Central Tendency

Central Tendency is a fundamental concept in statistics that describes the center point or typical value of a dataset. It helps in summarizing data sets with a single value that represents the entire data. The three main measures of central tendency are:

1. Mean: The average of all data points, calculated by adding them up and dividing them by the number of points.
2. Median: The middle value in an ordered dataset. It is a measure of central tendency that is not affected by outliers.
3. Mode: The most frequently occurring value in a dataset, highlighting the most common data point.

1.0Central Tendency Definition

Central Tendency in Statistics is a concept used to identify the central or average value within a data set. It provides a single value that represents the typical or central value of the data distribution. The main three measures of central tendency are the mean, median, and mode.

2.0Types of Central Tendency

The three main types of Central Tendency are:

1. Mean: Also known as the average, the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

2. Median: The median is determined as the central value within a dataset once the values have been arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. It is less affected by extreme values and is useful for skewed distributions.

3. Mode: The mode is the value that appears most often in the data set.It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

These measures offer important insights into the central tendencies of a dataset, helping researchers and analysts understand its characteristics and make informed decisions.

Arithmetic Mean

1. For Ungrouped Distribution

The arithmetic mean, often simply called the mean, is a measure of central tendency of statistics that represents the average of a set of numerical values. It is calculated by adding up all the values in the data set and then dividing the sum by the total number of values.

The formula for calculating the arithmetic $mean(\overline{\mathrm{x}})$ of a data set x₁, x₂, ..., x_n with n observations is:

$\overline{\mathrm{x}}=\frac{{\mathrm{x}}_1+{\mathrm{x}}_2+\dots +{\mathrm{x}}_{\mathrm{n}}}{\mathrm{n}}$

In simpler terms, you add up all the values and then divide by how many values there are. The mean provides a single value that summarizes the central tendency of the data set. It is widely used in various fields for summarizing and analyzing data. However, it can be influenced by extreme values (outliers) in the data set, so it's important to consider the context of the data when interpreting the mean.

2. For Ungrouped Frequency Distribution 

For ungrouped frequency distributions, where each data point has a corresponding frequency (how often it occurs), the arithmetic mean is calculated slightly differently. 

Let's say we have a set of n data points x₁, x₂, ..., x_n with corresponding frequencies f₁, f₂, ..., f_n. To find the $mean(\overline{\mathrm{x}})$ , you follow these steps:

1. Multiply each data point by its respective frequency.

2. Sum up all these products.

3. Divide the sum by the total frequency (sum of all frequencies).

The formula for calculating the mean for ungrouped frequency distributions is:

$\overline{\mathrm{x}}=\frac{{\mathrm{f}}_1{\mathrm{x}}_1+{\mathrm{f}}_2{\mathrm{x}}_2+\dots .+{\mathrm{f}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}{\left({\mathrm{f}}_1+{\mathrm{f}}_2+\dots ..+{\mathrm{f}}_{\mathrm{n}}\right)}$

$\bar{x}=\frac{{\sum }_{i=1}^n{f}_i{x}_i}{{\sum }_{i=1}^n{f}_i}\text{ or }\bar{x}=\frac{{\sum }_{i=1}^n{f}_i{x}_i}{N}\text{ Where }N={\sum }_{i=1}^n{f}_i$

In other words, you are finding the weighted average of the data points, where each data point is weighted by its frequency. This accounts for the fact that some values may occur more frequently than others in the data set.

Two Interesting Properties of A.M. 

1. The sum of deviations of data points from the arithmetic mean is always zero. Mathematically, this is represented as $\sum (\mathrm{x}-\overline{\mathrm{x}})=0$.
2. The arithmetic mean is sensitive to extreme values. Any significantly large or small value can skew the mean upward or downward.

Weighted Mean (W.M.)

If ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots \dots ,{\mathrm{x}}_{\mathrm{n}}$ are n values of variate $\overline{\mathrm{X}}$ and their weightages are ${\mathrm{w}}_1,{\mathrm{w}}_2,\dots \dots ,{\mathrm{w}}_{\mathrm{n}}$ respectively, then their weighted mean is -

$\text{ W.M. }=\frac{{\mathrm{w}}_1{\mathrm{x}}_1+{\mathrm{w}}_2{\mathrm{x}}_2+\dots \dots .+{\mathrm{w}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}{{\mathrm{w}}_1+{\mathrm{w}}_2+\dots \dots .+{\mathrm{w}}_{\mathrm{n}}}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}}$

Median

The median serves as an additional measure of central tendency, similar to the mean, but it represents the middle value of a data set when arranged in ascending or descending order. To find the median:

1. Arrange the data set in either ascending or descending order.
2. When the number of observations (n) is odd, the median corresponds to the value positioned at the center of the ordered list.
3. If the total number of observations (n) is even, the median is calculated as the arithmetic mean of the two middle values.
4. If n is the number of observations:
5. For Ungrouped Distribution and Ungrouped Frequency Distribution

• For odd $n:\text{ Median }=\text{ Value of }{\left(\frac{\mathrm{n}+1}{2}\right)}^{th}\text{observation. }$
• For even $n:\text{ Median }=\frac{\text{ Value of }{\left(\frac{n}{2}\right)}^{th}\text{observation }+\text{ Value of }{\left(\frac{n}{2}+1\right)}^{th}\text{ observation }}{2}$.

The median is often preferred over the mean when the data set contains outliers or is skewed, as it is less affected by extreme values. It provides a better representation of the typical value in such cases.

2. For Grouped Frequency Distribution

When dealing with a grouped frequency distribution, where data is organized into intervals or classes along with their corresponding frequencies, finding the median involves a slightly different approach.

Here's how to find the median for a grouped frequency distribution:

1. Find the Median Class: Determine the class interval containing the median. This is usually the class where the cumulative frequency exceeds half of the total frequency.
2. Calculate Median: Use the formula for finding the median within a class interval:
3. $\text{ Median }=l+\frac{\left(\frac{N}{2}-F\right)}{f}\times h$

where:

l = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency (c. f) of the class before the median class

f = Frequency of the class interval that contains the median.

h = Width of the class interval 

This formula calculates the median by finding a value within the median class, considering the width of the class interval and the cumulative frequencies.

If the median class is not a continuous interval (e.g., if it's a single value), you can still use finding a value within that interval using similar principles.

Once you find the median, it represents the middle value of the data set. It's a useful measure of central tendency, especially for data sets with continuous values.

Mode

Mode is the value of the variate, which has maximum Frequency. 

(i) for ungrouped distribution

The value of the variate which repeated the maximum number of times.

(ii) for ungrouped frequency distribution. 

The value of the variate which has the maximum frequency.

(iii) for grouped frequency distribution.

First, we find the class with the maximum frequency. This is a modal class, then

$\text{ Mode }=\mathrm{I}+\frac{\left({\mathrm{f}}_0-{\mathrm{f}}_1\right)}{\left(2{\mathrm{f}}_0-{\mathrm{f}}_1-{\mathrm{f}}_2\right)}\times \mathrm{h}$

Where l = lower limit of modal class.

f₀ = frequency of modal class.

f₁ = frequency of the class preceding modal class.

f₂ = frequency of the class succeeding modal class.

h = class interval of modal class.

3.0Relation Between Central Tendency

In a moderately asymmetrical distribution following relation between mean, median and mode of a distribution. 

It is known as the Empirical Formula. 

Mode = 3 Median – 2 Mean

4.0Relative Position of Mean, Median, Mode

Relative Position of Arithmetic Mean(Mean), Median and Mode:-

If we denote: 

Arithmetic Mean = M_e 

Median = M_i 

Mode = M_o 

Then, their relative magnitudes are such that either M_e > M_i > M_o or M_e < M_i < M_o (with suffixes occurring in alphabetical order). It's noteworthy that the median always falls between the arithmetic mean and the mode.

5.0Solved Examples

Example 1: Mean of n observation $1^2,2^2,3^2,\dots ,n^2\text{ is }\frac{46n}{11}$. Then value of n is:

(A) 11 (B) 12   (C) 23 (D) 22  

Ans. (A)

Solution:

$\bar{x}=\frac{\sum x_i}{\mathrm{n}}\Rightarrow \frac{46n}{11}=\frac{1^2+2^2+\dots \dots n^2}{n}$

$\Rightarrow \frac{46n}{11}=\frac{n(n+1)(2n+1)}{6n}$

Solving

n = 11

Example 2: Find the A.M of the following frequency distribution

Solution:

Let a = 900, h = 20

∵  $\bar{x}=a+\frac{\sum f_i{u}_i}{N}\times h$

₌ $900+\frac{(-40)}{100}\times 20$

= 900 – 8 

= 892

Example 3: Find the weighted mean of first n natural numbers when their weights are equal to their squares respectively

Solution:

$WeightedMean=\frac{1.1^2+2.2^2+\dots .+n\cdot n^2}{1^2+2^2+\dots .+n^2}=\frac{1^3+2^3+\dots .+n^3}{1^2+2^2+\dots .+n^2}=\frac{[n(n+1)/2{]}^2}{[n(n+1)(2n+1)/6]}=\frac{3n(n+1)}{2(2n+1)}$ 

Example 4: Find the median of the following frequency distribution.

Solution:

Here  $\frac{\mathrm{N}}{2}=\frac{30}{2}=15$

Median class is 20.5 – 30.5

∴ Median = $20.5+\frac{(15-11)}{8}\times 10$

= 20.5 + 5 = 25.5

Example 5: Find the mode of the following frequency distribution.

Solution:

Here the variate 7 has maximum frequency.

∴ Mode = 7

Example 6: Find the mode of following frequency distribution.

Solution:

Here model class is 20 – 30 (which have maximum frequency)

$\text{ Mode }=l+\frac{\left({\mathrm{f}}_0-{\mathrm{f}}_1\right)}{\left(2{\mathrm{f}}_0-{\mathrm{f}}_1-{\mathrm{f}}_2\right)}h$

= $\text{ Mode }=l+\frac{\left({\mathrm{f}}_0-{\mathrm{f}}_1\right)}{\left(2{\mathrm{f}}_0-{\mathrm{f}}_1-{\mathrm{f}}_2\right)}h$

= 20 + 2.5 

= 22.5

6.0Practice Questions

1. If the mean of the numbers 27 + x, 31 + x, 89 + x, 107 + x,156 + x is 82, then the mean of

130 + x, 126 + x, 68 + x, 50 + x, 1 + x is

(A) 75 (B) 157 (C) 82 (D) 80

2. Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is

(A) 48 (B) 82$\frac{1}{2}$ (C) 50 (D) 80

3. The mean weight per student in a group of seven students is 55 kg. If the individual weights of 6 students are 52, 58, 55, 53, 56 and 54; then weights of the seventh student is

(A) 55 kg (B) 60 kg (C) 57 kg (D) 50 kg

4. The mode of the distribution

(A) 5 (B) 6 (C) 8 (D) 10

5. If the difference between mean and mode is 63, the difference between mean and median is 

(A) 189 (B) 21 (C) 31.5 (D) 48.5