How is it that a random samples gives a fairly accurate representation of public opinion?

Answer 1

The main point here is that the Sample Mean can be used to estimate the Population Mean.

What I mean by that is that on average, the Sample Mean is a good estimator of the Population Mean.

There are two reasons for this, the first is that the Bias of the estimator, in this case the Sample Mean, is zero. A Bias other than zero overestimates or underestimates the Population Mean depending on its value.

Bias = Expected value of estimator - mean.

This can be expressed as EX(pheta) - mu (pheta)

As the Sample Mean has an expected value (what value it expects to take on average) of itself then the greek letter mu which stands for the Sample Mean will provide a Bias of 0.

Bias = mu - mu = 0

Secondly as the Variance of the the Sample Mean is mu/(n-1) this leads us to believe that the Variance will fall as we increase the sample size.

Variance is a measure of the dispersion of values collected from the centre of the data. Where the centre of the data is a fixed value equal to the median.

Put Bias and Variance together and you get the Mean Squared Error which is the error associated with using an estimator of the Population Mean.

The formula for Mean Squared Error = Bias^2 + Variance

With our estimator we can see that as the Bias = zero, it has no relevance to the error and as the variance falls as the sample size increases then we can conclude that the error associated with using the sample mean will fall as the sample size increases.

Conclusions:

The Random Sample of public opinon will on average lead to a true representation of the Population Mean and therefore the random samle you have will represnt the public opinion to a fairly high degree of accuracy. Finally, this degree of accuracy will rise incredibly quickly as the sample size rises thus leading to a very accurate representation (on average)