Normal Distribution & Central Limit Theorem
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Overview
When exploring the concepts of probability and statistical inference, two pivotal concepts we encounter are the Normal Distribution and the Central Limit Theorem (CLT).
🌟Normal Distribution
What is normal distribution?
Normal distribution is the most important probability distribution in the field of statistics and probability, because it accurately describes many practical and significant real-world quantities such as noise, voltage, electrons and other quantities which are results from many small independent random terms.
It is also called a Gaussian distribution beacuse of the main contribution of Johann Carl Friedrich Gauss--the Prince of Mathematics--to a normal distribution.
The probability density function of normal distribution:
Here, the parameter μ is the mean or expectation of the distribution, and parameter σ is its standard deviation, the variance of the distribution is σ².
The normal distribution is often referred as X ~ N(μ , σ²);
The simplest case of a normal distribution is known as the standard normal distribution, it follows X ~ N (0, 1), where μ=0, σ²=1.
Because of the shape of the pdf, it is also called the Bell Curve.
The pdf's shape of normal distribution
The pdf's shape of normal distribution
You can play the shiny app to shrink the shape of normal distribution's pdf later.
The empirical rule
The empirical rule
The empirical rule, also called the three-sigma or 68-95-99.7 rule, is a statistical rule which states that for normally distributed data, almost all observed data will fall within three standard deviations σ of the mean or average µ of the data.
Specifically, 68% of observations fall within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ) of the mean.
🌟Central Limit Theorem (CLT for short)
What is the Central Limit Theorem?
The central limit theorem is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both alternatives are concerned with drawing finite samples size n from a population with a known mean, μ, and a known standard deviation, σ.
The first alternative, CLT for averages, says that if we collect samples of size n with a "large enough n," calculate each sample's mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape.
The second alternative, CLT for sums, says that the distribution of a sum of independent random variables from a given population converges to the normal distribution as the sample size increases, regardless of what the population distribution looks like.
Why is CLT useful?
The CLT is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally-distributed in most cases. This allows for easier statistical analysis and inference.
For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over a period of time.
The key assumptions for the Central Limit Theorem are as follows:
1. Independence: The random variables in the sample must be independent of each other.
2. Identical Distribution: The random variables are identically distributed means each observation is drawn from the same probability distribution with the same mean μ and the same variance σ².
3. Sample Size: Sample Size (n) should be sufficiently large, typically (n>30), for the Central Limit Theorem (CLT) to provide accurate approximations.
The Central Limit Theorem Implications
The Central Limit Theorem Implications
One real life application on CLT
One real life application on CLT
Since our country is a large population country, it is unable to gather national data on the body weight of citizens to establish the average body weight in our country.
Therefore, we could conduct surveys of 1,000 groups, each consisting of 50 people. Subsequently, we will calculate the average weight of the first group, followed by the second group, and continue this process through to the last group.
According to the Central Limit Theorem, the distribution of these average values will be normal. In the end, by taking the mean of the averages calculated from these 1,000 groups, we anticipate that this final average will closely represent the national average body weight.
🌟Relationship between normal distribution and CLT
Why is normal distribution so normal in nature?
The Central Limit Theorem tells us that if we take enough sample means from a fixed sampling distribution, the distribution will approach a normal distribution.
Macroscopic phenomena in nature often result from the accumulation of microscopic phenomena. The physical quantities we measure often come from the superposition of many small contributions, regardless of the distribution of these small contributions themselves. Due to the Central Limit Theorem, the macroscopic physical quantities become a normal distribution.
Also , there is a video of galton board experiment for you to watch.
You can see that after many balls have fallen in, the curve eventually becomes bell-shaped.
Why is that?
Because you can consider every ball falls in the number n hole as an event, by CLT for sums, the sum of these independent random variables converges to the normal distribution as the sample size increases.
Instruction
In the Shiny app, you can explore the properties :
Normal distribution
a. You can explore that the normal curve is symmetric about its mean;
b. You can explore that the parameters μ and σ completely characterize the normal distribution.
First, you can choose the mean = 0 and the standard deviation = 1 to see the curve of the standard normal distribution.
Then keep the standard deviation = 1, choosing the mean to change the symmetric axis of the curve of the normal distribution.
Then also keep the mean = 0, choosing the standard deviation to change the shape of the curve .
You will find that when the standard deviation becomes bigger the shape is a little bit fatter ;
the standard deviation becomes smaller, the curve changed to be thinner compared with the standard normal distribution.
Second, you can choose the "Frequency distribution" to see the histogram of the sample from a normal distribution, choose the "Overlay Normal Density" to see the probability density function of normal distribution based on the former histogram.
Central Limit Theorem
You can choose the distribution firstly, and then you can choose sample size from 10 to 100; and if the sample size is greater than 30, the sample mean distribution is approaching the normal distribution. And below the graph, there are two values: sample mean and population mean. We can find this two values are close to each other, so when we estimate the population mean we can use the sample mean.
The previous shiny app seems to have a very simple explanation of the central limit theorem, so we use the shiny app to give an example of uniform distribution about the detail.
The parent distribution is the only uniform distribution. (We can see this distribution is not normal distribution.)
Then we can slide the sample size and number of samples.(We can see the single the sample.)
The sampling distribution is showing.(Approaching the normal distribution)
Task 1 Normal Distribution Exercise
1.There are two parameters in the normal distribution μ and σ, ( ) the shape of the normal curve will be flat.
A. larger μ B. smaller μ C. larger σ D. smaller σ
Answer: C; For the parameter σ will affect the shape of the normal distribution, you can use Shiny app to verify the answer.
Answer: C; For the parameter σ will affect the shape of the normal distribution, you can use Shiny app to verify the answer.
2. Given that the random variables X ~ N (3 , σ2 )，then find the value of P(X < 3)?
2. Given that the random variables X ~ N (3 , σ2 )，then find the value of P(X < 3)?
A. 0.2 B. 0.25 C. 0.3 D. 0.5
Answer: D; By looking the graph of the normal distribution, we can find that the symmetric axis is μ=3, so P(X < 3)=P(X > 3)=0.5
3. Suppose two normal distributions N (μ1, σ12) ( σ1>0)and N (μ2, σ22)( σ2>0), and the probability density function are shown as follows then ( )
3. Suppose two normal distributions N (μ1, σ12) ( σ1>0)and N (μ2, σ22)( σ2>0), and the probability density function are shown as follows then ( )
A. μ1 < μ2 , σ1 < σ2
B. μ1 < μ2 , σ1 > σ2
C. μ1 > μ2 , σ1 < σ2
D. μ1 > μ2 , σ1 > σ2
Answer: A; According the property of the normal distribution, we can find the answers μ1 < μ2 , σ1 < σ2.
Task 2 CLT Exercise
1. The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.
a. In words, X = _____________
b. In words, 𝑥¯ = _____________
c. 𝑋 ~ __ (__，__)
d. How is it possible for the standard deviation to be greater than the average?
Answer:
a. Χ = the yearly income of someone in a third world country
b. 𝑥¯= the average salary from samples of 1,000 residents of a third world country
c. 𝑋 ∼𝑁(2000, 8000/sqrt(1000))
d. Very wide differences in data values can have averages smaller than standard deviations.
2. Traffic Flow "Time Headway"
"Time headway" in traffic flow is the elapsed time between when one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let Xi denote the time headway, in seconds, for a randomly chosen pair of consecutive cars on a freeway. Then f(x)= 0.15e^{−0.15(x−0.5)}, for x≥0.5, and f(x)=0 otherwise. E(Xi)=7.167 and Var(Xi)=44.44.
Question: Use the CLT to find the approximate distribution of total seconds of time headway for 50 randomly selected pairs of consecutive cars on a freeway.
Answer:
🫧🪻🪻🪻~End of learning normal distribution & CLT~🪻🪻🪻🫧
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