Changes to Standard Deviation?

In summary, Standard Deviation has changed from \sqrt{\frac{\Sigma(x_i - \overline{x})^2}{n}} to \sqrt{\frac{\Sigma(x_i - \overline{x})^2}{n - 1}} due to the need for an unbiased estimator for the population variance. Using n-1 instead of n in the denominator provides the appropriate correction for this. Degrees of freedom are important to consider in more advanced statistics like ANOVA.
  • #1
The Bob
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Changes to Standard Deviation??

How many of you know that Standard Deviation has changed.

It used to be:[tex]\sqrt{\frac{\Sigma(x_i - \overline{x})^2}{n}}[/tex]

And now it is:[tex]\sqrt{\frac{\Sigma(x_i - \overline{x})^2}{n - 1}}[/tex]

It is the Variance of data but square rooted:

[tex]s^2 = \frac{\Sigma(x_i - \overline{x})^2}{n - 1}[/tex] convertd to: [tex]s = \sqrt{\frac{\Sigma(x_i - \overline{x})^2}{n - 1}}[/tex]

Not really anything important, just wanted people to know and comment (if necessary) on the fact that it has changed.

The Bob (2004 ©)
 
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  • #2
Actually, both formulae are used... I forget the reasons for using n instead of n-1, though.
 
  • #3
It depends on what you are using for the mean. If you know the mean, then you divide by n. If you estimate the mean from the sample, then you use n-1, because the estimated mean has a statistical error.
 
  • #4
Hurkyl said:
Actually, both formulae are used... I forget the reasons for using n instead of n-1, though.
I do understand that both are still used but I didn't realize why until:
mathman said:
It depends on what you are using for the mean. If you know the mean, then you divide by n. If you estimate the mean from the sample, then you use n-1, because the estimated mean has a statistical error.
- Mathman came along and said why.

Cheers guys.

The Bob (2004 ©)
 
  • #5
mathman said:
It depends on what you are using for the mean. If you know the mean, then you divide by n. If you estimate the mean from the sample, then you use n-1, because the estimated mean has a statistical error.
Why does using n-1 instead correct the error? Is this negligible for large values of n?
 
  • #7
n-1

n-1 is used for samples in order to adjust for the variability of the data set which does not included all possible events.

using n tends to produce an undersestimate of the population variance. So we use n-1 in the denominator to provide the appropriate correction for this tency.

to sum up:
when using populations, use n as the denominator.
else
use n-1

hope that helps!
 
  • #8
The factor (n-1) is used to make the sample variance an "unbiased estimator" of the population variance.

There's no particular reason you need an unbiased estimator, though. For example, if you want to minimize mean squared error, it turns out that it's much better to use (n+1) instead of (n-1) (in case of a normal distribution). See http://www-laplace.imag.fr/Jaynes/prob.html [Broken], chapter 17.
 
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  • #9
The denominator is not the sample size, but the number of degrees of freedom. Initially the two are equal, but when you do the mean (sum over n) you "fix" or "lose" one degree of freedom. So when you then go to use the mean in the sd calculation, you have only n-1 degrees of freedom left.

Degrees of freedom are a thorny thing to teach, and they only become essential to consider in things like ANOVA, so they are frequently skipped in teaching simple statistics.
 

What is standard deviation and why is it important?

Standard deviation is a measure of variability or spread in a data set. It tells us how much the data points deviate from the mean. It is important because it helps us understand the distribution of the data and make comparisons between different sets of data.

What causes changes to standard deviation?

Standard deviation can change due to changes in the data values, such as the addition or removal of outliers. It can also change if the mean changes, as standard deviation is calculated based on the mean. Changes in the sample size can also affect standard deviation, as larger sample sizes tend to have smaller standard deviations.

How do changes to standard deviation affect our interpretation of data?

Changes to standard deviation can affect our interpretation of data by indicating the level of variability in the data set. A smaller standard deviation means the data points are closer to the mean, while a larger standard deviation means the data points are more spread out. This can impact our understanding of the data's distribution and any conclusions we draw from it.

Can standard deviation be negative?

No, standard deviation cannot be negative. It is a measure of variability and cannot be less than 0. A negative value for standard deviation would not make sense in the context of a data set.

How is standard deviation calculated?

Standard deviation is calculated by taking the square root of the variance. The variance is calculated by finding the average of the squared differences between each data point and the mean. This value is then squared to get the standard deviation.

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