Average Deviation Calculator

The mean absolute deviation is a measure of the dispersion of a set of data. It gives the average distance between each value and a central point (usually the arithmetic mean). The smaller the mean absolute deviation, the more concentrated the data are around the mean; the larger it is, the more spread out the data are.

This calculator will find the average deviation from the mean or median of a set of numbers and show the work. Enter your numbers in the text box, separated by commas, spaces, semicolons, etc. Select an average type to get a complete breakdown.

What indicates the average deviation:

Each value in a data set is a certain distance away from the mean. The average deviation takes all of those distances and combines them into one number that tells you how far values are typically away from the center of the data.

The average deviation is also called the mean absolute deviation and abbreviated MAD. The "absolute" part is important because it uses the absolute value of each distance before averaging them. Without using the absolute values, positive and negative deviations from the mean would cancel out, and the sum would always be zero.

The mean deviation can never be negative and is expressed in the same units as the original data. For example, if you have a temperature series, then the average deviation will also be in degrees. A zero mean deviation means that all numbers are equal.

Formula to calculate average deviation:

In standard deviation (relative to the mean), absolute differences between each value and the mean are summed up and divided by the number of values.

where xᵢ is each individual value, x̄ is the mean and n is the number of values. Replacing the mean with the median gives you the average deviation from the median which this calculator also displays.

Average and median: which reference value to choose?

The only choice that changes the result is which measure of central tendency you use as your reference point. If you use the mean as your reference point, then it's called standard deviation. If you use the median as your reference point, then the result is less sensitive to outliers because the median is not affected much by adding a single extreme value.

An interesting conclusion here is that the average deviation is always smallest when calculated relative to the median. For any data set, the result based on the median will be smaller than the result based on the mean. So the median is a natural reference point when few outliers could affect results.

If the data is nearly symmetrical and contains no outliers, use the mean. If the data has a tendency or contains extreme values that could affect the outcome, choose the median.

Calculation of average deviation:

The formula is very simple but in practice it only consists of four steps.

First, the mean or median is calculated to determine the "center". Then each value is subtracted from this center and thus obtains the deviation for each value. Next, the absolute value of each deviation is taken, which removes any negative signs.

Finally, you add up all of these absolute deviations and divide the result by the number of values. This average is the mean deviation.

Example calculation:

Let's take an example of a data set that is loaded into this calculator and consider eight values: 1, 2, 2, 3, 4, 6, 10, and 12. The sum is 40, so the mean is 5.

The absolute deviations from the mean of 5 are 4, 3, 3, 2, 1, 1, 5, and 7. The sum of these is 26.

Now here we're measuring the absolute deviations from the median. For eight values, the median is the average of the two middle values, so three and four, which gives us 3.5.

The absolute deviations from the median of 3.5 are 2.5, 1.5, 1.5, 0.5, 0.5, 2.5, 6.5 and 8.5, so the sum is 24.

The result based on the median (3.0) is less than the result based on the mean (3.25). This is consistent with what we would expect from the theorem. While the long "tail" of the data set pulls the average up to 5, the median stays at 3.5. So the median gives a more accurate representation of the typical spread.

Comparison of mean deviation and standard deviation.

Both the mean absolute deviation and standard deviation give information about spread but treat distances differently. The mean absolute deviation takes the absolute value of each distance while the standard deviation squares each distance, averages them, then takes the square root.

Because squaring gives more weight to values that are further from the mean, standard deviation reacts more strongly to outliers. The mean absolute deviation treats each distance proportionally to its size, which leads to more stable results and is easier for many people to understand as "the average of distances."

For data that follows a normal distribution (bell curve) they are related to each other by a specific ratio.

Thus the mean absolute deviation of normally distributed data is about four-fifths of the standard deviation. In formal statistical analyses, the standard deviation is still more often used because it is easier to work with mathematically. For a general audience, however, the mean absolute deviation is usually a clearer summary.

Median absolute deviation (MAD)

A related concept that is worth knowing about is the median absolute deviation (MAD). Instead of averaging the absolute deviations from the median, it takes the median of those deviations.

As the median is used as a central value and once as a summary indicator and another time being extremely insensitive to outliers and often used in robust statistics for identifying anomalies. If you open up the associated measurements, you can check out the average deviation along with your own data.

Relative average deviation

The average deviation is expressed in the units of the data, which makes it difficult to compare two sets of data with very different spreads. The relative average deviation solves this problem by dividing the average deviation by the mean and expressing the result as a percentage.

A relative average deviation of 20 percent means that each value is typically about one-fifth of the mean away from the mean. Because it's a pure ratio, monthly precipitation and daily stock prices can be compared on the same scale.

Use of average deviation

In forecasting and supply chain planning, the average deviation of forecast error is a key accuracy indicator usually referred to as the mean absolute deviation of the forecast. The lower this value is, the closer the forecast is to actual reality.

In finance, the average absolute deviation measures how far prices or returns typically deviate from their average value and provides a clear picture of volatility. Compared to standard deviation, it is less sensitive to extreme values on any given day.

In manufacturing and quality management a small average absolute deviation indicates a more stable process, while an increase in the average absolute deviation may indicate possible inaccuracies in the measurements. Teachers use it to show how spread out or concentrated student performance is around the mean score of the class.

Used symbols:

The symbols used in the above formulas are shown in the table below.

This calculator is for general educational and analytical purposes only. For formal statistical work you should check which measures of central tendency or variability are most appropriate to answer your particular question.