Mean Absolute Deviation on Calculator
Mean absolute deviation (MAD) is a measure of variability that describes how far data values typically fall from the mean. Unlike variance and standard deviation, MAD uses the absolute value of each deviation, making the result easier to interpret in the original units of the data.
For students, researchers, and data analysts, understanding how to calculate mean absolute deviation provides another useful way to describe the spread and consistency of a dataset.
What Is Mean Absolute Deviation?
Mean absolute deviation is the average of the absolute differences between each observation and the dataset’s mean.
The formula is: MAD=∑i=1n∣xi−xˉ∣/n
Where:
- xi = individual data value
- xˉ = arithmetic mean of the dataset
- ∣xi−xˉ∣ = absolute deviation from the mean
- n = number of observations
- ∑ = sum of all absolute deviations
The absolute value is important because positive and negative deviations would otherwise cancel each other out.
A smaller MAD indicates that observations tend to be closer to the mean, while a larger MAD indicates greater dispersion.
Mean Absolute Deviation vs. Standard Deviation
MAD and standard deviation both describe variability, but they calculate dispersion differently.
Mean absolute deviation averages the absolute distances from the mean: MAD=∑∣xi−xˉ∣/n
Standard deviation is based on squared deviations from the mean.
MAD is often easier to explain because it remains in the same units as the original data and directly represents the average absolute distance from the mean.
For example, if a dataset measures income in dollars, a MAD of $500 means the observations are, on average, $500 away from the mean in absolute terms.
Example: Calculate MAD on a TI-84 Calculator
Consider the following dataset: 8, 13, 14, 16, 19, 24
We can use a TI-84 calculator to calculate the mean absolute deviation.
Step 1: Enter the Data
Press:
STAT → 1:Edit
Enter the observations into L1:
81314161924
Your calculator should now contain the six observations in L1.
Step 2: Calculate the Mean
First calculate the mean of the dataset.
On the home screen, enter:
mean(L1)
On a TI-84, you can access the mean function through the statistical calculation menus.
The mean is: xˉ=68+13+14+16+19+24 xˉ=15.667
So the dataset has a mean of approximately 15.67.
Step 3: Calculate the Absolute Deviations
Now calculate the distance of each observation from the mean.
The formula is: ∣xi−xˉ∣
For the dataset:
| Value | Deviation from Mean | Absolute Deviation |
|---|---|---|
| 8 | -7.667 | 7.667 |
| 13 | -2.667 | 2.667 |
| 14 | -1.667 | 1.667 |
| 16 | 0.333 | 0.333 |
| 19 | 3.333 | 3.333 |
| 24 | 8.333 | 8.333 |
The sum of the absolute deviations is approximately: 7.667+2.667+1.667+0.333+3.333+8.333=24
Therefore: MAD=624=4
So the mean absolute deviation is 4.
Step 4: Calculate MAD Directly on the TI-84
You can also use the TI-84 to perform the calculation using a second list.
Go to:
STAT → 1:Edit
Move to the top of L2 and enter a formula equivalent to:
abs(L1-mean(L1))
Press ENTER.
The calculator will calculate the absolute deviation for each value in L1 and place the results into L2.
The values will be approximately:
7.6672.6671.6670.3333.3338.333
Now calculate the mean of L2:
mean(L2)
The result is: 4
Therefore:
Mean Absolute Deviation = 4
How to Interpret the Result
The MAD of 4 means that the observations in this dataset are, on average, 4 units away from the mean.
Remember that MAD describes the typical absolute distance from the mean; it does not mean that every observation is exactly 4 units away.
For example, the value 8 is about 7.67 units below the mean, while 16 is only about 0.33 units above the mean.
Why Use Absolute Deviations?
Suppose we simply calculated the ordinary deviations: xi−xˉ
The positive and negative deviations would add up to zero because the arithmetic mean balances the data.
For this reason, we take the absolute value: ∣xi−xˉ∣
This converts every deviation into a nonnegative distance before calculating the average.
Mean Absolute Deviation Formula
For a dataset containing n observations, the general formula as mentioned above MAD.
The calculation can be summarized in four steps:
- Calculate the mean.
- Subtract the mean from every observation.
- Take the absolute value of each deviation.
- Calculate the average of those absolute deviations.
Mean Absolute Deviation for the Example
For the dataset: 8,13,14,16,19,24
the mean is: 15.667
The absolute deviations sum to: 24
Therefore: MAD=624=4
Thus: MAD=4
Important Note About MAD
Mean absolute deviation can be calculated around different measures of central tendency, such as the mean or median. In this tutorial, MAD is calculated around the arithmetic mean, consistent with the formula: MAD=n∑∣xi−xˉ∣
If a problem specifically asks for the mean absolute deviation from the median, the calculation will be different.
When Is Mean Absolute Deviation Useful?
MAD can be useful when you want a straightforward measure of typical variability.
For example, it can help answer questions such as:
- How far are observations typically from the average?
- How consistent are measurements?
- How much variation exists within a dataset?
- Which of two datasets has less variability?
- How much do individual observations differ from the central value?
Because MAD uses the same units as the original data, it can be particularly intuitive when explaining variability.
Conclusion
Mean absolute deviation (MAD) measures the average absolute distance between observations and the mean of a dataset.
The formula is: MAD=n∑∣xi−xˉ∣
Using the dataset: 8, 13, 14, 16, 19, 24
the mean is approximately 15.67, and the resulting mean absolute deviation is: 4
The TI-84 calculator can make the calculation easier by storing the original observations in L1, calculating the absolute deviations in L2, and then finding the mean of L2.
MAD is a useful complement to measures such as variance and standard deviation, particularly when you want a simple interpretation of the typical distance of observations from the mean.
