To estimate population mean, we apply the central limit theorem:

"Consider a population with mean `mu` and standard deviation `sigma`, the mean of a sample with size `n` is `bar x`. When `n` increases to infinite, `bar x` is normally distributed with mean `mu` and standard deviation `sigma//sqrt(n)`."

So the sample mean `bar x` can be used as unbiased estimator for the population mean `mu`.

On presenting the result of point estimation, we can include the value of standard error. In case of population mean, standard error is calculated by formula:

`SE=sigma/sqrt(n)`

(2)

In (2), `s` is the standard deviation of sample, `n` is the number of elements of sample.

Example : After weighing 30 cans manufactured by company C, the result shows that the mean weight of cans is 365,2 g, standard deviation of this parameter is 12,4 g.

So we estimate that the mean weight of cans manufactured by this company is 356,2 g with standard error:

  `SE=s/sqrt(n)=(12,4)/sqrt(30)=2,26`

And presents this result as:

  `bar W=356,20+-2,26` g

 

Sample proportion can be used to estimate population proportion. In this case, standard error can be calculated by:

`SE=sqrt((p(1-p))/n)`

(3)

In (3) `p` is sample proportion.

Example : Conducting a survey with 150 students, there are 48 students which go to university by bus. The proportion of this group of student is `48//150 = 0,32` (or 32%). Standard error of this estimation is:

  `SE=sqrt((0,32xx0,68)/150)=0,038`

So the proportion of student going to university by bus is `0,320+-0,038` (or `32,0+-3,8\ %`)

 

When we have already all the data of a population, its variance is calculated by:

`sigma^2=(sum_(i=1)^N (x_i-mu)^2)/N`

(4)

Whereas, variance of a sample is determined by:

`s^2=(sum_(i=1)^n (x_i-barx)^2)/(n-1)`

(5)

We can recognize that there are some difference between (4) and (5). The reason of this difference is that (5) is the unbiased estimate of variance.

When random variable conform to normal distribution, standard error of variance estimate is calculated by:

`SE=sqrt(2/(n-1))\ s^2`

(6)

Estimation for standard deviation

In theory, when we use formula (5) to calculate `s` and use this quantity as an estimate for standard deviation of population `sigma`, then it is a biased estimate. But the difference is small, so we can use `s` as estimate of `sigma`.

`s` is square root of `s^2`, but standard error of `s` is not square root of standard error of `s^2`. It's difficult and complicated to calculate exact standard error of `s`, but we can use this approximative formula:

`SE=s/sqrt(2(n-1))`

(7)