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

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?

• Attempt of a solution: (1) The normal disitribution is only an approximation of the distribution of the height. When we say that the heigth is distributed as a normal distribution, it would be more precise to say that the heigth is distributed approximately as a normal distribution (2) The reason is given in (1). It is important to realize that it is NOT correct to generate the values of the heights of the individuals in the sample and then calculate the population mean. The reason is that the distribution of the heights of the individuals is the true distribution. N(150,30) is only an approx – FredrikAa Aug 25 '16 at 10:46
• You say "Let us consider the distribution of heights of a population consisting of 100 people. Assume the population height has the normal distribution"... while a population can indeed be finite, a finite population cannot actually have a normal distribution; at best it might be approximately normal. – Glen_b Aug 25 '16 at 13:12

## 1 Answer

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.

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

• Thanks for your answer @Tim. I agree with you regarding the problem with my example. Is my comment correct? When I mean approximation, I mean that since the populations is finite the distrubition of the heights cannot follow a normal distribution, since there is only a finite amount of posible values ot the height. I do not follow your answer on question (3); let us say that the population size is 20. If you measure the heigths of the 20 individuals, you know the population height, right? What then does the weak law of large numbers tell you? Does it only apply to infinite populations? – FredrikAa Aug 25 '16 at 10:57
• @FredrikAa any theoretical distribution is only approximation of any real-life data. Nothing is exactly normally distributed, we use distribution as a model that approximates reality (see stats.stackexchange.com/questions/194558/…). This has nothing to do with population being finite. – Tim Aug 25 '16 at 11:17
• Of course if you drawn huge number of random values from the normal distribution as your simulated population and then drawn huge (but smaller) sample from it, then it will be closer to normal then with dealing with small samples. But the problem with your example is that your population is infinite and normally distributed, and you drawn sample of size 100 from it. The sample of size 100 was not your population unless you wanted to simulate example where your sample is your population, but then it was not distributed as N(150, 30) but mean of the population is mean(example_data) exactly. – Tim Aug 25 '16 at 11:19