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In a large data set, to find out to what extent there are variations in relation to a mean score, it is best to use the mean absolute deviation and the standard deviation . The standard deviation is the measure of the dispersion of the results in a data set. To find the total variability of our data set we simply add the deviation of each score from the mean.
The mean deviation of a score can be calculated by dividing the total (total variability of the data set) by the number of scores . The absolute deviation and the standard deviation are measures of dispersion that make it possible to deduce, depending on the measure used, the variation of a score with respect to the mean.
Absolute deviation and mean absolute deviation
The easiest way to calculate the deviation of a score from the mean is to take each of the scores and find the mean. As an example, we will work with the average score of a group of 100 students that appear in the following table.
The mean score of this group of 100 students is 58.75 out of 100. Using the example of the student with 60 out of 100 points, the deviation of this score from the mean is 1.25. This value results from subtracting the student’s score, which is 60, from the mean, which is 58.78. It is important to note that scores above the mean have positive deviations, while scores below the mean will have negative deviations.
On the other hand, if we end up having positive and negative signs, by adding all these deviations they would cancel out, giving us a total deviation of zero. If, for the example, our interest is focused on knowing what the deviation of a score is, but not in what range the mean is, then we can simply dispense with the minus sign and focus our attention on the value that would give us the absolute deviation.
Adding all these absolute deviations and dividing them by the total number of scores we get the mean absolute deviation . Therefore, for our 100 students in this example, the mean absolute difference is 12.81. The formula to obtain it is the following:
Where:
- MAD = mean absolute deviation
- ∑ = sum of.
- X= sample (the score for this example).
- µ= mean
- N = number of values.
So:
- DMA = 1281/100
- DMA = 12.81
Standard deviation
The standard deviation is a measure of the dispersion of results in a data set. In general, this measure is used to find out the variability of the population for the data being measured. However, because we are often only presented with data from a sample, we can estimate the population standard deviation from the sample standard deviation. These two standard deviations, that is, the sample standard deviation and the population standard deviation, are calculated differently.
Sample or population standard deviation when to use each?
Normally we are interested in knowing the standard deviation of the population because our population contains all the values we need. Therefore, we would calculate the population standard deviation if we have the entire population, or if we have a sample from a larger population but are only interested in that sample and do not want to generalize our results to the entire population.
However, the standard deviation is not exempt from being able to provide samples with which we can generalize a population. Therefore, if you only have a sample but want to make a statement about the standard deviation of the population from which it was drawn, you should use the sample standard deviation. Confusion about which standard deviation to use can often arise, since the name “sample” standard deviation is mistakenly interpreted as the standard deviation of the sample itself rather than as an estimate of the standard deviation of a population taking as a sample basis.
The formula for the sample standard deviation is as follows:
Where:
- s = standard deviation of the sample.
- ∑ = sum of.
- X= sample.
- x¯ = sample mean.
- n = number of scores in the sample.
What to consider when calculating the standard deviation
To begin with, it is important to keep in mind that the standard deviation is a measure of dispersion that is used, along with the mean, to reduce continuous data, but not categorical data. In the same way, it is only appropriate to use these forms of data quantification when there is certainty that the continuous data have neither values out of the typical nor biases in a higher percentage.
In conclusion, the mean deviation or mean absolute deviation is calculated in a similar way to the standard deviation, but uses absolute values. This is done to avoid the problem of negative differences between data points and their means. In practice, absolute value means that we must remove any negative signs in front of a number and treat all numbers as positive (or zero).
Sources
- Castillo, O. (2009). Applied Statistics . Dispersion measures .
- Flatiron School. (2015). Measures Of Dispersion .
- Lopez, J. (2017). Standard or typical deviation . Economipedia.com
- Mendizabal, M (2017). How is the absolute deviation calculated ?
