What Is Mean Absolute Deviation Formula Steps and Solved Examples
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
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.
Mean:
x̅ = {∑i=1 n xᵢ}/n
[{5+4+5+3+5+1+5}/{7}]
= 4
The mean of the given data is 4.
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
Mode:
The most frequent data is the mode i.e., 5.
FAQs on Mean Absolute Deviation Explained with Formula and Examples
1. What is Mean Absolute Deviation (MAD)?
The Mean Absolute Deviation (MAD) is the average of the absolute differences between each data value and the mean of the dataset. It measures how spread out the data values are around the mean.
- MAD uses absolute values, so negative deviations do not cancel out positive ones.
- It gives a clear idea of overall variability.
- A smaller MAD means the data is more closely clustered around the mean.
2. What is the formula for Mean Absolute Deviation?
The formula for Mean Absolute Deviation is MAD = (1/n) Σ |xᵢ − x̄|, where x̄ is the mean of the data.
- n = total number of data values
- xᵢ = each individual value
- x̄ = mean of the dataset
- |xᵢ − x̄| = absolute deviation from the mean
3. How do you calculate Mean Absolute Deviation step by step?
To calculate Mean Absolute Deviation, find the mean, compute absolute deviations, and then average them.
- Step 1: Find the mean (x̄) of the dataset.
- Step 2: Subtract the mean from each value.
- Step 3: Take the absolute value of each difference.
- Step 4: Add all absolute deviations.
- Step 5: Divide by the total number of values (n).
4. Can you give an example of Mean Absolute Deviation?
Yes, for the data set 2, 4, 6, the Mean Absolute Deviation is 4/3 ≈ 1.33.
- Mean = (2 + 4 + 6)/3 = 4
- Absolute deviations: |2−4|=2, |4−4|=0, |6−4|=2
- Sum of deviations = 2 + 0 + 2 = 4
- MAD = 4/3 ≈ 1.33
5. What does Mean Absolute Deviation tell you?
The Mean Absolute Deviation tells you the average distance of data values from the mean. It helps measure the spread or variability of a dataset.
- Low MAD → data points are close to the mean.
- High MAD → data points are more spread out.
- It is useful for comparing consistency between datasets.
6. What is the difference between Mean Absolute Deviation and Standard Deviation?
The key difference is that MAD uses absolute values while standard deviation uses squared deviations.
- MAD formula: (1/n) Σ |xᵢ − x̄|
- Standard deviation formula: √[(1/n) Σ (xᵢ − x̄)²]
- Standard deviation gives more weight to large deviations.
- MAD is simpler and easier to interpret.
7. Is Mean Absolute Deviation always positive?
Yes, Mean Absolute Deviation is always non-negative because it uses absolute values of deviations. Since absolute values cannot be negative:
- MAD is either zero or positive.
- MAD equals zero only when all data values are the same.
8. How is Mean Absolute Deviation used in real life?
The Mean Absolute Deviation is used to measure consistency and variability in real-world data. Common applications include:
- Analyzing test scores to check score spread.
- Measuring financial market stability.
- Evaluating forecast accuracy in business.
9. What is the Mean Absolute Deviation for grouped data?
For grouped data, the Mean Absolute Deviation is calculated using class midpoints and frequencies. The formula is MAD = (1/N) Σ f|x − x̄|.
- f = frequency of each class
- x = class midpoint
- N = total frequency
- x̄ = mean of grouped data
10. What are common mistakes when calculating Mean Absolute Deviation?
A common mistake in calculating Mean Absolute Deviation is forgetting to take absolute values before averaging. Other errors include:
- Using the wrong mean value.
- Not dividing by the total number of observations (n).
- Confusing MAD with standard deviation.
