comparing population variance vs variance of sample means I have a data set as 
lab6data <- c(2,5,4,6,7,8,4,5,9,7,3,4,7,12,4,10,9,7,8,11,8,
              6,13,9,6,7,4,5,2,3,10,13,4,12,9,6,7,3,4,2)

variance of the population is 9.522436 mean of the data set is 6.625
Then I took 30 random samples from the data set which has 5 elements without replacements. 
Mean of the sample mean is 6.6 and variance of the sample means is 1.873103 now I need to compare these population variance & mean with the mean and variance of sample means. 
Is there any theory to do that? Please help me to compare them and to ensure the values I got are correct.
 A: There is definitely theory to this, depending on what you want to conclude from your comparison. 
If you would want to check whether the sample is representative for the entire population for instance. 
This link gives a superficial idea: https://www.statpac.com/manual/index.htm?turl=compareasamplemeantoapopulationmean.htm
The general idea would be to calculate the probability that such differences occur by random chance, or that they have a structural cause. For this I would refer you to theory on hypothesis testing, p-values & t-testing. 
Hope this helps!
A: I wrote a little R code to visualize each sample distribution. This shows the law of large numbers visually by sample distribution (there are other, arguably better ways to visualize this). 
Basically, we see the probability density of each sample (n = 30). The white vertical line is the population mean. All sample means cluster around the population mean, and if we ran this an infinite number of times, we would end up with sample mean identical to the population mean. 

Since I found this on Stack Overflow, here's the code:
require(tidyverse)
set.seed(123)

map_df(1:30,function(x){
  a <- sample(lab6data,5,replace = F)
  as_data_frame(a)
}) %>% 
  mutate(sample = as.factor(flatten_dbl(map(1:30,function(x){
    rep(x,5)
  })))) %>% 
  ggplot(aes(value,color = sample,fill = sample))+
    geom_density(aes(color  = sample),show.legend = F,alpha = .25)+
  geom_vline(xintercept = 6.6,color = 'white')+
  scale_x_continuous(breaks = seq(1,20,1))+
scale_fill_grey()+
scale_color_grey()+
      theme_bw()+
  theme(panel.grid = element_blank(),
        panel.border = element_blank())

