Measuring mean absolute deviation is an easy way to understand the degree of variation across statistical data points. In this article, we’ll define mean absolute deviation; discuss how it differs from its more common counterpart, standard deviation; and show how to calculate it in four quick steps.
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What is MAD in math? In math and statistics, MAD stands for mean absolute deviation.
Mean absolute deviation (MAD) is a measure of the average absolute distance between each data value and the mean of a data set. Similar to standard deviation, MAD is a parameter or statistic that measures the spread, or variation, in your data.
Mean Absolute Deviation Definition - A measure of the average absolute distance between your data points and the mean of your data set. To calculate MAD, we measure the absolute distance (or absolute deviation) between each data point and the mean. We then take the average of those distances. This average distance is what we call the mean absolute deviation.
The Mean Absolute Deviation formula is then:
M A D = ∑ ∣ x i − x ˉ ∣ n MAD = \frac \right |> M A D = n ∑ ∣ x i − x ˉ ∣ Let’s look at two quick examples. Below are two sets of data. What is the mean absolute deviation in each case?
| Data Set A | Data Set B |
| 3, 3, 7 ,7 | 1, 2, 7, 10 |
Before we calculate MAD, notice that the mean, or average, of both data sets is 5. In Data Set A, however, the data is more closely clustered around the mean. This tells us that MAD must be smaller for Data Set A than it is for Data Set B since the distances between the data values and the mean are smaller. We can see this more clearly by plotting our data onto dot plots.
Using these dot plots, we can visualize the deviations.
For Data Set A, we see that every single data point deviates from the mean by 2. Calculating MAD in this case is easy; the average of four twos is just two!
M A D A = ( 2 + 2 + 2 + 2 ) / 4 = 2 MAD_A=(2+2+2+2)/4 = 2 M A D A = ( 2 + 2 + 2 + 2 ) /4 = 2For Data Set B, we see that the data is more varied. The absolute deviations are 4, 3, 2, 5. The average of these absolute deviations is 3.5.
M A D B = ( 4 + 3 + 2 + 5 ) / 4 = 3.5 MAD_B = (4+3+2+5)/4 =3.5 M A D B = ( 4 + 3 + 2 + 5 ) /4 = 3.5
Concretely, what do these figures tell us? In the first instance, MAD tells us that, on average, the values in Data Set A deviate from the mean by 2.
In the second case, MAD tells us that, on average, the values in Data Set B deviate from the mean by 3.5.
As we mentioned earlier, MAD and standard deviation are both measures of spread — they tell you how much variation there is in your data relative to the mean. So what then is the difference between MAD and standard deviation, and why would you want to use one of these measures over the other?
While MAD measures the average absolute deviation, standard deviation takes the square root of the average squared deviations.
∑ ∣ x i − x ˉ ∣ n \frac \right |> n ∑ ∣ x i − x ˉ ∣ ∑ ( x i − x ˉ ) 2 n − 1 \sqrt)^2>> n − 1 ∑ ( x i − x ˉ ) 2 At this point, you may be wondering why we bother taking absolute values or squares of the deviations. Good question! Whenever you measure deviations from a mean, you run into the problem of dealing with positive and negative values.Some data points lie above the mean, while others lie below it, so if you measure the deviations as each data value minus the mean ( ∣ x i − x ˉ ∣ \left | x_i-\bar \right | ∣ x i − x ˉ ∣ ), you’ll end up with positive deviations for values that are greater than the mean and negative deviations for values that are less than the mean. The negative deviations partially or wholly cancel out the positive deviations, and this interferes with the goal of calculating the average deviation.
MAD and standard deviations deal with the problem of having positive and negative deviations in different ways. When you calculate MAD, you take the absolute value of the deviations, which turns all the deviations positive. When you calculate standard deviation, you square all the deviations, which turns the deviations positive but changes the units of the deviations into squared units. As an added step, when calculating standard deviation, you take the square root of the average square deviations to get the units back into their original form.
Although standard deviation seems like the clunkier of the two measurements, it’s actually the more common one. Because calculating standard deviation involves squared deviations, it’s more sensitive to values that are farther away from the mean. You can think of standard deviation as a weighted average that gives less weight to the values that lie close to the mean and more weight to the data points that lie farther away from the mean. For this reason, standard deviation tends to be larger than MAD.
To understand mean absolute deviation, you also need to know the following terms:
Assume you have the following data.
To calculate mean absolute deviation without the use of a dot plot, follow these easy steps. Recall that the MAD formula is:
M A D = ∑ ∣ x i − x ˉ ∣ n MAD = \frac \right |> M A D = n ∑ ∣ x i − x ˉ ∣ 1. Calculate the mean of the data, x ˉ \bar x ˉ
− 2 + 1 + 5 + 8 4 = 3 \frac = 3 4 − 2 + 1 + 5 + 8 = 32. Find the absolute deviation between each data point and the mean, ∣ x i − x ˉ ∣ \left | x_i-\bar \right | ∣ x i − x ˉ ∣
∣ − 2 − 3 ∣ = 5 |-2 - 3| = 5 ∣ − 2 − 3∣ = 5 ∣ 1 − 3 ∣ = 2 |1-3| =2 ∣1 − 3∣ = 2 ∣ 5 − 3 ∣ = 2 |5-3| = 2 ∣5 − 3∣ = 2 ∣ 8 − 3 ∣ = 5 |8-3| =5 ∣8 − 3∣ = 53. Take the summation of the distances you found in Step 2, ∑ ∣ x i − x ˉ ∣ \sum \left | x_i-\bar \right | ∑ ∣ x i − x ˉ ∣
5 + 2 + 2 + 5 = 14 5+2+2+5 = 14 5 + 2 + 2 + 5 = 144. Take the average deviation by dividing the sum found in Step 3 by the total number of data points, n. ∑ ∣ x i − x ˉ ∣ n \frac \right |> n ∑ ∣ x i − x ˉ ∣
M A D = 14 4 = 3.5 MAD = \frac<14> = 3.5 M A D = 4 14 = 3.5Outlier (from the co-founder of MasterClass) has brought together some of the world's best instructors, game designers, and filmmakers to create the future of online college.
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