measures of Central Tendency - Mean, Median and Mode

The numerical expressions which represent the characteristics of a group are called Measures of Central Tendency.
Three measures of central tendency are:
i. Mean
ii. Median
iii. Mode

Arithmetic Mean

The arithmetic mean is the sum of all observations in the data divided by the number of observations.'
Arithmetic Mean of n numbers x₁, x₂, x₃, ... , x_n =

The Greek letter ∑ (called sigma) represents the sum of numbers.
The arithmetic mean may be computed by any of the following methods:
i. Direct method
ii. Short-cut method
iii. Step-deviation method

Direct method

If a variable X takes values x₁, x₂, x₃ .... , x_n with corresponding frequencies f₁, f₂, f₃ ,... f_n respectively, then the arithmetic mean of these values is given by,
Mean =

Short-cut method

This method is used to overcome the difficulty faced in calculations where large quantities are involved.
Let x₁, x₂ ,......., x_n be values at a variable x with corresponding frequencies f₁, f₂,.......,f_n respectively.
Takingthe  derivative at an arbitrary point 'A', we have
Mean = A +  where A = Assumed mean and d = x - A

Step-deviation method

Sometimes, during the application of the shortcut method for finding the arithmetic mean of the derivative d is divisible by a common number i (say).
In such cases, arithmetic is reduced to a great extent by taking u_i = t = then
Mean = A +

Median

• Median is the value of the middle observation(s).
• The median is to be calculated only after arranging the data in ascending order or descending order.

To find the Median for raw and arrayed data

To find the median of raw data, arrange the raw data in ascending or descending order. Then, observe the number of variables in the data. Let it be n. Then find the median as follows.
(a) If n is odd, then the  variate is the median.
(b) If n is even, then the mean of the n^th/2 and variates is the median, i.e.,

Median for tabulated data

• Construct a cumulative frequency distribution table
• If there are n terms in the given distribution, then use the table to find the value of (n/2)^th or term, which is the median of the given distribution.

Median for grouped data (both continuous and discontinuous)

• Draw a cumulative frequency curve (Ogive).
• If there are n terms in the given distribution, then use the ogive to find the value of (n/2)^th or  term, which is the median of the given distribution.

Mode

• The mode of a statistical dataset is the value of that variate which has the maximum frequency.
• The mode for ungrouped data is the value that occurs most often.
• The mode may be greater than, less than or even equal to the mean.

Mode for tabulated data

• In the case of a grouped frequency distribution, a class with the maximum frequency is called as the modal class.

To find the mode of a group frequency distribution (using a histogram)
Steps:

• Draw a histogram of the given distribution.
• Inside the highest rectangle, which represents the maximum frequency (or modal class), draw two lines AC and BD diagonally from the upper corners C and D of adjacent rectangles.
• The point of intersection is M. Now ML is perpendicular to the horizontal axis.
• The value of point L on the horizontal axis represents the value of the mode.
  

Quartiles

• The three variates which divide the data of a distribution in four equal (quarters) are called quartiles.
• If there are n terms arranged in an ascending order, then
  Lower Quartile (Q₁) = (n/4)^th term, if n is even or  term, if n is odd.
• If there are n terms arranged in an ascending order, then
  Upper Quartile (Q₃) = (3n/4)^th term, if n is even or  term, if n is odd.
• Q₂ is called the middle quartile, and the median is the second quartile.
• Lower Quartile is the value which cuts off the lowest 25% of the data. It is denoted by Q₁.
• Upper Quartile is that value which cuts off the highest 25% of the data. It is denoted by Q₃ and is the 75^th percentile.
• The difference between the greatest variate and the smallest variate in a distribution is called the range of the distribution.
• The difference between the lower quartile and the upper quartile is the Interquartile Range, and it is equal to Q₃ - Q₁.
• The interquartile range is always positive, as Q₃ > Q₁
• Semi-interquartile range is equal to 1/2(Q₃ - Q₁).