Population and the mean, standard deviation and the distribution of a population charactertistic

Question

I have some questions regarding the statistical notion of "population'', the mean, standard deviation and the distribution of a population characteristic.

Background:

According to Wikipedia, ``A statistical population can be a group of actually existing objects". Let us consider the distribution of heights of a population consisting of 100 people. Assume the population height has the normal distribution with mean $\mu=150$ and standard deviation $\sigma=30$. Let us generate possible actual heights of the individuals in the population using R:

n=100
mu=150
sigma=30
set.seed(5)
example_data=rnorm(n,mu,sigma)

It is then possible to make a plot of the cdf:

plot(ecdf(example_data))

which gives the folowing plot:

The theoretical cumulative distribution function is given in the following plot

We can calculate the populations mean and variance as

$\mu=\frac{\sum_{i=1}^nx_i}{n}=150.9491$

and

$\sigma^2=\frac{\sum_{i=1}^n(x_i-\mu)^2}{n}=796.1643$

The population standard deviation is then

$\sigma=\sqrt{796.1643}=28.21638$

Questions:

1) we see that the plot of the empirical cdf and the plot of the cdf are not exactly the same. They have similar shape but do not coincide at all points along the abscissa. How can we then say that the population height is distributed as the specified normal distribution?

2) When we calculate the population mean and standard deviation, we do not get the true values 150 and 30. Why are then the population mean and variance defined as they are? Is it only in an infinite population the expression for the population mean, variance etc. give the true values?

3) According to the weak law of large numbers the sample mean converges to the true mean as the number of observations approaches infinity. How can then the expression for the population mean give the true value of the mean?

4) Can then a population consist of a finite amount of individuals or are finite sets of individuals just samples?

Answer 1

1) we see that the plot of the empirical cdf and the plot of the cdf are not exactly the same. They have similar shape but do not coincide at all points along the abscissa. How can we then say that the population height is distributed as the specified normal distribution?

This is because ecdf computes empirical cdf, i.e. cumulative distribution function of your sample, not the true cdf of the population (that you correctly given on your plot of theoretical cdf).

2) When we calculate the population mean and standard deviation, we do not get the true values 150 and 30. Why are then the population mean and variance defined as they are? Is it only in an infinite population the expression for the population mean, variance etc. give the true values?

Because you calculate sample mean and sample standard deviation. As ecdf, they are estimates of the true parameters.

3) According to the weak law of large numbers the sample mean converges to the true mean as the number of observations approaches infinity. How can then the expression for the population mean give the true value of the mean?

It does not. It gives you estimate of the true mean that approaches your true mean as your sample size approaches infinity.

4) Can then a population consist of a finite amount of individuals or are finite sets of individuals just samples?

Certainly yes. You can have population of all patients that visited your local dentist dr. Henry last year. This population is obviously finite. You can draw sample of such population and calculate it's mean and standard deviation that will be estimates of the true mean and true standard deviation. Obviously, if you calculated mean on the whole population it would differ from the one estimated on sample taken from it.

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The problem with your example is that you are confusing sample with population. In your example you have drawn sample of size 100 from the infinite population. For finite population example you should rather do something like below:

pop <- rnorm(1000, 150, 30) # sample your population from hyper-population
mean(pop) # this is mean of your population
sd(pop)   # this is sd of your population

sam <- sample(pop, 100) # take sample of size 100 from your population
mean(sam) # this is mean of your sample
sd(sam)   # this is sd of your sample

Check also:
What is the difference between a population and a sample?