Review and intuition why we divide by n-1 for the unbiased sample | Khan Academy

This is essential reading. A book of similar stripe became a cornerstone in my personal growth. "Game Theory and the Pursuit of Algorithmic Fairness" by Jack Frostwell

Why do we not use |Xi - x̄ | instead of (Xi - x̄ )² ?

What if you take the 3 highest values?

whos this man? he knows so much and explains so majestic. I wonder why he does not have a statue in the main square of my city ? he deserve a few

That was unclear.

Thank you very much for your video, it was very very good at explaining. But I have one more question, If descriptive statistics do not try to generalize to a population (since there is no uncertainty in descriptive statistics), then why does the sample standard deviation try to best estimate the population mean? Yet it is still considered a descriptive statistic

thank you sal :4)

N-1 is "better", but it is still very flawed

Just gotta say, you're videos are awesome. Glad they exist.

So instead of the sample lying somewhere much lower than the true population mean, what if it's lying much higher? Would it be correct to use n+1 instead of n-1 in order to deliberately make the sample variance smaller?

After 3 videos, I finally understood this n-1. Basically when we consider a sample from our population and calculate the mean for it, it may or may not be as close to the overall population mean (which is thr mean that matters) so to lower the possibility of a highly distinct sample mean/variance we use n-1 to reach at least near the population mean...

A very interesting and important discussion. I made a break in the middle and thought about it by myself. I have a rather short explanation: If the sample size n is very small, such as 3, the variance calculated for the sample has more chance to be very different from the actual variance. The smaller the n is, the more effect has this '-1' on the result.
Why do we use '-1' and not some other values like '-2', I think it is just a tradition. For the smallest sample size of 2, this unbiased variance can still be calculated. However, it is not really purely 'unbiased', just relatively 'unbiased'.

Doesn't explain the point sal, sample could've been among the higher than mu values only; in that case this would be completely opposite, we should've divided by n+1 then

This is not explained at all.

But the same can be there for the other end where we would overestimate it?

Awesome video! Thank you!

Is n-1 mathematically derived?

Could we justify doing something else, e.g. using "0.85n" to build in conservativeness even for large n?

Hi
How is this S2 variance of sample different from the sigma squared /n formula ( population variance /n) which is also the sample variance

thanks

Much better than what my school teacher taught me

Because of the upper and lower boundaries, samples are biased to be less spread, compared to the population mean, which is typically more centralized.

What is bogus logic....khan academy is jack of all trade,master of none

I get the math.... What I don't get is how you're able to write with the drawing/annotation feature so freakin' nicely?!?!? Either you missed your calling as a steady-handed microsurgeon or there is some sort of stabilization assistance with the program you're using.

by this logic it can be n+1 also ig

Let's say a report comes out that mentions standard deviation. How are we supposed to know which formula was used to calculate that standard deviation.

What if the sample mean is far greater than the population mean, then would you not divide by n+1 in order that your sample mean is not an overestimate?

The analogy you’re using is probably not very convincing/intuitive enough. Because there’s also a likelihood that the sample is over-estimating the population mean, so why don’t we divide it by n+1?

This is terrible. Still no explanation of why it is unbiased if using n-1.

I would like to know why we use the square of the difference between x and xbar, and not the absolute value of the difference?

NOT one of Khan Academy's shining moments. You're other video (thanks Dhiraj Budhrani) is MUCH better (with the simulation & a mathematical explanation!).

Starts at 5.00

So this means that the n-1 of the sample variance equation was just an arbitrarily chosen value because it's empirically closer to the actual population variance? Or is there any equation or a logical path in deriving the n-1? I kinda see that it's the former but kinda feel that there might be a theory that could explain why n-1 is the most appropriate and not any other value and that it's just a natural consequence of our math. Anyone who does have one, please tell me!
Thank you for the video Khan Academy! It was very informative!

what if all the samples you took were greater than the mean? then you would be overestimating even more if you divide by n-1

I can't understand why we would underestimate variance in general this way. Let's take population [0, 10, 20] and its sample [0, 20]. They have the same mean 10, and variance of the population is (100 + 100 + 0) / 3, while variance of the sample is (100 + 100) / 2, so we overestimate the variance.

So I guess the biased variance is better if your sample is still close to the entire population

I had the intuition that overestimation and underestimation would compensate each other. Why is it not the case?

Why isn't this video on the statistics playlist?

this does not give an explanation for why it is exactly n-1.

still dont get it. yes you would be underestimating it if u take the sample cluster below the mean. but if the cluster is above the mean? you would be overestimating it! seems arbitrary to me.

So why minus - 1? Why not - 2 ? Or minus 6,345 % ? This is still not an explanation of the n - 1 :-(.

Didn't say anything about n-1, misleading title.

I love you, fuck the rest of explanations on internet, this made me understand

If you want a more technical explanation/proof, Wikipedia Bessel's Correction. This video has some good intuition though.

It has to do with the fact that on an interval with N points there are N-1 smallests subintervals. Consider for example the interval [1,4] on the natural number line in which case N=4 You can subdivide it only in [1,2] [2,3] [3,4] which is 3 not 4 smallest subintervals.

The most common question seems to be why n-1 and not n-2 or n-3424342 (any other number). The way I understand it comes from the definition of unbiased estimators (look it up on wikipedia), in a nutshell an unbiased estimator is one whose expected value equals the value it is estimating. n-1 is known as bessel's correction (also on wikipedia). Here you can see that E[S^2]=sigma^2, hence it is unbiased. This makes sense; if you take enough samples and average them, you get true pop value.

I GET IT! I had to work out the proof and think about it really hard, but I get it! I have an intuition for why n-1 makes sense! Message me with your questions, because I don't think I can explain it easily in the comment boxes.