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# Mean absolute deviation calculator for ungrouped data

## Mean absolute deviation for ungrouped data

Mean absolute deviation is another measure of dispersion. MAD is an absolute measure of dispersion.

Let $x_i, i=1,2, \cdots , n$ be $n$ observations. The mean of $X$ is denoted by $\overline{x}$ and is given by

 $$\begin{eqnarray*} \overline{x}& =&\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*}$$
The mean absolute deviation about mean is

$MAD =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|$

where,

• $n$ total number of observations
• $\overline{x}$ sample mean

Mean absolute deviation is minimum when taken about median.

## Coefficient of Mean absolute deviation

The coefficient of mean absolute deviation is a relative measure of dispersion. The coefficient of quartile deviation is used to study and compare the degree of variation for two or more data set having different units of measurements.

The coefficient of mean absolute deviation about mean is given by

 \begin{aligned} \text{Coeff. of MAD about mean}&= \frac{MAD \text{ about mean}}{mean}\\ &=\frac{MAD \text{ about mean}}{\overline{X}} \end{aligned}

where,

• MAD about mean mean absolute deviation about mean
• $\overline{x}$ sample mean

## Mean Absolute Deviation Calculator for ungrouped data

Use this calculator to find the variance and standard deviation for ungrouped (raw) data.

Mean Absolute Deviation Calculator
Enter the X Values (Separated by comma,)
Results
Number of Obs. (n):
Sample Mean : ($\overline{x}$)

## How to calculate mean absolute deviation for ungrouped data?

Step 1 – Enter the $x$ values separated by commas

Step 2 – Click on "Calculate" button to get mean absolute deviation about mean for ungrouped data

Step 3 – Gives the output as number of observations $n$

Step 4 – Calculate the sample mean

Step 5 – Calculate the mean absolute deviation about mean

## Mean absolute deviation Example 1

The age (in years) of 6 randomly selected students from a class are : 22,25,24,23,24,20.

Compute mean absolute deviation about mean.

#### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
22 -1 1
25 2 2
24 1 1
23 0 0
24 1 1
20 -3 3
Total 138 8

The sample mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{138}{6}\\ &=23 \text{ years} \end{aligned}
The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{8}{6}\\ &=\frac{8}{6}\\ &= 1.3333 \text{ years} \end{aligned}

## Mean absolute deviation Example 2

The systolic blood pressure (in mmHg) of 10 randomly selected patients are :

123, 128, 136, 112, 143, 114, 104, 137, 145, 150.

Compute mean absolute deviation about mean.

#### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
123 -6.2 6.2
128 -1.2 1.2
136 6.8 6.8
112 -17.2 17.2
143 13.8 13.8
114 -15.2 15.2
104 -25.2 25.2
137 7.8 7.8
145 15.8 15.8
150 20.8 20.8
Total 1292 130

The sample mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1292}{10}\\ &=129.2 \text{ mmHg} \end{aligned}

The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{130}{10}\\ &=\frac{130}{10}\\ &= 13 \text{ mmHg} \end{aligned}

## Mean absolute deviation Example 3

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.

Find mean absolute deviation about mean.

#### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
75 -5.55 5.55
80 -0.55 0.55
72 -8.55 8.55
78 -2.55 2.55
82 1.45 1.45
85 4.45 4.45
73 -7.55 7.55
75 -5.55 5.55
97 16.45 16.45
87 6.45 6.45
84 3.45 3.45
76 -4.55 4.55
73 -7.55 7.55
79 -1.55 1.55
99 18.45 18.45
86 5.45 5.45
83 2.45 2.45
76 -4.55 4.55
78 -2.55 2.55
73 -7.55 7.55
Total 1611 117.2

The sample mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1611}{20}\\ &=80.55 \text{ mg/dl} \end{aligned}

The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{117.2}{20}\\ &=\frac{117.2}{20}\\ &= 5.86 \text{ mg/dl} \end{aligned}

## Mean absolute deviation Example 4

Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:

65,76,64,73,74,80, 71, 68,66, 81, 79, 75, 70, 62, 83,63, 77, 78.

#### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
65 -7.5 7.5
76 3.5 3.5
64 -8.5 8.5
73 0.5 0.5
74 1.5 1.5
80 7.5 7.5
71 -1.5 1.5
68 -4.5 4.5
66 -6.5 6.5
81 8.5 8.5
79 6.5 6.5
75 2.5 2.5
70 -2.5 2.5
62 -10.5 10.5
83 10.5 10.5
63 -9.5 9.5
77 4.5 4.5
78 5.5 5.5
Total 1305 102

The sample mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1305}{18}\\ &=72.5 \text{ mmHg} \end{aligned}

The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{102}{18}\\ &=\frac{102}{18}\\ &= 5.6667 \text{ mmHg} \end{aligned}

## Mean absolute deviation Example 5

The following data are the heights, correct to the nearest centimeters, for a group of children:

126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
137, 138, 141, 143, 144, 146, 147, 152, 154, 161 

Calculate mean absolute deviation about mean for the above data.

#### Solution

$x_i$ $(x_i-xb)$ $|x_i-xb|$
126 -13.25 13.25
129 -10.25 10.25
129 -10.25 10.25
132 -7.25 7.25
132 -7.25 7.25
133 -6.25 6.25
133 -6.25 6.25
135 -4.25 4.25
136 -3.25 3.25
137 -2.25 2.25
137 -2.25 2.25
138 -1.25 1.25
141 1.75 1.75
143 3.75 3.75
144 4.75 4.75
146 6.75 6.75
147 7.75 7.75
152 12.75 12.75
154 14.75 14.75
161 21.75 21.75
Total 2785 148

The sample mean of $X$ is

 \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{2785}{20}\\ &=139.25 \text{ cm} \end{aligned}

The mean absolute deviation about mean is

 \begin{aligned} MAD & =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|\\ &=\frac{148}{20}\\ &=\frac{148}{20}\\ &= 7.4 \text{ cm} \end{aligned}

## Conclusion

In this tutorial, you learned about formula for mean absolute deviation (MAD) for ungrouped data and how to calculate mean absolute deviation for ungrouped data. You also learned about how to solve numerical problems based on mean absolute deviation for ungrouped data.