FINALLY! Why we divide by N-1 for Sample Variance and Standard Deviation

The best and simplest explanation of why we divide the sample variance by n-1. This step-by-step explanation is clear and concise and makes sense! :)

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**** Check out some of our other mini-lectures:

Watch my complete 40 min lecture on Simple Regressions:

Ever wondered why we divide by N-1 for sample variance?

Simple Introduction to Hypothesis Testing:

A Simple Rule to Correctly Setting Up the Null and Alternate Hypotheses:

The Easiest Introduction to Regression Analysis:

Super Easy Tutorial on Calculating the Probability of a Type 2 Error:

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[QuantConceptsE]

Hi my viewers! Are you in need of an online tutor? If so, check out the video description for details 😊

[SansSariph]

thank you - watched four videos explaining n-1 and this is the first one that clicked. combination of demonstrating degrees of freedom and explaining the sample mean error helped a lot.

[karleisheim]

First of all, thanks for the video. I still don't get, why in the example with the population mean at the end, you said that depending on the value of X4 (10/20/50), the results differ. I mean if you change X4, then also the mean changes and the result will be 0 as for the sample variance or what am I seeing wrong? Thanks for any help :)

[file]

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

[RoyalRiku]

Great vid! From what I gathered, the population mean is predetermined/fixed and no calculation of estimation is needed which would have developed a constraint, leading to all observations contributing to DoF. However, I understand the example at 5:20 is meant to show the significance of each observation on the numerator of population variance, but I think to those who are mathematically inclined see this as a contradiction to the tautology of the sum of (observation minus their mean) = 0 instead of a simple demonstration of the influence each population observation has :)

[PedroRibeiro-zsgo]

Boy that was a very interesting explanation of degrees of freedom! Thanks!

[chuckbecker]

These videos are superbly brilliant, THANK YOU!

[kowtharhassan]

Thank you v much but my Q is why do u assume that population mean is 10? I mean this assumption is based on what? May be it is not compatible with reality of that population? Actually can we ever measure the mean of weight for instance of a whole population of a country? By the time we are done measuring or entering data of all population probably their weights have changed.

[ejomaumambala]

You didn't fulfil the promise of the title. Ok, the dof is n-1. That doesn't explain why we divide by that quantity.

[jiahaoliu]

omg you literally saved my life thank you so much!

[YohanesRonald]

Why do we lose just one observation for calculating sample variance? Is it because degree of freedom n-1 in calculating sample mean? Thank you for your answer. I like how you explain stat in layman manner.

[rafaelportela]

Great explanation of degrees of freedom! Congrats.

For those trying to get a deeper explanation on the matter, try this wikipedia link:


Best of luck!

[bowtangey]

Using the sample mean will always result in a smaller numerator than using population mean? No, what if the sample mean equals the population mean? (Say the sample is the three numbers 9, 10, and 11.)

Also, the argument of n-1 around 4:24 seems like a bit of hocus-pocus. True, given the sample mean and n-1 of the sample points, you can calculate the last sample point. But getting the sample mean itself did require all n points in the first place! Am I missing something?

[lhodgins]

I don't understand how sample variance has degrees of variance of n-1: surely if you knew X1, X2, and X3, you wouldn't be able to calculate the variance without knowing X4, and the value X4 took would affect the value of the sample variance?

[СергейПоклонский-хъ]

still makes no sense for me. say you have 4 and 2. their mean is 3. technically 4 and 2 do deviate from 3 only by 1. however, any software will tell you 1.41. makes absolutely no sense for me. i have watched like 10 videos and it seems also tutors do not completely understand it.

[flagshipbuilds]

Dividing by (n-1) is used for samples; unlike for population.

[ThePhysics]

"one observation does not contain critical input"

So we pretend its not there and divide with n-1 observations but we still take that observation and sum it up in the numerator in the calculation of variance.
That does not make sense.

We divide with n-1 instead of n because we have n-1 observations with critical input, but why does that last observation still go into the sum of squares? Why doesn't the sum in the calculation of variance also ignore one element and goes from 1 to n-1 instead of 1 to n...
This explanation does not make sense.

[hardikvegad]

can someone explain me this in a single line...if possible

[asr]

I give up. Maybe it requires much more math & longer tutorials to understand why it's (n-1)

[alekssandroassisbarbosa]

"always result in a smaller numerator"? I'm crazy or this is a lie. if you not divide by n-1 you'll get a different result, ok, but it's not the point here. why you said that? I think it's the point for me to understand it. Is this n-1 not better for predict variation from mean of samples that you really find the population mean possible to calculate? Imagine a case you don't know the population mean, how would you say there's some degree of freedom if you have not even noticed that the first or last observation. Ok, if we are dealing with last observation we are not dealing with statistics.. but the doubt is still the same... why you said that? Maybe i answered this question inside my brain but I really wanna be sure