How to Calculate Mean Absolute Deviation (with ungrouped data)

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When working with data, there are different ways to look at the range and spread of values in a set, with the most common being the mean. Most of us have learned that to calculate the mean, you need to find the sum of the set and then divide it by the number of values in the group. The more advanced form of math is calculating the mean absolute deviation. This calculation shows how close the values in the set are to the average. To calculate the mean absolute deviation from the mean of the data set, calculate the absolute deviation of each data point from the mean, and then average those deviations.

Table of Contents

Calculate the average

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Collect and count data. For any set of data, the mean is a measure of the central value. Depending on the type of data, the mean can indicate the central value of the set. To find the mean, you first need to collect data through experimentation or from the given problem itself. ^{[1] X Research Source}

For example, we have a data set consisting of the values 6, 7, 10, 12, 13, 4, 8 and 12. This set is small enough to count, you can easily see that there are 8 numbers in the set. .
In statistics, the variable $WOMEN$ ${displaystyle N}$ $WOMEN$ or $n$ ${displaystyle n}$ $n$ commonly used to represent the number of values in a data set.

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Find the sum of the values in the data set. The first step in averaging is summing all the data points. In statistical notation, each value is represented by the variable

x

{displaystyle x}

x

. The capital letter sigma in Greek means sum of values and is denoted by

Σ

x

{displaystyle Sigma x}

{displaystyle Sigma x}

. For the above example, the sum of the values would be: ^{[2] X Research Source}

$Σ$ $x$ $=$ $6$ $+$ $7$ $+$ $ten$ $+$ $twelfth$ $+$ $13$ $+$ $4$ $+$ $8$ $+$ $twelfth$ $=$ $72$ ${displaystyle Sigma x=6+7+10+12+13+4+8+12=72}$ ${displaystyle Sigma x=6+7+10+12+13+4+8+12=72}$

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Perform division to find the average. To find the average, divide the total by the number of values. The mean is usually denoted by .

μ

{displaystyle mu }

{displaystyle mu }

, which is the Greek letter mu. So, the average of the dataset in the example is: ^{[3] X Research Source}

$μ$ $=$ $Σ$ $x$ $WOMEN$ $=$ $72$ $8$ $=$ $9$ ${displaystyle mu ={frac {Sigma x}{N}}={frac {72}{8}}=9}$ ${displaystyle mu ={frac {Sigma x}{N}}={frac {72}{8}}=9}$

Calculate mean absolute deviation

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Tabulator. Table with 3 columns will help to maintain the order of values and you can calculate more easily. Please write the first column heading as

x

{displaystyle x}

x

, the second column is

x

-

μ

{displaystyle x-mu }

{displaystyle x-mu }

and the third column is

|

x

-

μ

|

{displaystyle |x-mu |}

{displaystyle |x-mu |}

. ^{[4] X Research Sources}

Fill in the data points of the problem in the first column.

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Calculate the deviation of each data point. In the second column with the title

x

-

μ

{displaystyle x-mu }

{displaystyle x-mu }

, you will calculate the deviation or difference between each data point and the average of the set of values. Just subtract the average from each value in the data. ^{[5] X Research Sources}

In the example of the article, the offsets would be:
- $6$ $-$ $9$ $=$ $-$ $3$ ${displaystyle 6-9=-3}$ ${displaystyle 6-9=-3}$
- $7$ $-$ $9$ $=$ $-$ $2$ ${displaystyle 7-9=-2}$ ${displaystyle 7-9=-2}$
- $ten$ $-$ $9$ $=$ $first$ ${displaystyle 10-9=1}$ ${displaystyle 10-9=1}$
- $twelfth$ $-$ $9$ $=$ $3$ ${displaystyle 12-9=3}$ ${displaystyle 12-9=3}$
- $13$ $-$ $9$ $=$ $4$ ${displaystyle 13-9=4}$ ${displaystyle 13-9=4}$
- $4$ $-$ $9$ $=$ $-$ $5$ ${displaystyle 4-9=-5}$ ${displaystyle 4-9=-5}$
- $8$ $-$ $9$ $=$ $-$ $first$ ${displaystyle 8-9=-1}$ ${displaystyle 8-9=-1}$
- $twelfth$ $-$ $9$ $=$ $3$ ${displaystyle 12-9=3}$ ${displaystyle 12-9=3}$
To check if these results are correct, you can sum the values in the deviation column. If this sum is zero, you are correct. If the sum is non-zero, it’s likely that the mean is incorrect, or that you miscalculated one or more deviations. Let’s retrace each calculation.

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The absolute value of the deviation. When calculating the deviation of each data point from the mean, we only care about the magnitude of the difference and not the direction (negative or positive). At this point, you need to find the absolute value of the offset. The absolute value is denoted by two vertical dashes | |. ^{[6] X Research Source}

In mathematics, absolute value is used to measure distance or size, not in terms of direction.
To find the absolute value, simply remove the negative sign from each number in the second column and fill in the third column as follows:
$x$ $.$ $.$ $.$ $.$ $.$ $($ $x$ $-$ $μ$ $)$ $.$ $.$ $.$ $.$ $.$ $|$ $($ $x$ $-$ $μ$ $)$ $|$ ${displaystyle x\dots..(x-mu )\dots..|(x-mu )|}$ ${displaystyle x.....(x-mu ).....|(x-mu )|}$
$6\dots\dots\dots.$ $-$ $3\dots\dots\dots.3$ ${displaystyle 6\dots\dots\dots-3\dots\dots\dots.3}$ ${displaystyle 6.........-3..........3}$
$7\dots\dots\dots.$ $-$ $2\dots\dots\dots.2$ ${displaystyle 7\dots\dots\dots.-2\dots\dots\dots.2}$ ${displaystyle 7..........-2..........2}$
$10\dots\dots\dots\dots1\dots\dots\dots\dots1$ ${displaystyle 10\dots\dots\dots1\dots\dots\dots.1}$ ${displaystyle 10.........1..........1}$
$12\dots\dots\dots3\dots\dots\dots.3$ ${displaystyle 12\dots\dots\dots3\dots\dots\dots.3}$ ${displaystyle 12.........3..........3}$
$13\dots\dots\dots4\dots\dots\dots.4$ ${displaystyle 13\dots\dots\dots4\dots\dots\dots.4}$ ${displaystyle 13.........4..........4}$
$4\dots\dots\dots.$ $-$ $5\dots\dots\dots5$ ${displaystyle 4\dots\dots\dots.-5\dots\dots\dots.5}$ ${displaystyle 4..........-5..........5}$
$8\dots\dots\dots.$ $-$ $1\dots\dots\dots1$ ${displaystyle 8\dots\dots\dots-1\dots\dots\dots.1}$ ${displaystyle 8.........-1..........1}$
$12\dots\dots\dots3\dots\dots\dots.3$ ${displaystyle 12\dots\dots\dots3\dots\dots\dots.3}$ ${displaystyle 12.........3..........3}$

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Calculate the average of the absolute deviation values. After completing the third column of the table, calculate the average of the absolute values in the third column. Similar to when you are looking for the mean of the data set, take the sum of the deviations divided by the number of values. ^{[7] X Research Sources}

Continuing this example, the average absolute deviation would be:
- $3$ $+$ $2$ $+$ $first$ $+$ $3$ $+$ $4$ $+$ $5$ $+$ $first$ $+$ $3$ $8$ $=$ $22$ $8$ $=$ $2$ $,$ $75$ ${displaystyle {frac {3+2+1+3+4+5+1+3}{8}}={frac {22}{8}}=2.75}$ ${displaystyle {frac {3+2+1+3+4+5+1+3}{8}}={frac {22}{8}}=2.75}$

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Explain the results. The mean absolute deviation is a measure of how closely the values are in grouped data. This value answers the question: “How close are the data points to the mean?” ^{[8] X Research Sources}

As in the data set above, the mean is 9 and the mean distance from the mean is 2.75. Note that some values will be closer to the average than 2.75, others will be further away. However, that is the average distance.

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Calculate the average

Calculate mean absolute deviation

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