2026-10-25
The Central Limit Theorem states that as sample size increases, the sampling distribution of the mean approaches a normal distribution.
In other words: If you were to take a histogram of the Sample Means as the Same Size went up, the histogram would slowly get closer to a normal distribution.
In more simple terms: The more we sample the closer we get to the population value
Sample Mean: the Mean of the Sample we’re observing from the population
For Example, When Given:
Population: (5, 4, 3, 2, 3)
Sample 1: (5, 4, 3)
The Sample and Population Means are:
\(\mu = (5 + 4 + 3 + 2 + 3)/5 = 3.4\)
\(\bar{x} = (5 + 4 + 3)/3 = 4\)
Normal Distribution: What is most commonly known as our bell shaped curve. The Normal Distribution is characterized when the mean = mode = median and is often seen through standardization.
The graph below acts as an example:
Let’s use an example. The following dataframe is a visual example of how many points each Big 12 team scored each week.
## Arizona ArizonaSt Baylor BYU UCF Cincinnati Colorado Houston ## Week1 61 48 45 41 57 38 31 7 ## Week2 22 30 12 18 45 27 10 12 ## Week3 7 31 31 34 35 27 28 33 ## Week4 0 22 31 38 0 34 38 0 ## Week5 23 0 28 34 21 41 48 0 ## Week6 22 35 21 0 13 0 0 30 ## Week7 19 27 0 41 13 19 28 0 ## Week8 7 14 59 38 35 24 34 14 ## Week9 26 0 38 37 24 23 34 17 ## Week10 12 42 37 0 56 0 0 24 ## Week11 0 35 0 22 31 24 41 0 ## Week12 27 24 49 13 0 17 49 3 ## Week13 28 28 20 23 21 15 21 10 ## Week14 7 49 45 30 14 13 52 18 ## Championship 0 45 0 0 0 0 0 0 ## BowlGames 0 31 31 36 0 0 14 0 ## IowaSt Kansas KansasSt OaklahomaSt TCU TexasTech Utah WestVirginia ## Week1 21 48 41 44 34 52 49 12 ## Week2 20 17 34 39 45 16 23 49 ## Week3 0 20 31 45 34 66 38 34 ## Week4 52 28 9 19 42 30 22 32 ## Week5 20 27 42 20 38 44 10 0 ## Week6 43 31 0 0 19 28 0 38 ## Week7 28 0 31 14 0 0 19 16 ## Week8 38 42 45 35 13 35 7 18 ## Week9 0 27 29 28 35 34 14 31 ## Week10 22 0 19 21 34 23 0 0 ## Week11 36 45 14 13 38 27 21 31 ## Week12 34 17 0 0 0 0 24 35 ## Week13 31 37 41 48 49 56 28 31 ## Week14 29 17 21 0 20 52 28 15 ## Championship 19 0 0 0 0 0 0 0 ## BowlGames 42 0 44 0 34 26 0 37
The following assumptions can be made:
During a BYE Week, a team scored 0 points
The zero point rule is included during the post-season, even if a team didn’t play
This dataframe starts at week 1, any international games prior to the season are not counted
To start with, the dataframe is flattened and a random seed is set. This can be seen in the following code:
flattened <- unlist(Big12Football2024) set.seed(47)
Then we move onto generating samples. In this case, we will be taking samples of:
n = 5
n = 10
n = 15
CLT: 5 Samples: This slide shows the code and process of getting the first set of samples. In this case, we are taking 5 random samples and getting it’s sample mean. This process is repeated 100 times, and the means are recorded as shown here:
sample_means5 <- replicate(100, {
random_sample5 <- sample(flattened, size = 5, replace = TRUE)
mean(random_sample5)
})
We can also calculate the standard error using:
\(s / \sqrt{n}\)
in this case where standard deviation is already calulated as 16.5:
\(16.5 / \sqrt{5} = 3.3\)
A histogram can be created from the previous sample means using the following code:
df5 <- data.frame(sample_means5 = sample_means5)
ggplot(df5, aes(x = sample_means5)) +
geom_histogram(bins = 10, fill = "skyblue", color = "white") +
labs(
title = "Distribution of 100 Sample Means (n = 5)",
x = "Sample Mean Scores",
y = "Frequency"
) +
theme_minimal()
The histograms for n = 5, n = 10 and n = 15 can be found below:
The visual evidence in the previous slide shows that the histograms are getting closer to normal distribution as the sample size increases.
Additionally, let’s compare the means of the sample means to the population mean:
## n = 5: 24.556
## n = 10: 24.046
## n = 15: 23.11333
## population: 23.20703
Another visual example of this difference can be found on the following graph:
Overall, what did we learn from this example?
As n increased, the mean of the sample means became closer to the population mean.
Additionally, through the histograms, we saw that the higher n became, the closer it got to the normal distribution
Overall, this example showed how CLT works in action and proves that as n increases for each sample, the sample becomes closer to the population.