What is the average absolute deviation for the set of numbers?
A website captures information about each customer's order. The total dollar amounts of the last 8 orders are listed in the table below. What is the mean absolute deviation of the data? Show
There are many measurements of spread or dispersion in statistics. Although the range and standard deviation are most commonly used, there are other ways to quantify dispersion. We will look at how to calculate the mean absolute deviation for a data set. DefinitionWe begin with the definition of the mean absolute deviation, which is also referred to as the average absolute deviation. The formula displayed with this article is the formal definition of the mean absolute deviation. It may make more sense to consider this formula as a process, or series of steps, that we can use to obtain our statistic.
VariationsThere are several variations for the above process. Note that we did not specify exactly what m is. The reason for this is that we could use a variety of statistics for m. Typically this is the center of our data set, and so any of the measurements of central tendency can be used. The most common statistical measurements of the center of a data set are the mean, median and the mode. Thus any of these could be used as m in the calculation of the mean absolute deviation. This is why it is common to refer to the mean absolute deviation about the mean or the mean absolute deviation about the median. We will see several examples of this. Example: Mean Absolute Deviation About the MeanSuppose that we start with the following data set: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. The mean of this data set is 5. The following table will organize our work in calculating the mean absolute deviation about the mean.
We now divide this sum by 10, since there are a total of ten data values. The mean absolute deviation about the mean is 24/10 = 2.4. Example: Mean Absolute Deviation About the MeanNow we start with a different data set: 1, 1, 4, 5, 5, 5, 5, 7, 7, 10. Just like the previous data set, the mean of this data set is 5.
Thus the mean absolute deviation about the mean is 18/10 = 1.8. We compare this result to the first example. Although the mean was identical for each of these examples, the data in the first example was more spread out. We see from these two examples that the mean absolute deviation from the first example is greater than the mean absolute deviation from the second example. The greater the mean absolute deviation, the greater the dispersion of our data. Example: Mean Absolute Deviation About the MedianStart with the same data set as the first example: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. The median of the data set is 6. In the following table, we show the details of the calculation of the mean absolute deviation about the median.
Again we divide the total by 10 and obtain a mean average deviation about the median as 24/10 = 2.4. Example: Mean Absolute Deviation About the MedianStart with the same data set as before: 1, 2, 2, 3, 5, 7, 7, 7, 7, 9. This time we find the mode of this data set to be 7. In the following table, we show the details of the calculation of the mean absolute deviation about the mode.
We divide the sum of the absolute deviations and see that we have a mean absolute deviation about the mode of 22/10 = 2.2. Fast FactsThere are a few basic properties concerning mean absolute deviations
Common UsesThe mean absolute deviation has a few applications. The first application is that this statistic may be used to teach some of the ideas behind the standard deviation. The mean absolute deviation about the mean is much easier to calculate than the standard deviation. It does not require us to square the deviations, and we do not need to find a square root at the end of our calculation. Furthermore, the mean absolute deviation is more intuitively connected to the spread of the data set than what the standard deviation is. This is why the mean absolute deviation is sometimes taught first, before introducing the standard deviation. Some have gone so far as to argue that the standard deviation should be replaced by the mean absolute deviation. Although the standard deviation is important for scientific and mathematical applications, it is not as intuitive as the mean absolute deviation. For day-to-day applications, the mean absolute deviation is a more tangible way to measure how spread out data are. What is the mean absolute deviation of a set of numbers?Mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set. Mean absolute deviation helps us get a sense of how "spread out" the values in a data set are.
What is the mean absolute deviation of the data set 7 10 14 and 20?And the answer for this is 12.75.
What is the average deviation of the data set?The average deviation of a set of scores is calculated by computing the mean and then the specific distance between each score and that mean without regard to whether the score is above or below the mean. It is also called an average absolute deviation.
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