Why standard deviation is preferred over mean deviation?

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Discussion Overview

The discussion revolves around the comparison between standard deviation and mean deviation as methods to measure dispersion and variability in data. Participants explore the mathematical properties, conceptual similarities, and differences between the two measures, as well as their applications in probability and statistics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants note that standard deviation is preferred because it has more prominent roles in probability and statistics compared to mean deviation.
  • One participant suggests that the squaring of deviations in standard deviation makes larger deviations more significant, which could be seen as beneficial since large deviations are less likely.
  • There are claims that standard deviation and mean deviation should yield similar results, but others argue that they are defined differently and do not necessarily have the same value.
  • Some participants highlight that while standard deviation provides specific percentages of data within certain ranges (e.g., 68%, 95%, 99% for normal distributions), this property does not apply universally to all data distributions.
  • It is mentioned that for non-normally distributed populations, mean deviation and standard deviation may not maintain a consistent proportional relationship.
  • Participants discuss the mathematical properties of standard deviation, such as differentiability and its connection to correlation and covariance, which are not as neatly associated with mean deviation.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between standard deviation and mean deviation, with some asserting similarities in concept while others emphasize their distinct definitions and properties. The discussion remains unresolved regarding the extent to which the two measures can be considered equivalent or interchangeable.

Contextual Notes

Participants note that the properties of standard deviation, such as its mathematical behavior and association with correlation, contribute to its preference over mean deviation. However, the discussion acknowledges that these properties may not apply uniformly across all types of data distributions.

Shehbaj singh
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I was studying in my math textbook about these two methods to measure dispersion and variability in data. I was able to understand mean deviation that it's the average by how much a quantity deviates from mean but I was unable to understand standard deviation. Also, as standard deviation squares deviation it makes big deviation bigger. So, please help to understand why it's preferred over mean deviation.
 
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Hello Shehbaj, :welcome:

Squares are a lot more comfortable to deal with than absolute values. But the real reason is that standard deviations have a much more prominent role in probability and statistics than mean deviations. Your 'squares make big deviations bigger' is in fact an argument pro: big deviations are a lot less likely than small deviations.

If you are -- as I guess -- just being introduced to this matter, my advice is to accept it for the moment and exercise patience. It'll become pretty obvious after a while.

:smile:
 
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Okay. Thanks for your suggestion . I am somewhat weird student and gets excited very often in topics of maths.
 
Being excited in maths is not weird in itself, but claiming you're weird because of it is.
 
Standard deviation should be the same as mean deviation. Because the standard deviation is how much a set of values vary from the mean of the values.

Apart from this, I'm not sure.
 
mathexam said:
Standard deviation should be the same as mean deviation. Because the standard deviation is how much a set of values vary from the mean of the values.
Certainly not !
 
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BvU said:
Certainly not !

Can you explain?

What I think it is, is that standard deviation represents the answer differently but they are similar in concept. I know standard deviation also deal with how much the numbers vary from the mean of the group.
 
The two are clearly defined. There is no reason they should have the same value.
 
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Yeah, I believe it's similar in concept but defined differently as output. With standard deviation, 1 standard deviation entails 68% of the data, 2 standard deviations entails 95% of the data, while 3 is over 99% of the data. Mean deviation probably holds a different value in terms of this.
 
  • #10
mathexam said:
Yeah, I believe it's similar in concept but defined differently as output. With standard deviation, 1 standard deviation entails 68% of the data, 2 standard deviations entails 95% of the data, while 3 is over 99% of the data. Mean deviation probably holds a different value in terms of this.
It is not automatically true that 68% of the points in a data set will lie within one standard deviation of the mean. That result holds for data sets that follow a "normal" distribution. Yes, it is true that for a normal distribution, the mean deviation and the standard deviation will be in proportion. But for a population that is not normally distributed, the mean deviation and the standard deviation need not be in that same proportion.
 
  • #11
jbriggs444 said:
It is not automatically true that 68% of the points in a data set will lie within one standard deviation of the mean. That result holds for data sets that follow a "normal" distribution. Yes, it is true that for a normal distribution, the mean deviation and the standard deviation will be in proportion. But for a population that is not normally distributed, the mean deviation and the standard deviation need not be in that same proportion.

A bound is possible though. For example, for any distribution with mean and variance, it is known that at least 75% of the data lies within 2 standard deviations of the mean.

Anyway, you will often hear people say that we work with standard deviation instead of mean deviation because it has nicer properties mathematically, such as differentiablity. This is only part of the story. The real answer is the concept of a correlation and a covariance. This is a very natural concept which measures the degree of linear association between two data sets. This leads automatically to the concept of a variance. The mean deviation does not have a nice associated "correlation measure".
 
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  • #12
Shehbaj singh said:
I was studying in my math textbook about these two methods to measure dispersion and variability in data. I was able to understand mean deviation that it's the average by how much a quantity deviates from mean but I was unable to understand standard deviation. Also, as standard deviation squares deviation it makes big deviation bigger. So, please help to understand why it's preferred over mean deviation.
It is because the standard deviation has nice mathematical properties and the mean deviation does not.

The variance is the square of the standard deviation. The sum of the variances of two independent random variables is equal to the variance of the sum of the variables. This is fundamental.
 

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