Central Limit Theorem

Central Limit Theorem | Meaning and Importance | Business Statistics | Management Notes

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The central limit theorem states that the distribution of sample means approximates a normal distribution as the sample size gets larger (assuming that all samples are identical in size), regardless of population distribution shape (Ganti, 2019).

The Theorem is exactly what the shape of the distribution of means will be when we draw repeated samples from a given population. Specifically, as the sample sizes get larger, the distribution of means calculated from repeated sampling will approach normality (Mishra, 2018).

Importance of Central Limit Theorem in Statistics

Central Limit Theorem is important in Statistics because it allows us to use the normal distribution to make inferences concerning the population mean.

It is important in Statistics because it guarantees that, when it applies, the samples that are drawn are always randomly selected.
It is important in Statistics because it enables reasonably accurate probabilities to be determined for events involving the sample average when the sample size is large enough regardless of the distribution of the original value.

Numerical Illustration

Given Information;

Exam Scores: 79, 64, 84, 82, 92 and 77

Numerical Answer (a)

Population mean (µ) =∑X/N = (79 + 64+ 84 +82+ 92 + 77) = 478/6 =79.66

Therefore, the Population mean is 79.66.

Numerical Answer (b)

Number of samples for selecting two test grades = ⁿC_r= ⁶C₂ = 15

Therefore, the number of samples for selecting two test grades is 15.

Numerical Answer(c)

Possible samples of size 2 and their mean value
Samples	Sample Mean (x̅)
79,64	71.5
79,84	81.5
79,82	80.5
79,92	85.5
79,77	78
64,84	74
64,82	73
64,92	78
64,77	70.5
84,82	83
84,92	88
84,77	80.5
82,92	87
82,77	79.5
92,77	84.5
Total	1195

Numerical Answer (d)

Calculated Population mean (µ) = ∑X/N =478/6= 79.66
Sample mean (µ_x) = ∑means/ samples = 1195/15 =79.66

Therefore both the Population mean and sample mean are equal.

Numerical Answer (e)

If I were a student, I would not like this arrangement because if we take into account the concept of Central Limit Theorem which says that as the number of samples considered go on increasing, the tendency of the sample is more representative of the population would go higher i.e. the sample distribution has higher tendency to follow the normal distribution.

After dropping of the lowest score, the population mean is given by:

Population mean (µ) = (79 + 84 +82+ 92 + 77) / 5 = 82.8.

References

Ganti, A. (2019, April 19). (CLT). Retrieved from Investopedia: https://www.investopedia.com/terms/c/central_limit_theorem.asp

Mishra, M. (2018, June 19). Understanding The CLTm. Retrieved from https://towardsdatascience.com/understanding-the-central-limit-theorem-642473c63ad8

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Smirti

(Founder of Management Notes)
MBA,BBA.

I am Smirti Bam, an enthusiastic edu blogger with a passion for sharing insights into the dynamic world of business and management through this website.

I hold a MBA degree from Presidential Business School, Kathmandu, and a BBA degree with a specialization in Finance from Apex College,

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Central Limit Theorem – Meaning and Importance | Business Statistics