Central Tendency in Statistics- Mean, Median, Mode

Central Tendencies are the numerical values that are used to represent a large collection of numerical data. These numerical values are called measures of central tendency or average values. A central or average value of any statistical data or series is the variable's value representative of the entire data or its associated frequency distribution.

Some commonly used measures of central tendency are:

• Mean
• Median
• Mode

Mean

The mean is a statistical measure used to determine the average value in a dataset. It represents the central or typical value of the data. While the arithmetic mean is the most common, there are also other types like the geometric mean and harmonic mean. These are calculated differently, depending on the situation. It is sensitive to outliers, i.e, if there are extreme values in our dataset then it could cause a different mean rather than the typical mean.

Note: Outliers are data points that differ significantly from the majority of the data, often appearing as extreme values

1. Arithmetic Mean: This is just the sum of all the values in the dataset, divided by how many values there are. It's the most commonly used average.

\bar{x} = \frac{\sum x_i}{N}

Where:

• \Sigma x_i is the sum of all the observations.
• N is the total number of observations.

2. Geometric Mean: This one is calculated by multiplying all the values together and then taking the nnn-th root of the total, depending on how many values you have.

\text{G.M.} = \left( \prod_{i=1}^n x_i \right)^{\frac{1}{n}}

​Where:

• x_1,x_2,\dots,x_n ​ are the values.
• n is the number of values.

3. Harmonic Mean: This is the average of the reciprocals of the values, which is helpful when dealing with things like rates or speeds.

\text{H.M.} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}

Where:

• x_1,x_2,\dots,x_n ​ are the values.
• n is the number of values.

Mean for Ungrouped Data

Mean for ungrouped data, also known as the arithmetic mean (\bar{x}), is calculated by summing up all individual values in the dataset and dividing the sum by the total number of values. This provides a single representative value that reflects the central tendency of the data. It's commonly used to understand the average value or typical value of a given set of observations.

Formula for same is,

\bold{\bar{x} = \frac{\sum x_i}{N}}
OR
Mean = Sum of all Observations ÷ Total number of Observations

Example: If there are 5 observations, which are 27, 11, 17, 19, and 21 then the mean (\bar{x}  ) is given by

\bar{x} = (27 + 11 + 17 + 19 + 21) ÷ 5
⇒ \bar{x} = 95 ÷ 5
⇒ \bar{x} = 19

Mean for Grouped Data

Mean (\bar{x}) is defined for the grouped data as the sum of the product of observations (x_i) and their corresponding frequencies (f_i) divided by the sum of all the frequencies (f_i).

\bold{\bar{x} = \frac{\sum f_i x_i}{\sum f_i}}

Example: If the values (x_i) of the observations and their frequencies (f_i) are given as follows:

then Arithmetic mean (\bar{x}) of the above distribution is given by

\bar{x} = (4×5 + 6×10 + 15×8 + 10×7 + 9×10) ÷ (5 + 10 + 8 + 7 + 10)
⇒ \bar{x} = 360 ÷ 40
⇒ \bar{x} = 9

Properties of Mean (Arithmetic)

There are various properties of Arithmetic Mean, some of which are as follows:

• Sum of Deviations: The sum of deviations from the mean is always zero:

\sum (x_i - \bar{x}) = 0

• Adding a Constant: If a constant a is added to each observation, the new mean becomes:

\bar{x'} = \bar{x} + a

• Subtracting a Constant: If a constant a is subtracted from each observation, the new mean becomes:

\bar{x'} = \bar{x} - a

• Multiplying by a Constant: If each observation is multiplied by a, the new mean becomes:

\bar{x'} = \bar{x} \times a

• Dividing by a Constant: If each observation is divided by a, the new mean becomes:

\bar{x'} = \frac{\bar{x}}{a}

Median

The median is the middle value in a set of numbers when they’re arranged in order, either from smallest to largest or vice versa. If there’s an odd number of numbers, the median is the one right in the middle. If there’s an even number, the median is the average of the two middle numbers.

Median of Ungrouped Data

To calculate the Median, the observations must be arranged in ascending or descending order. If the total number of observations is N then there are two cases

Case 1: N is Odd

Median = Value of observation at [(n + 1) ÷ 2]^th Position

\text{Median} = \left\lfloor \frac{(n+1)}{2} \right\rfloor^{th} \text{Term}

Case 2: N is Even

Median = Arithmetic mean of Values of observations at (n ÷ 2)^th and [(n ÷ 2) + 1]^th Position

\text{Median} = \frac{\left[ \left( \frac{n}{2} \right)^{th} \text{ Term} + \left\{ \left( \frac{n}{2} \right) + 1 \right\}^{th} \text{ Term} \right]}{2}

Example 1: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20, 32 then the Median is given by 

Arranging data in ascending order: 20, 22, 23, 25, 26, 28, 31, 32, 36, 38
N = 10 which is even then
Median = Arithmetic mean of values at (10 ÷ 2)th and [(10 ÷ 2) + 1]th position
⇒ Median = (26 + 28) ÷ 2
⇒ Median = 27

Example 2: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20 then the Median is given by

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 36, 38
N = 9 which is odd then
⇒ Median = Value at 5^th position
⇒ Median = 26

Median of Grouped Data

Median of Grouped Data is given as follows:

\bold{Median =l+ \frac{N/2 - c_f}{f} \times h}

where,

• l is Lower limit of median class
• n is Total number of observations
• c_f is Cumulative frequency of the preceding class
• f is Frequency of each class
• h is Class size

Example: Calculate the median for the following data.

Solution:

Create the following table for the given data.

Class	Frequency	Cumulative Frequency
10 - 20	5	5
20 - 30	10	15
30 - 40	12	27
40 - 50	8	35
50 - 60	5	40

As n = 40 and n/2 = 20,
Thus, 30 - 40 is the median class.

l = 30, c_f = 15, f = 12, and h = 10

Putting the values in the formula
Median = 30 + (20 - 15)/12) × 10
⇒ Median = 30 + 4.17
⇒ Median = 34.17

So, median value for this data set is 34.17

Mode

Mode is the value of the observation that occurs with the highest frequency. In other, that observation of the data occurs the maximum number of times in a dataset.

Mode of Ungrouped Data

Mode of Ungrouped Data can be simply calculated by observing the observation with the highest frequency. Let's see an example of the calculation of the mode of ungrouped data.

Example: Find the mode of observations 5, 3, 4, 3, 7, 3, 5, 4, 3.

Solution:

Create a table with each observation with its frequency as follows:

xi	5	3	4	7
fi	2	4	2	1

Since 3 has occurred a maximum number of times i.e. 4 times in the given data;
Hence, Mode of the given ungrouped data is 3.

Mode of Grouped Data

Formula to find the mode of the grouped data is:

\bold{Mode = l +\left [\frac{f_1-f_0}{2f_1-f_0-f_2}\right]×h}

where,

• l is the lower class limit of modal class
• h is the class size
• f₁ is the frequency of modal class
• f₀ is the frequency of class which proceeds the modal class
• f₂ is the frequency of class which succeeds the modal class

Example: Find the mode of the dataset which is given as follows.

Solution:

As class interval with the highest frequency is 40-50, which has a frequency of 16. Thus, 40-50 is the modal class. 
Thus, l  = 40 , h = 10 , f₁ = 16 , f₀ = 12 , f₂ = 10 

Plugging in the values in formula \bold{Mode = l +\left [\frac{f_1-f_0}{2f_1-f_0-f_2}\right]×h}     , we get
Mode = 40 + (16 - 12)/(2 × 16 - 12 - 10) × 10
⇒ Mode = 40 + (4/10)×10
⇒ Mode = 44

Therefore, mode for this set of data is 44.

Differences between Mean, Median and Mode

Lets see the differences between mean, median and mode,

Empirical Relation Between Measures of Central Tendency

Three central tendencies are related to each other by the empirical formula which is given as follows:

2 × Mean + Mode = 3 × Median

Formula is used to calculate one of the central tendencies when two other central tendencies are given.