Mean absolute deviation calculator for grouped data
- 1 Mean absolute deviation Calculator
- 2 Coefficient of Mean absolute deviation
- 3 Mean Absolute Deviation Calculator for grouped data
- 4 How to use mean absolute deviation calculator for grouped data?
- 5 Example #1 - Calculate Mean absolute deviation
- 6 Example #2 - Find Mean absolute deviation
- 7 Example #3 - Compute Mean absolute deviation
- 8 Example #4 - Get Mean absolute deviation for frequency distribution
- 9 Conclusion
Mean absolute deviation Calculator
Use this calculator to find the Mean Absolute Deviation using frequency distribution (uniform or discrete),frequencies and grouped data
Mean absolute deviation is another measure of dispersion. MAD is an absolute measure of dispersion.
Let $(x_i,f_i), i=1,2, \cdots , n$
be given frequency distribution. The mean of $X$ is denoted by $\overline{x}$ and is given by
$$ \begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*} $$
The mean absolute deviation about mean is given by
$$MAD =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|$$
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 grouped data
Mean Absolute Deviation Calculator | |
---|---|
Type of Frequency Distribution | DiscreteContinuous |
Enter the Classes for X (Separated by comma,) | |
Enter the frequencies (f) (Separated by comma,) | |
Results | |
Number of Observation (n): | |
Sample Mean : ($\overline{x}$) | |
Mean Absolute Deviation : (MAD) | |
How to use mean absolute deviation calculator for grouped data?
Step 1 - Select type of frequency distribution (Discrete or continuous)
Step 2 - Enter the Range or classes (X) seperated by comma (,)
Step 3 - Enter the Frequencies (f) seperated by comma
Step 4 - Click on "Calculate" for mean absolute deviation
Step 5 - Gives output as number of observation (N)
Step 6 - Calculate the sample mean
Step 7 - Calculate mean absolute deviation
Example #1 - Calculate Mean absolute deviation
A librarian keeps the records about the amount of time spent (in minutes) in a library by college students. Data is as follows:
Time spent | 30 | 32 | 35 | 38 | 40 |
---|---|---|---|---|---|
No. of students | 8 | 12 | 20 | 10 | 5 |
Calculate mean absolute deviation about mean.
Solution
$x_i$ | $f_i$ | $f_i*x_i$ | $(x_i-xb)$ | $|x_i-xb|$ | $f_i|x_i-xb|$ | |
---|---|---|---|---|---|---|
30 | 8 | 240 | -4.62 | 4.62 | 36.95 | |
32 | 12 | 384 | -2.62 | 2.62 | 31.42 | |
35 | 20 | 700 | 0.38 | 0.38 | 7.64 | |
38 | 10 | 380 | 3.38 | 3.38 | 33.82 | |
40 | 5 | 200 | 5.38 | 5.38 | 26.91 | |
Total | 55 | 1904 | 136.74 |
The mean absolute deviation about mean is given by
$$ \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}| \end{aligned} $$
where,
$N$
total number of observations$\overline{x}$
sample mean
The mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{1904}{55}\\ &=34.62 \text{ minutes} \end{aligned} $$
The mean absolute deviation about mean is
$$ \begin{aligned} MAD & =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{136.74}{55}\\ &= 2.49 \text{ minutes} \end{aligned} $$
Example #2 - Find Mean absolute deviation
The number of defective items in successive groups of fifteen items were counted as they came off a production line. The results ccan be summarized as follows:
No. of defective | 0 | 1 | 2 | 3 | 4 | $>$ 4 |
---|---|---|---|---|---|---|
Frequency | 57 | 57 | 18 | 5 | 3 | 0 |
Calculate mean absolute deviation about mean.
Solution
$x_i$ | $f_i$ | $f_i*x_i$ | $(x_i-xb)$ | $|x_i-xb|$ | $f_i|x_i-xb|$ | |
---|---|---|---|---|---|---|
0 | 57 | 0 | -0.86 | 0.86 | 48.86 | |
1 | 57 | 57 | 0.14 | 0.14 | 8.14 | |
2 | 18 | 36 | 1.14 | 1.14 | 20.57 | |
3 | 5 | 15 | 2.14 | 2.14 | 10.71 | |
4 | 3 | 12 | 3.14 | 3.14 | 9.43 | |
Total | 140 | 120 | 97.71 |
The mean absolute deviation about mean is given by
$$ \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}| \end{aligned} $$
where,
$N$
total number of observations$\overline{x}$
sample mean
The mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{120}{140}\\ &=0.86 \text{ } \end{aligned} $$
The mean absolute deviation about mean is
`
$$
\begin{aligned}
MAD & =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\
&= \frac{97.71}{140}\
&= 0.7 \text{ }
\end{aligned}
$$
Example #3 - Compute Mean absolute deviation
The following table gives the distribution of weight (in pounds) of 100 newborn babies at certain hospital in 2012.
Weight (in pounds) | 3-5 | 5-7 | 7-9 | 9-11 | 11-13 |
---|---|---|---|---|---|
No.of babies | 10 | 30 | 28 | 18 | 14 |
Calculate mean absolute deviation about mean.
Solution
Class Interval | $x_i$ | $f_i$ | $f_i*x_i$ | $x_i-xb$ | $|x_i-xb|$ | $f_i|x_i-xb|$ | |
---|---|---|---|---|---|---|---|
3-5 | 4 | 10 | 40 | -3.92 | 3.92 | 39.2 | |
5-7 | 6 | 30 | 180 | -1.92 | 1.92 | 57.6 | |
7-9 | 8 | 28 | 224 | 0.08 | 0.08 | 2.24 | |
9-11 | 10 | 18 | 180 | 2.08 | 2.08 | 37.44 | |
11-13 | 12 | 14 | 168 | 4.08 | 4.08 | 57.12 | |
Total | 100 | 792 | 193.6 |
The mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{792}{100}\\ &=7.92 \text{ pounds} \end{aligned} $$
The mean absolute deviation about mean is
$$ \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{193.6}{100}\\ &= 1.94 \text{ pounds} \end{aligned} $$
Example #4 - Get Mean absolute deviation for frequency distribution
The following data shows the distribution of maximum loads in short tons supported by certain cables produced by a company:
Maximum load | No. of Cables |
---|---|
9.25-9.75 | 2 |
9.75-10.25 | 5 |
10.25-10.75 | 12 |
10.75-11.25 | 17 |
11.25-11.75 | 14 |
11.75-12.25 | 6 |
12.25-12.75 | 3 |
12.75-13.25 | 1 |
Compute mean absolute deviation about mean for the above frequency distribution.
Solution
Class Interval | $x_i$ | $f_i$ | $f_i*x_i$ | $x_i-xb$ | $|x_i-xb|$ | $f_i|x_i-xb|$ | |
---|---|---|---|---|---|---|---|
9.25-9.75 | 9.5 | 2 | 19 | -1.59 | 1.59 | 3.18 | |
9.75-10.25 | 10 | 5 | 50 | -1.09 | 1.09 | 5.46 | |
10.25-10.75 | 10.5 | 12 | 126 | -0.59 | 0.59 | 7.1 | |
10.75-11.25 | 11 | 17 | 187 | -0.09 | 0.09 | 1.56 | |
11.25-11.75 | 11.5 | 14 | 161 | 0.41 | 0.41 | 5.72 | |
11.75-12.25 | 12 | 6 | 72 | 0.91 | 0.91 | 5.45 | |
12.25-12.75 | 12.5 | 3 | 37.5 | 1.41 | 1.41 | 4.22 | |
12.75-13.25 | 13 | 1 | 13 | 1.91 | 1.91 | 1.91 | |
Total | 60 | 665.5 | 34.6 |
The mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{665.5}{60}\\ &=11.09 \text{ tons} \end{aligned} $$
The mean absolute deviation about mean is
$$ \begin{aligned} MAD &=\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|\\ &= \frac{34.6}{60}\\ &= 0.58 \text{ tons} \end{aligned} $$
Conclusion
In this tutorial, you learned about formula for mean absolute deviation (MAD) for grouped data and how to calculate mean absolute deviation for grouped data. You also learned about how to solve numerical problems based on mean absolute deviation for grouped 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 grouped data with examples and your thought on this article.