Difference in formula between sample and population variance?
Question
Explain the difference with example
solved
1
Statistics
4 years
10 Answers
5760 views
Grand Master 0
Answers ( 10 )
Simply put, We don’t have the concept of degrees of freedom when it comes to parameters (Population Variance) however for Statistics we lose a degree of freedom for every assumed parameter.
Hence the only difference is denominator we used for calculation of both the variances. In Population, we divide by number of observations, however, in the sample, we divide by degrees of freedom which is the number of observations minus 1 (as we calculate the variance given the mean, in this, we have lost one degree of freedom)
To put it simply (n−1) is a smaller number than (n). When you divide by a smaller number you get a larger number. Therefore when you divide by (n−1) the sample variance will work out to be a larger number.
Let’s think about what a larger vs. smaller sample variance means. If the sample variance is larger than there is a greater chance that it captures the true population variance. That is why when you divide by (n−1) we call that an unbiased sample estimate. Whereas dividing by (n) is called a biased sample estimate.
Because we are trying to reveal information about a population by calculating the variance from a sample set we probably do not want to underestimate the variance. Basically by just dividing by (n) we are underestimating the true population variance, that is why it is called a biased estimate.
Basically comes down to calculating a biased vs. unbiased sample variance estimate.
When it comes to calculating variance in sample we divide it by (n-1) instead of n as we do it in population. When it comes to sample , we take only a portion of all the samples and that somehow underestimates the effect of all parameters which can be proved by experiments.
So to negate this effect , we divide it by a slightly lesser value, in this case ( n-1) , so when the denominator shrinks, the overall value of the variance increases and it comes closer to the value of population variance.
We divide with n while calculating the population variance and with (n-1) while calculating the sample variance where n is the total number of observations. We are actually trying to calculate the variance of a sample and estimating this variance to the population. So, we use a slightly lesser value in the denominator so that the variance of the sample increases and can be estimated to the population variance.
Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data.
Population variance = the sum of x minus the mean squared divided by N
Sample Variance = the sum of x minus the mean squared divided by N-1
1) There is only one little difference in the calculation of variance and it is at the very end of it. For both population and sample variance, I calculate the mean, then the deviations from the mean, and then I square all the deviations. I sum all the squared deviations up. So far it was the same for both population and sample variance. When I calculate population variance, I then divide the sum of squared deviations from the mean by the number of items in the population BUT for sample variance, I divide it by the number of items in the sample less one.
2) As a result, the calculated sample variance (and therefore also the standard deviation) will be slightly higher than if we would have used the population variance formula.
3) The purpose of this little difference it to get a better and unbiased estimate of the population‘s variance (by dividing by the sample size lowered by one, we compensate for the fact that we are working only with a sample rather than with the whole population)
4) This is also called Bessels’ correction
DEFINITIONS:
1) VARIANCE is defined and calculated as the average squared deviation from the mean
2) STANDARD DEVIATION is calculated as the square root of the variance
3) A POPULATION is defined as all members of a specified group
4) A SAMPLE is a part of a population that is used to describe the characteristics (e.g. mean or standard deviation) of the whole population. The size of a sample can be less than 1%, or 10%, or 60% of the population, but it is never the whole population
Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data.
The main difference between population variance and sample variance relates to calculation of variance. Variance is calculated in five steps. First mean is calculated, then we calculate deviations from the mean, and thirdly the deviations are squared, fourthly the squared deviations are summed up and finally this sum is divided by number of items for which the variance is being calculated. Thus variance= Σ(xi-x-)/n. Where xi = ith. Number, x- = mean and n = number of items..
Now, when the variance is to be calculated from population data, n is equal to the number of items. Thus if variance in blood pressure of all the 1000 people is to be calculated from data on blood pressures of all the 1000 people, then n = 1000. However when the variance is calculated from sample data 1 is to be deducted from n before dividing the sum of the squared deviations. Thus in the above example if sample data have 100 items, the denominator would be 100 – 1 = 99.
Due to this, the value of variance calculated from sample data is higher than the value that could have been found out by using population data. The logic of doing that is to compensate our lack of information about the population data.
Population variance refers to the calculation of variance by taking into account each and every data point in the population whereas Sample variance refers to the calculation of variance by taking into account every data point within the sample.
However, it is not practically possible to get the data of every data point in a relatively large population set, therefore, we *estimate* the variance of the population with the help of our sample.
Since we are estimating the population variance from the sample, we probably do not want to underestimate the variance. So, we divide the sum of squared deviations from the mean by N-1 instead of N.
As a result, the calculated sample variance (and therefore also the standard deviation) will be slightly higher than if we would have used the population variance formula. The purpose of this little difference it to get a better and unbiased estimate of the population‘s variance.
When we are dealing with larger samples say larger than 30 then we may use the same formula as used for population variance because it will not make any major difference in the value of variance
We can use n_ 1 in denominator for large samples too but using n as denomitor for small samples will lead to error