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Reviewing the population mean, sample mean, population variance, sample variance and building an intuition for why we divide by n-1 for the unbiased sample variance
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Thanks! Very intuitive explanation.
what a brilliant explanation of the concept. Cheers
Not quite convinced!
The proof from 8:11 onwards does not seem rigorous. Correct me if I'm wrong.
The sum of squares of the distances the 3 samples have to pop. mean, divided by 3, could potentially be an over-estimation of the pop. variance. Because you could have picked 3 samples that are all far from the pop. mean, whereas in the population there are points much closer to the pop. mean.
Which means: even the sum of the squares of the distances the 3 samples have to the sample mean, divided by 3, is always smaller than the sum of squares of the distances the 3 samples have to pop. mean, divided by 3, it is still not enough to show that the sum of the squares of the distances the 3 samples have to the sample mean, divided by 3, is always smaller than the pop. variance.
But you haven't explained it ! LOL
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?
6:10 Why we divide by n – 1 in variance
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
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.
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
8:40 I think you should not represent the true variance and the sample variance on the same number line you drew for the population points. Also the consequence of your putting them together is you're visualizing the distance between the sample variance and the population variance on the same number line, resulting in your conclusion that because the sample points are far from the population mean, the variance is far too. Ponder over it, you'll realize.
Love your lectures BTW 😃
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
9:08 – You are just as likely to be overestimating, you just chose to pick the bottom points rather than the top ones. This offers literally NO explanation, let alone an intuitive one, as to why I should expect there to be a downwards bias.
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?
starts at 5:05
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.