Central Limit Theorem

Dec 03, 2020

central limit theorem is at the heart of hypothesis testing.Let's understand central limit theorem with some examples

1. Let say, In a computer Science department of a university, there are 10 sections divided for students and in each section, there are 100 students.so the population is 1000 students if i am not wrong.Our task is to calculate average height of students of computer Science department. i think the solution is simple what i will do, i will collect the height of each student and find the average.right?little time consuming but yes right. Now think, what if the data is humongous(very large)? Does this approach make sense.this will take too much time and efforts .So what can we do instead? let's took an alternate approach

• wee will note down height of 30 students(sample) from each section
• calculate the individual mean of these samples.
• calculate the mean of these sample means.(this value gives the approximate mean weight of the students).
• probability of sample mean of students will resemble a bell shaped curve(Normal distribution).

Given a dataset from a unknown distribution(It can be uniform, binomial or may be completly random), the sample mean will approximate the normal distribution.

Examples

Assumptions behind CLT

• Data must follow randomized condition.
• samples should be independent to each other.(one sample should not influence the other sample).
• Sample size should be sufficient large. How will we check? sample size > 30 means sufficient when population is symmetric.

statistical significance

• Analyzing data involves statistical methods like hypothesis testing and constructing confidence intervals. These methods assume that the population is normally distributed. In the case of unknown or non-normal distributions, we treat the sampling distribution as normal according to the central limit theorem
• If we increase the samples drawn from the population, the standard deviation of sample means will decrease. This helps us estimate the population mean much more accurately

Practical applications

• Political/election polls are prime CLT applications. These polls estimate the percentage of people who support a particular candidate. You might have seen these results on news channels that come with confidence intervals. The central limit theorem helps calculate that
• Confidence interval, an application of CLT, is used to calculate the mean family income for a particular region

Source Analytics Vidhya

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