Replacement of Squaring in Variance Equation: Benefits?

In summary, the variance equation sums up the distances between data values and the mean of the set. The squared distances serve a purpose, possibly to keep the numbers positive, but statisticians typically use mean squared deviation instead of mean absolute deviation because it is more useful and directly related to the Normal Distribution. The mean squared deviation also follows a nice law, while the mean absolute deviation does not. Both measures have been studied and used, but the mean squared deviation is more commonly used due to its many benefits. When discussing these measures, it is important to clarify which of the three meanings is being referred to and whether it is being discussed as a random variable or a specific value.
  • #1
vanmaiden
102
1
The variance equation basically sums up all the distances between each data value and the mean of the set. The interesting thing is that each distance and squared for a reason that I believe is to make the distance positive, but why don't the statisticians just take the absolute value of each distance to give a smaller number? Is there some benefit to having a large number to work with? I mean, the smaller numbers nevertheless have decimals that can be used to compare magnitude and such.

Thank you.
 
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  • #2


vanmaiden said:
why don't the statisticians just take the absolute value of each distance to give a smaller number?

That's a good question and I don't think it has a simple answer. There are many reasons why the mean squared deviation is very useful. It is directly related to a parameter in the often used Normal Distribution, while the "Mean Absolute Deviation" (which is what you are proposing as an alternative) is not. If X and Y are independent random variables then variances obey the law Var(X+Y) = Var(X) + Var(Y), but I don't think the Mean Absolute Deviation obeys such a nice law.

The Mean Absolute Deviation has been studied and used, so you can't really say that statisiticans haven't tried it.

If you want to talk about things like the mean squared deviation or the mean absolute deviation, you need to be clear which of the 3 different meanings you are discussing. Each of these things can be 1) A parameter of a probability distribution, 2) a statistic computed from a sample or 3) a formula for estimating a parameter in a probability distribution by using values from a sample. Each of those 3 things can be discussed as 1) a random variable or 2) a specific value of random variable.
 

Related to Replacement of Squaring in Variance Equation: Benefits?

1. What is the purpose of replacing squaring in the variance equation?

The purpose of replacing squaring in the variance equation is to provide a more accurate measure of variability in a set of data. Traditional variance equations square the differences between each data point and the mean, which can result in a distorted measure of variability. By replacing squaring with absolute values, the new equation provides a more robust and reliable measure of variance.

2. How does replacing squaring in the variance equation benefit data analysis?

Replacing squaring in the variance equation benefits data analysis by reducing the impact of extreme values on the overall measure of variability. This allows for a more accurate representation of the data and can lead to more reliable conclusions and insights.

3. Are there any limitations to using the replacement of squaring in the variance equation?

While replacing squaring in the variance equation can provide more accurate results, it may not be appropriate for all types of data. For example, if the data has a natural symmetry or follows a specific distribution, traditional squaring may be more appropriate. It is important to consider the nature of the data before deciding to replace squaring in the variance equation.

4. Can the replacement of squaring in the variance equation be used with any statistical test?

Yes, the replacement of squaring in the variance equation can be used with any statistical test that requires the calculation of variance. This includes t-tests, ANOVAs, and regression analysis. However, it is important to note that the interpretation of the results may differ slightly from traditional variance calculations.

5. Is there any research supporting the use of the replacement of squaring in the variance equation?

Yes, there is research that supports the use of the replacement of squaring in the variance equation. Studies have shown that using absolute values in the variance equation can provide more accurate and robust results, particularly in cases where there are extreme values or outliers in the data. However, more research is needed to fully establish the benefits and limitations of this approach.

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