See Part I to get started.
Central Limit Theorem: Statement & Assumptions
Suppose we are taking repeated samples of size ‘n’ from a population with any kind of probability distribution. Then, the Central Limit Theorem states that given a high enough sample size, the following properties hold true:
- Sampling distribution’s mean = Population mean (μ)(μ), and
- Sampling distribution’s standard deviation (standard error) = σ/√nσ/√n, such that for n ≥ 30, the sampling distribution tends to a normal distribution for all practical purposes.
In the next section, we will try to understand the workings of the CLT with the help of simulations in Python.
Demonstration of CLT in action using simulations in Python with examples
The main point demonstrated in this section will be that for a population following any distribution, the sampling distribution (sample mean’s distribution) will tend to be normally distributed for a large enough sample size.
We will consider two examples and check whether the CLT holds.
Example 1: Exponentially distributed population
Suppose we are dealing with a population which is exponentially distributed. Exponential distribution is a continuous distribution that is often used to model the expected time one needs to wait before the occurrence of an event.
The main parameter of exponential distribution is the ‘rate’ parameter λλ, such that both the mean and the standard deviation of the distribution are given by (1/λ)(1/λ).
The following represents our exponentially distributed population:
We can see that the distribution of our population is far from normal! In the following code, assuming that λλ=0.25, we calculate the mean and the standard deviation of the population:
# Importing necessary libraries
import pandas as pd, numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# rate parameter for the exponentially distributed population
rate = 0.25
mu = 1/rate
# Population standard deviation
sd = np.sqrt(1/(rate**2))
print(‘Population mean:’, mu)
print(‘Population standard deviation:’, sd)
Population mean: 4.0
Population standard deviation: 4.0
Stay tuned for the next installment, in which Ashutosh Dave will see how the sampling distribution looks for this population and will consider two cases, i.e. with a small sample size (n= 2), and a large sample size (n=500).
Visit QuantInsti for additional details and to download full code: https://blog.quantinsti.com/central-limit-theorem/.
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