Statistics Questions: Exploring Standard Deviation & Chebychev's Rule

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In summary, there are many concepts in statistics that may seem arbitrary, but they are derived from mathematical principles. The equation for standard deviation involves squaring the deviations to make them positive and dividing by (n-1) instead of n to provide a more accurate estimate for sample data. Chebychev's rule also has a proof, which may be difficult to understand for some. The use of standard deviation is based on the concept of normal distributions, where the mean and variance are the first two moments that determine the shape of the distribution. Other probability distributions may have different numbers of moments, and some may not have any moments at all.
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Moose352
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It seems to me that a lot the concepts in statistics are rather arbitrary and don't seem be mathematically derived. For example, how is the equation for standard deviation derived? The textbook says that the standard deviation is the mean of all of the deviations of the values in the sample and since all the deviations add up to zero, the values are sqaured to get rid of the negative. I understand that, but why doesn't it just take the absolute value? Why is the square-root taken only after everything has been summed? Furthermore, why is it divided by (n-1) and not n?

Also, can anyone explain the proof for Chebychev's rule?

Thanks very much.
 
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  • #2
Hello Moose,

The standard deviation formula can be derived, my Maths teacher showed me. Unfortunately I am unable to derive it so I will not be much help there. The reason we divide by n-1 sometimes is to give as a most accurate or unbiased estimate for sample data. There is also proof for that, which I am also unable to do. Hope this helped.

Regards,

Daniel
 
  • #3
Thanks repugno. It's good to know that there is a proof, but I will not be convinced until i see it.
 
  • #4
Lol .. tried to give you some Mathematical proof, seems that I can't get the Latex code right. You're on your own now. :D
 
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  • #5
Ah! Please try again. I can't find any other proof. I think the real problem is I haven't yet found a concept (of course, granted that I haven't learned much) that the current definition of SD exclusively works for.
 
  • #6
A large amount of statistics is based on the notion of normal distributions.

For normal distributions it is possible to, for example, show that a certain fraction of the results are within a standard deviation of the peak.
 
  • #7
I completely understand. But why does it have to be based on that specific definition standard distribution. Can not those fractions be recalculated based on another definition of the standard deviation?
 
  • #8
Very early in the history of statistics they did use absolute values. But the math of those is difficult: they are not "analytic functions". Squares on the other hand are polynomials, easy to work with. In fact the real number here is the variance, the square of the standard deviation (or rather, the standard deviation is the square root of the variance).

Any probability distribution that has moments has the mean as its first moment, and the variance as its second moment (essentially and with some tech fiddles). What are moments? well in one sense they are the parameters that determine the equation of the probability curve. The normal curve is distinguished because it has only those two moments; it is a two parameter curve. You tell me where the mean is and what the standard deviation is and I will give you the formula for that notmal curve and be able to draw it. Any other probability distribution that has moments at all - some don't - will be determined if you specify all of its moments, which may be an infinite number.
 
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Originally posted by selfAdjoint
Any other probability distribution that has moments at all - some don't - will be determined if you specify all of its moments, which may be an infinite number.
Wow, that sounds interesting. Is this like a Taylor expansion of a function? If the distribution has no moments, is it trivial?
 

1. What is standard deviation and why is it important in statistics?

Standard deviation is a measure of how spread out the data is from the average or mean. It is important in statistics because it helps us understand the variability of the data and make comparisons between different sets of data.

2. How do you calculate standard deviation?

To calculate standard deviation, you first find the mean of the data set. Then, for each data point, find the difference between that data point and the mean, square the difference, and add all of these values together. Finally, divide this sum by the total number of data points and take the square root to get the standard deviation.

3. What is Chebychev's Rule and how is it used in statistics?

Chebychev's Rule, also known as the Empirical Rule, states that for any data set, regardless of its shape, at least 75% of the data falls within two standard deviations from the mean and at least 89% falls within three standard deviations from the mean. It is used in statistics to help determine the likelihood of certain values falling within a certain range.

4. Can standard deviation ever be negative?

No, standard deviation cannot be negative. It is a measure of variability and therefore must be a positive value.

5. How does increasing or decreasing standard deviation affect the shape of a data set's distribution?

Increasing standard deviation will result in a wider spread of data points, making the distribution more spread out or "flatter". Decreasing standard deviation will result in a narrower spread of data points, making the distribution more peaked or "skinnier".

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