Mean Deviation: Formula, Calculation and Types

Mean Deviation is a statistical measure that is used to determine the typical deviation from the mean value of the provided data set.

• It demonstrates the average distance between all of the observations and the middle.
•  Each deviation is an absolute deviation since it has an absolute value, meaning that the negative signs should be ignored. 
• Additionally, the deviations from the mean on both sides must be equal.
• The deviation is a metric used in statistics and mathematics to determine the discrepancy between an observed value and an expected value of a variable. 
• The deviation can be defined as the distance from the center point. 

Also read: Mean Absolute Deviation  

Key terms: Mean, Median, Mode, Class Mark, Statistics, Probability Distribution, Random Variable, Data Set

---

---

What is Mean Deviation ?

[Click Here for Sample Questions]

Mean Deviation is a statistical metric that calculates the standard deviation of a given data collection. 

Different data series, including continuous data series, discrete data series, and individual data series, can be used to compute the mean deviation.

Mean Deviation Diagram

---

Mean Deviation Formula

[Click Here for Previous Year Questions]

The formula for Mean Deviation Formula is as follows-

∑ |X – \(\bar{X}\)| / N

• X = represents each value in the set of data.
• \(\bar{X}\) = indicates the data set's average value.
• N= number of data values overall
•  | | = omits the sign and reflects absolute value.

Also Read:

Class intervals are the main component of this kind of grouped data. The continuous frequency distribution indicates how often an observation is repeated within each interval.

The formula is given below: 

MAD = \(\frac{\sum _1^n f_i |x_i - \bar{x}|}{\sum _1^n f_i}\)

The specific data points in this sort of data are listed, along with their frequency of occurrence. For this the formula is given below:

MAD = \(\frac{\sum _1^n f_i |x_i - \bar{x}|}{\sum _1^n f_i}\)

Data that has been structured and categorized into groups is referred to as grouped data. Continuous and discrete frequency distributions are used to group data.

Also read: Difference Between Relation and Function

---

Mean Deviation for Frequency Distribution

[Click Here for Sample Questions]

We group the data and describe the frequency distribution of each group in order to provide it in a more condensed format. Class intervals are the names of these teams.

Data can be divided in two different ways :

1. Grouped data
2. Ungrouped data 

Mean Deviation Formula for Grouped Data

When data is organized and classified into groups it is known as grouped data. The mean deviation formulas for grouped data are under 2 categories that are given below:

• Discrete Frequency Distribution
• Continuous Frequency Distribution

Also Check: Mean Deviation Continuous Frequency Distribution

Mean Deviation for Discrete Distribution Frequency

Discrete refers to something that is distinct or non-continuous. The frequency (number of observations) in this type of distribution is discrete by nature. A representation of data is called a discrete distribution of frequency if it contains values x1, x2, x3,... xn that occur with frequencies of f1, f2,... fn, correspondingly.

The steps below are used to determine the mean deviation for grouped data, particularly for data with discrete distributions:

Step I: The measure of central tendency that will be used to determine mean deviation is calculated. Consider this measure be = a. The calculation is as follows if this measure is the mean: 

\(\bar{X}\) = \(\frac{\sum^n_{i=1} x_if_i}{\sum^n_{i=1} f_i}\)

⇒ \(\bar{X}\) = \(\frac{1}{N} \)\(\sum_{i=1}^n\) x_if_i

Where,

N = \(\sum_{i=1}^n f_i\)

If the measure is median, the given set of data is arranged in ascending order, the cumulative frequency is computed, and the observations whose cumulative frequency is equal to or just greater than N/2 are taken as the median for the given discrete frequency distribution. It is then obvious that this value falls in the middle of the frequency distribution.

• Step II: Determine each observation's absolute departure from the central tendency value determined in Step I.
• Step III: Using the formula, determine the mean absolute deviation from the central tendency measure.

M.A.D (a) = \(\frac{\sum _1^n f_i |x_i - a|}{N}\)

When Central Tendency is Mean then:

M.A.D (\(\bar{x}\)) = \(\frac{\sum _1^n f_i |x_i - \bar{x}|}{N}\)

When central tendency is median then:

M.A.D (M) = \(\frac{\sum _1^n f_i |x_i - M|}{N}\)

Mean Deviation for Continuous Frequency Distribution

Such a type of grouped data consists of class intervals. The frequency of repetition of an observation within each interval is given by the continuous frequency distribution. The mean deviation formula is as follows:

MAD = \(\frac{\sum _1^n f_i |x_i - \bar{x}|}{\sum _1^n f_i}\)

fi is the frequency of repetition of x_i.x_i denotes the mid value of the class interval.

Read More: Mean of Grouped Data 

Mean Deviation Formula for Ungrouped Data

Data that is not sorted or classified into groups and remains in raw form is known as ungrouped data. To calculate the mean deviation for ungrouped data the formula is as follows:

MAD = \(\frac{\sum _1^n|x_i - \bar{x}|}{n}\)

Here, 

X_i represents the ith observation,x¯ represents the central point (mean, median, or mode), and 'n' is the number of observations in the data set.

---

Mean Deviation about Mean

[Click Here for Previous Year Questions]

The expected value of a data collection is another name for the mean. The sum of all observations divided by the total number of observations is the straightforward definition of mean.The formula for the following is given below:

• Ungrouped data MAD = \(\frac{\sum_1^n |x_i - \mu|}{n}\)

where, mean is μ = \(\frac{x_1 +x_2+...+x_n}{n}\)

• Continuous and discrete frequency distribution MAD = \(\frac{\sum^n_1 f_i |x_i - \mu|}{\sum^n_1 f_i}\)

Where, the mean of grouped data is \(\frac{\sum^n_1 x_if_i}{\sum^n_1 f_i}\)

---

Mean Deviation about Median

[Click Here for Sample Questions]

The median is the number in the middle of a sorted, ascending or descending list of numbers. 

For Individual Data

For Discrete Data

The median of discrete data is calculated as above

For Continuous Data

where,

• c.f = the average frequency before the median class
• l = a lower median class value
• f = the median class's frequency
• h = the median class's length

Also read: Difference between Sequence and Series

---

Mean Deviation about Mode

[Click Here for Previous Year Questions]

The mode is the value that appears the most frequently in a particular data set. The following are the formulas to determine mean deviation from the mode:

Mean deviation For Individual Data

 

Formula for mean deviation in individual data is :

where, mode = most frequent value in the set of data

Mean Deviation For Discrete Data

The formula for mean deviation in discrete data is as follows:

Mean Deviation for For Continuous Data 

The formula for mean deviation in continuous data is :

where,

l = decrease in the modal class's value

h = the modal class's size

f = how often the modal class occurs

f1 =frequency of the modal class's preceding class

f2 = the likelihood that a class will follow the modal class

Also read: Statistics Revision Notes

---

Calculating Mean Deviation 

[Click Here for Sample Questions]

Steps to calculate Mean Deviation with example with the help of a question below:

Determine the mean deviation from the mean for a dataset with the values 3, 5, 7, and 9.

• We determine the dataset's mean, which is (3+5+7+9)/4 = 6
• Then, we calculate the absolute values of each value in the data set by subtracting it from the mean, as in |3-6| = 3, |5-6| = 1, |7-6| = 1 , and |9-6| = 3.
• In other words, 3 + 1 + 1 + 3 Equals 8.
• The mean deviation is then calculated by dividing this amount by the dataset's (4) total number of values. The solution is 2 (8/4).

---

Advantages and Disadvantages of Mean Deviation 

[Click Here for Previous Year Questions]

The advantages and disadvantages of mean deviation are described as follows:

Advantages of Mean Deviation

The advantages of mean deviation are:

• The calculation and understanding of mean deviation are simpler in nature.
• All the components of the data set are taken into account while calculating mean deviation.
• This statistical measure has the lowest sample volatility when compared to other measures.
• Because it is based on departures from the mean, it is a valuable comparison metric.

Disadvantages of Mean Deviation

Mean deviation have some disadvantages:

• As mean deviation can be estimated in relation to the mean, median, and mode, it is not precisely defined.
• As we employ the absolute value, we see both adverse and favorable signs. The outcome may be inaccurate as a result.
• The mean deviation from the various averages (Mean, Median and Mode) will vary, it is not a statistic with a clear definition.
• It cannot be algebraically processed any further.
• The variations in the sampling have a big effect on it.

---

Difference between Mean Deviation and Standard Deviation

[Click Here for Sample Questions]

The difference between mean deviation and standard deviation are as follows :

Also check: Difference Between Variance and Standard Deviation 

---

Things to Remember 

• The average deviation from the mean value of the provided data set is determined using the mean deviation, which is defined as a statistical metric.
• Mean deviation is straightforward to compute and comprehend. As a result, mean deviation is used frequently in daily life by numerous working individuals from various industries.
• The central tendency is measured by the mean deviation. It explains how far out from the middle, on average, all of the observations are.
• The deviation is a measurement used in statistics and mathematics to find the discrepancy between an observed value and an expected value of a variable.

Also read:

---

Previous Year Questions

• The mean and variance of 7 observations are 8 and…
• A bag contains (2n+1) coins. It is known that n of these coins have a head on ….[KCET 2015]
• A and B each select one number at random from the distinct numbers 1,2,3 ...[WBJEE 2019]
• Two dice are thrown simultaneously, the probability of ….[KCET 2016]
• A bag contains 2n coins out of which n−1n−1 are unfair with ….[BITSAT 2019]
• The mean and variance of 20 observations are found to… (JEE Main 2020)
• The variance of the numbers 2, 3, 11 and x is…
• Two cards are drawn at random from a pack of 52 cards ….(KCET 2016)
• Probability of four sons to a couple is…..(NEET 2001)