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 7.5 — 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?
Mean absolute deviation (MAD) measures the average distance of data values from the mean. The formula is \[\text{MAD} = \frac{1}{n}\sum_{i=1}^{n} |x_i – \bar{x}|.\] You find the mean, then take the absolute value of each deviation from the mean, then average those absolute deviations. MAD answers “on average, how far off the mean are my data points?”
How do you calculate MAD step by step?
For data \(72, 85, 91, 88, 73\): (1) Mean is \((72 + 85 + 91 + 88 + 73)/5 = 81.8\). (2) Deviations from mean: \(-9.8, 3.2, 9.2, 6.2, -8.8\). (3) Absolute deviations: \(9.8, 3.2, 9.2, 6.2, 8.8\). (4) Sum of absolute deviations: \(37.2\). (5) MAD \(= 37.2 / 5 = 7.44\). On average, scores are 7.44 points from the mean.
Why take the absolute value of deviations?
Because deviations from the mean always sum to zero (positive and negative deviations cancel). Without absolute values, the “average deviation” would always be 0 – useless for measuring spread. Taking absolute values strips the signs and makes all deviations positive distances, so they don’t cancel.
How is MAD different from standard deviation?
Both measure spread around the mean. MAD averages the absolute deviations; standard deviation averages the squared deviations and then takes a square root. Squaring penalizes large deviations more heavily, so standard deviation is more sensitive to outliers than MAD. For most data, MAD and standard deviation tell similar stories, but MAD is more intuitive to interpret.
Can MAD be negative?
No. MAD is the average of absolute values, which are always non-negative. MAD \(= 0\) only when every data point equals the mean (no variation at all). Any positive MAD means at least some values differ from the mean.
How does MAD relate to spread in a histogram?
A small MAD means data points cluster close to the mean – a narrow peak on the histogram. A large MAD means points spread far from the mean – a wide, flatter shape. Two data sets with the same mean but different MADs will have differently shaped histograms.
Walk me through a comparison example
Set A: \(\{18, 19, 20, 21, 22\}\). Mean \(= 20\). Absolute deviations: \(2, 1, 0, 1, 2\). MAD \(= 6/5 = 1.2\). Set B: \(\{10, 15, 20, 25, 30\}\). Mean \(= 20\). Absolute deviations: \(10, 5, 0, 5, 10\). MAD \(= 30/5 = 6\). Same mean, but Set B’s MAD is 5 times bigger – it’s much more spread out.
What’s the difference between MAD and range?
Range is just max minus min – uses only two data points and is extremely sensitive to outliers. MAD uses every data point and is a fuller measure of spread. For \(\{18, 19, 20, 21, 22, 100\}\), the range is 82, but most of the data is tightly packed. MAD would also be inflated by the outlier 100, but less dramatically than range.
How sensitive is MAD to outliers?
Less sensitive than standard deviation, more sensitive than the IQR. Adding a single extreme value to your data pulls the mean and inflates MAD. The IQR ignores extreme values entirely. For data with possible outliers, the IQR is the most resistant spread measure; MAD is a middle-ground choice.
Where does MAD show up on tests?
Common Core middle-school standards (grade 6 introduces MAD), state tests in grades 6-8, the SAT, ACT, GRE, and most college placement exams. Common question types: compute MAD from a small data set, compare MADs of two data sets, or interpret what a given MAD means about the spread of the data.
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