Step-by-Step Guide to Finding 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
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 Mastering Mean Absolute Deviation in Maths
1. What is Mean Absolute Deviation (MAD) in statistics?
Mean Absolute Deviation, often abbreviated as MAD, is a measure of variability in a dataset. It represents the average distance between each data point and the mean of the entire set. Essentially, it tells you, on average, how far each value is from the center of the data. A smaller MAD implies the data points are clustered closely together, while a larger MAD indicates they are more spread out.
2. What is the formula used to calculate Mean Absolute Deviation for ungrouped data?
The formula for calculating the Mean Absolute Deviation (MAD) for a set of ungrouped data is: MAD = (Σ|xᵢ - μ|) / N. In this formula:
- Σ represents the summation or 'sum of'.
- xᵢ stands for each individual data point in the set.
- μ (mu) is the mean (average) of the data set.
- |...| indicates the absolute value, which means we ignore any negative signs.
- N is the total number of data points in the set.
3. How do you calculate the Mean Absolute Deviation? Explain with a simple example.
To calculate the Mean Absolute Deviation, you follow four main steps. Let's use the data set {3, 6, 6, 7, 8, 11, 15, 16}.
Step 1: Calculate the mean (μ).
Mean = (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16) / 8 = 72 / 8 = 9.
Step 2: Find the absolute deviation for each data point.
This is the absolute difference between each value and the mean (9): |3-9|=6, |6-9|=3, |6-9|=3, |7-9|=2, |8-9|=1, |11-9|=2, |15-9|=6, |16-9|=7.
Step 3: Sum the absolute deviations.
Sum = 6 + 3 + 3 + 2 + 1 + 2 + 6 + 7 = 30.
Step 4: Divide the sum by the number of data points (N).
MAD = 30 / 8 = 3.75. The Mean Absolute Deviation is 3.75.
4. What does a large or small Mean Absolute Deviation value tell us about a set of data?
The value of the Mean Absolute Deviation provides direct insight into the consistency of the data.
- A small MAD indicates that the data points are very close to the mean. This suggests low variability and high consistency. For example, if the MAD of students' test scores is small, it means most students scored similarly to the class average.
- A large MAD indicates that the data points are spread far from the mean. This suggests high variability and low consistency. For instance, a large MAD for test scores would mean the scores were widely scattered, with some very high and some very low.
5. What is the main difference between Mean Absolute Deviation and Standard Deviation?
The primary difference lies in how they handle the deviations from the mean. Mean Absolute Deviation (MAD) calculates the average of the absolute differences from the mean. In contrast, Standard Deviation (SD) calculates the average of the squared differences from the mean and then takes the square root of that result. Because it squares the deviations, Standard Deviation gives much greater weight to outliers (very large or very small values) than MAD does, making it more sensitive to extreme data points.
6. Why is it necessary to use the absolute value when calculating mean deviation?
Using the absolute value is crucial because the sum of simple deviations from the mean (i.e., Σ(xᵢ - μ)) for any dataset is always zero. This is because the positive deviations (from values above the mean) perfectly cancel out the negative deviations (from values below the mean). By taking the absolute value of each deviation, we treat all deviations as positive distances, which allows us to measure the total spread and calculate a meaningful average dispersion.
7. In what real-world applications is Mean Absolute Deviation useful?
Mean Absolute Deviation is used in various fields where understanding data spread is important. For example:
- In business and finance, it can measure the volatility of a stock's returns or the error in sales forecasting. A lower MAD in a forecast indicates higher accuracy.
- In manufacturing, it can be used to monitor quality control by measuring how much the dimensions of a product vary from the required specification.
- In sports analytics, it can measure the consistency of a player's performance over a season.
8. Can Mean Absolute Deviation be calculated using the median instead of the mean?
Yes, it is possible and sometimes preferable to calculate the mean absolute deviation from the median instead of the mean. This measure is known as the Median Absolute Deviation. A key property is that the sum of absolute deviations from the median is the smallest possible, making this measure less sensitive to extreme outliers than the version calculated from the mean. However, in the CBSE syllabus, Mean Absolute Deviation is typically calculated with respect to the mean.
