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I have the following set of data:

Raw Scores   x-x̄        (x-x̄)²
-----------------------------
 7           -6            36
 8           -5            25
10           -3             9
14            1             1
26           13           169
-----------------------------
65            0           240
-----------------------------

From the raw scores we know that : $$ n=5 $$ and ∴ the mean, $$ x̄ = \frac{\sum}{n} = \frac{65}{5} = 13 $$

I then proceeded to complete the table above filling in $$ x-x̄ $$ & $$ (x-x̄ )² $$

I am now required to calculate the standard deviation & have the following eqation.

$$ \sigma = \sqrt{\frac{\sum(x_i-\overline{x})^2}{n}} = \sqrt{\frac{240}{5}} = 4 \sqrt{15} $$

However when I submit my data to wolfram and query standard deviatioin it returns the answer$$2\sqrt{15}$$

Maybe this is out of the scope of my learning but being curious I opened the step by step answer and I see that wolfram calculates the standard deviation by

$$ \sigma = \sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}} $$

My textbook accessible here and another textbook my teacher printed out have the eqation , I find it hard to believe that My teacher and 2 other textbooks can be wrong however Im pretty sure the developers at wolfram couldn't have got it wrong either.

What am I missing?

Ayvadia
  • 143

1 Answers1

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The difference is that you are calculating the true standard deviation, and Wolfram|Alpha is calculating the sample standard deviation.

Here's the difference, if you use the formula $$ \sigma = \sqrt{\frac{\sum(x_i-\overline{x})^2}{n}} $$ then you are answering the question "how much on average does my data differ from its mean?"

If you use the formula $$ \sigma_S = \sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}} $$ you are answering the question "if my data is a representative sample of a population, what is my best estimate of how much the population differs from its mean on average?"

nullUser
  • 27,877
  • Thanks This satisfies my curiosity will definitely upvote when I have the rep to do so straying from the topic , how did you format your eqations?I tried editing yor post to see the code behind it I can see that code but where will I find more examples?? – Ayvadia Oct 28 '13 at 05:10
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    See http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference for a short tutorial. – nullUser Oct 28 '13 at 05:12