Mean Absolute Deviation

Every day we come across a lot of information in the form of facts, numerical figures, tables, graphs, etc. These are provided by newspapers, television, magazines and other means of communication. These may relate to cricket batting or bowling averages, profits of a company, temperatures of cities, expenditures in various sectors of a five-year plan, polling results, and so on. These facts or figures, which are numerical or otherwise, collected with a definite purpose are called data. Data is the plural form of the Latin word datum. Of course, the word 'data' is not new for you. You have studied about data and data handling in earlier classes.

Mean Absolute Deviation

In simple words, ‘Mean’ refers to the average of the observation, and ‘Deviation’ refers to variation from previous data. Mean Absolute Deviation refers to the mean distance of each observation from the mean of the given data set.

Mean Absolute Deviation Formula:

Mean Absolute Deviation: \[\sum \frac{Absolute \: Values \: Deviation \: from \: Central \: Measure}{Total \: Number \; of \; Observations}\]

How to Calculate Mean Absolute Deviation ?

Steps to Find Mean Absolute Deviation:

Step 1: Find the mean of the given observations.

Step 2: Calculate the difference between each observation and the calculated mean.

Step 3: Evaluate the mean of the differences obtained in the second step.

Assume that the deviation from a central value is given as (x-a), where x is an observation of the data set. To determine the mean deviation, we need to find the average of all the deviations from a given data set. Since the measure of central tendency lies between the highest and lowest values of the data set, we can see that some deviations would be positive, and the rest would be negative. The sum of such variations would give a zero. Let us understand an example to make this point more transparent.

Examples:

Given Below is an Observation of Maximum Mark Scored By a Student:

x̅  = {∑_i=1 ⁿ nxᵢ}/n

= [{46+40+46+44}/{4}]

= 44

Mean Deviation = \[\sum \frac{Sum\;of\;Deviation\;from\;Mean}{Total\;Number\;of\;Observation}\]

Mean Deviation => 2 + (-4) + 2 = 0

The Mean Absolute Deviation can be calculated as:

Mean Absolute Deviation => \[\frac{2+|-4|+2}{4}\] = 2

This gives us a brief idea about the deviation of the observations from the measure of central tendency.

Central Tendency

The central tendency states the statistical measure representing the single value of the whole distribution or a dataset.

Measure Of Central Tendency

Various parameters give the measures of central tendency, but the most commonly used are mean, median, and mode.

Mean

Mean is generally used to measure central tendency. It represents the average of the assigned collection of data. It is suitable for discrete and continuous data.

It is equivalent to the sum of all the values in the set of data divided by the total number of values. Suppose we have ‘n’ set of data namely x1, x2, …...,xn.

\[x^{-}=\frac{x_{1}+x_{2}+....x_{n}}{n}=>x^{-}=\frac{\sum_{i=1}^{}nx_{i}}{n}\]

Median

Generally, the median depicts the mid-value of the given set of data when organized in a particular order.

Steps to find the median of a data set:

• The given data collection is arranged in ascending or descending order.

• If a quantity of values or observations in the given data is odd, then the median is given by \[(\frac{n+1}{2})^{th}\] observation.

• If the quantity of values or observations in the given data is even then the median is given by the average of \[(\frac{n}{2})^{th}\]and \[(\frac{n}{2}+1)^{th}\] observation.

Mode

It is the Most Frequently Occurring Data in the Data Set.

The maximum frequency observation is 78 since three students have scored 78 marks. Hence, so the mode of the given data collection is 78.

Examples

1. The provided chart presents the goals scored by different football players in a match. Find out the mean, median, and mode of the given data.

1. Mean:

x̅  = {∑_i=1 ⁿ xᵢ}/n

[{5+4+5+3+5+1+5}/{7}]

= 4

The mean of the given data is 4.

2. Median:

Since the number of items in the set of given data is odd in number, the median is \[(\frac{n+1}{2})^{th}\] observation.

Median = \[(\frac{7+1}{2})^{th}\] observation = 3

3. Mode:

The most frequent data is the mode i.e., 5.