Biased/unbiased Standard Deviation

Posted on 2002-06-10
Medium Priority
Last Modified: 2013-11-13
In standard deviation formula we sometimes divide by (N) and sometimes (N-1)
where N = number of data points.

Somewhere I read that 'N' or 'N-1' does not make difference for large datasets.
but when we calculate std. dev. for less than 20 data points, dividing by 'N' gives
a biased estimate and 'N-1' gives unbiased estimate.

Can someone explain with example..how does subtracting 1 help?
Question by:prashant_n_mhatre
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 2
  • 2

Expert Comment

ID: 7080054
standart deviation is the square root of the mean of the square of the deviation:

average = A

sample = x

deviation = x-A

square of deviation = (x-A)^2

mean of the square of the deviation = Sum((x-A)^2) / N

N = number of samples.

standart deviation = Sqrt(Sum((x-A)^2) / N)

That's all I know. And that is what I found on a web page:

"You use the N-1 if the estimate is unbiased". And the definition of bias is:

"A statistic is biased if, in the long run, it consistently over or underestimates the parameter it is estimating. More technically it is biased if its expected value is not equal to the parameter. A stop watch that is a little bit fast gives biased estimates of elapsed time. Bias in this sense is different from the notion of a biased sample. A statistic is positively biased if it tends to overestimate the parameter; a statistic is negatively biased if it tends to underestimate the parameter. An unbiased statistic is not necessarily an accurate statistic. If a statistic is sometimes much too high and sometimes much too low, it can still be unbiased. It would be very imprecise, however. A slightly biased statistic that systematically results in very small overestimates of a parameter could be quite efficient."

Hope it helps. I didnt get it very well...

Expert Comment

ID: 7089880
Okay - too long since I've done this stuff - but I can tell you for definite that you can derive the formula for standard deviation from a method called the Maximum Likelihood Estimator. This is essentially a (quite complex) method which will give you an estimator for a statistic for your data. Because it is complex, it can be difficult to solve for some statistics, but (relatively) easy for the mean and variance. As part of the derivation it can be found that while dividing by N given an unbiased estimator for a population, it would give a biased estimator for a sample. Dividing by N - 1 will solve the problem for a sample. If you really want, I can try to dig out some links for MLE, but quite honestly the logic ain't easy! Essentially in the calculation of an MLE there is also a bias element. You can trade off bias for accuracy (if memory serves).

I'm sorry the explanation isn't a simple one - but it's the best I can do without trying to relearn my college notes on the topic (and that's not worth 1000 points!!!).

Author Comment

ID: 7091078
Still it is not fully clear to me...let us keep this question open for few days !!!!

Accepted Solution

gd2000 earned 300 total points
ID: 7092624
Try the following links. Probably unlikely to explain things in clear terms (mainly because I'm not sure a wholly accurate explanation can be put in lay terms), but at least it will give you something to chew on (if you want to delve in further!).

http://www.asp.ucar.edu/colloquium/1992/notes/part1/node21.html (actually explains the reasons)

MLE for the mean: http://www.esg.montana.edu/eguchi/Biol504Fall2000/MaxLikelihoodsummary.pdf

Introduction to MLE: http://socserv.socsci.mcmaster.ca/jfox/Courses/soc740/MLE.pdf

The final link says that the property of an MLE estimator “asymptotically unbiased – but may be biased in finite samples”. This is essentially the reason (you need to multiply by n / (n - 1) to make the variance estimator unbiased for samples, you can see this from the bias term.

Author Comment

ID: 7124388
Thank you all...

The reson is explained very well in "Statistics In Plain English" book. Yep, it has to do with 'Sample' and 'population'. Generally the standard deviation calculated using sample is lower than population. To accomodate that, we divide it by N-1. Dividing by 1000 or 999 doesn't make much difference..but 10 or 9 numbers do...

I'd a look at Maximum Likelihood...I did study it long back....The links are very useful.

Featured Post

Free Tool: Site Down Detector

Helpful to verify reports of your own downtime, or to double check a downed website you are trying to access.

One of a set of tools we are providing to everyone as a way of saying thank you for being a part of the community.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Before You Read The Article Please make sure you understand these two concepts: Variable Scope (http://www.php.net/manual/en/language.variables.scope.php) and Property Visibility (http://www.php.net/manual/en/language.oop5.visibility.php).  And to …
Have you ever thought of installing a power system that generates solar electricity to power your house? Some may say yes, while others may tell me no. But have you noticed that people around you are now considering installing such systems in their …
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…
Suggested Courses

762 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question