How to Find Mean Absolute Deviation?
TL;DR: Two classes can have the same average and still feel completely different, one tightly clustered, one all over the map. Mean absolute deviation, or MAD, is how you measure that spread. Subtract the mean from each value, take the absolute value so you only care about distance (not direction), then average those distances. For the scores 72, 85, 91, 88, 73 with a mean of 81.8, the MAD comes out to about 7.4 (37.2 ÷ 5 = 7.44). That's your typical distance from the average.
Key takeaways:
- MAD \(= \dfrac{1}{n}\sum |x_i - \bar{x}|\) - average distance from the mean.
- Always non-negative; equals 0 only when every value is the mean.
- Smaller MAD = data clustered tighter; bigger MAD = more spread out.
- MAD uses absolute values, unlike variance (which squares deviations).
- Resistant to outliers compared to standard deviation (which squares).
To understand the mean of absolute deviation, let us split both words and try to understand their meaning. “Mean” refers to the average of observations, and deviation means departure or variation from a preset standard. Now, we can define mean deviation as the mean distance of each observation from the mean of the data.
A step-by-step guide to finding mean absolute deviation
The mean absolute deviation is the average deviation of data points from a central point. The center point can be the mean, median, mode, or any random point. The average is often considered the center point.
The formula of mean absolute deviation
There are two formulas for finding the mean absolute deviation. One is for ungrouped data, and the other is for grouped data.
Let \(x_1\), \(x_2\), …. \(x_n\) be the data set and let \(μ\) be its average of the ungrouped data. And, \(f\) is the frequency of the data point \(x_i\), for the grouped data. The mean absolute deviation formulas for the two types of data are as follows:
Mean absolute deviation for grouped data \(\color{blue}{=\frac{1}{n}\sum _{i=1}^n\:\left|x_i-μ\right|}\)
Mean absolute deviation for ungrouped data \(\color{blue}{=\frac{\sum f\left|x-x_i\right|\:}{\sum f\:}}\)
Finding Mean Absolute Deviation , Example 1:
Find mean absolute deviation for the following data set: \(300, 142, 356, 560, 459, 217, 220\)
Solution:
First, find mean of the data \((μ)=\frac{sum \ of \ the \ data}{total \ number \ of \ data \ entires}\)
\(=\frac{300+142+ 356+ 560+ 459+ 217+ 220}{7}\)
\(=\frac{2254}{7}=322\)
Now, using mean deviation formula: \(\color{blue}{=\frac{1}{n}\sum _{i=1}^n\:\left|x_i-μ\right|}\)
\(=\frac{|300-322|+|142-322|+|356-322|+|560-322|+|459-322|+|217-322|+|220-322|}{7}\)
\(=\frac{|-22|+|-180|+|34|+|238|+|137|+|-105|+|-102|}{7}\)
\(=\frac{22+180+34+238+137+105+102}{7}\)
\(=\frac{818}{7}\)
\(=116.86\)
Exercises for Finding Mean Absolute Deviation
Find the mean absolute deviation of the data.
- \(\color{blue}{86, 93, 88, 85, 89, 95, 85, 83}\)
- \(\color{blue}{29, 24, 15, 29, 41, 35, 65, 49, 46}\)
- \(\color{blue}{3.25}\)
- \(\color{blue}{11.78}\)
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Frequently Asked Questions
What is mean absolute deviation and how do you find it?
Mean absolute deviation (MAD) is the average distance between each data value and the mean. Find the mean, subtract it from each value, drop any negative signs, then average those distances. For 4, 6, 8, 10, 12, the mean is 8, the distances are 4, 2, 0, 2, 4, and the MAD is 12 ÷ 5 = 2.4.
How do you find mean absolute deviation in 6th grade?
Sixth graders usually follow four steps: find the mean, find how far each number is from the mean, add those distances, and divide by how many numbers there are. Try the ages 9, 11, 12, 14, 14. The mean is 60 ÷ 5 = 12, the distances are 3, 1, 0, 2, 2, and the MAD is 8 ÷ 5 = 1.6 years.
How do you find mean absolute deviation on a dot plot?
First list the values the dots stand for, counting each dot as one data point. A dot plot with one dot at 1, two at 3, one at 5 and one at 8 gives the data 1, 3, 3, 5, 8, with a mean of 20 ÷ 5 = 4. The distances from 4 are 3, 1, 1, 1, 4, which add to 10, so the MAD is 10 ÷ 5 = 2.
How do you find mean absolute deviation on a TI-84?
Enter the data in L1 with STAT, Edit. On the home screen, type mean(abs(L1 − mean(L1))) and press ENTER. You will find mean( under 2nd, STAT (LIST), MATH, and abs( under MATH, NUM. For the scores 72, 85, 91, 88, 73, the calculator returns 7.44, the average distance of those scores from their mean of 81.8.
Can mean absolute deviation be negative?
No. MAD is an average of distances, and a distance is never negative, so the smallest possible MAD is 0. A MAD of 0 means every value equals the mean, as in 7, 7, 7, 7. If you get a negative answer, you probably forgot to take the absolute value of the negative deviations before adding them.
How do you find mean absolute deviation in Excel?
Excel has a built-in function for it called AVEDEV. If your data are in cells A1 through A5, type =AVEDEV(A1:A5) in an empty cell and press Enter. For the scores 72, 85, 91, 88, 73, Excel returns 7.44. Google Sheets has the same AVEDEV function, so the formula works there too.
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