empirical rule

We have seen how to calculate the normal distribution with the various other forms of computations like, empirical rule, z-score, Gaussian models. We have also learned about what is the empirical rule, what is z-score, and what is Gaussian models.

In our today’s article we will learn how to use and solve the problems of normal distribution using the Central Limit Theorem.

According to the Central Limit Theorem, the huge is the sample size, the sample distribution of the sample means it will move to the normal distribution, and doesn’t care about the structure of the population distribution. This statement is true for the sample size which is above 30. You must be thinking what is the meaning of this now. So it indicates that if you have the larger samples in the size the graph of those samples will be more likely normal distribution.

Graphically the Central Limit Theorem is as in the below picture.

This graph is of the rolling of the fair die. The more times you roll the die, the graph which you will get will be much more like to the normal distribution.

Central Limit Theorem: Means

The Central Limit Theorem is the important constituent of the average of the sample means, which will be the mean of the population. In simpler words, the summation of all the sample means and then find out the average of it, it will be the actual population mean. In a similar way if you calculate the average standard deviation you will get the actual standard deviation of the population. This is a very useful event that helps us to predict the characteristics of the population.

Procedure For Central Limit Theorem Problems

The Central Limit Theorem problem constitutes the phrase “Greater Than”. So the steps to solve this problem is as follows:

  • First identifying the problem
    • Calculate the mean µ
    • Calculate the standard deviation σ
    • Determine the population size
    • Determine the sample size (n)
    • The number associated with the “Greater Than” X̅
  • Now you have to draw the graph and plot the mean in the graph and also shade the area above the X̅
  • By using the below formula calculate the z-score

Z = (X̅- µ)/(σ / √n)

  • With the help of the z-score you can find out the area with the help of the z tables.
  • Now, subtract the z-score value from 0.5
  • Now, convert the above decimal figure into the percentage. This will be your final answer.

Actual Example

One of the welfare recipients group receives the SNAP good sake of $110 weekly with the standard deviation value of $20. If randomly the 25 people sample is observed, then what will be the probability that the sake will be more than $120.


Here, X̅ = 120

µ = 110

σ = 20

and    n = 25

∴ we can calculate the probability using Z formula

Z = (X̅- µ)/(σ / √n)

∴ Z = (120 – 110) / (20 / √25)

= 10/4

∴ Z = 2.5

Therefore, according to the z table value of 2.5 will be 49.38%, so adding 50% to this value the actual probability will be 99.38%

Conclusion: In this technology news article, I have explained about the Central Limit Theorem and the steps involved in the calculation of the probability. We have also seen one actual example for the same.


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