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 : ($\overline{x}$) | |
Mean Absolute Deviation : (MAD) | |
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.
Find mean absolute deviationabout mean.
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.
To learn more about other descriptive statistics measures, please refer to the following tutorials:
Let me know in the comments if you have any questions on Mean absolute deviation calculator for ungrouped data with examples and your thought on this article.