How To Approximate the Results of Your Sample Set (Empirical Rule vs. Chebyshev’s Formula)

Author(s): Ibrahim Israfilov

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The empirical rule is a powerful instrument to capture the distribution of your observations within the dataset. However, it’s not the only way to do it, particularly when your dataset is not normally distributed.

Index:
1. Problem Description
2. Review Statistics
3. Empirical Rule
4. Chebyshev Theorem

Problem Description

Imagine that you have 60 students in math class and you have monitored the timing of how much it takes students to finish an exercise.
You get the next results. Average : 7', Variance: 5'
We would love to know what is the maximum number of students who spent more than 15' for an exercise.

Review Statistics

Before going to the solution let’s do a quick recap of the basic statistical notions. The average is calculated following the next formula.

Average takes the sum of all the observations in the dataset and divides by the number of the observations

The variance shows the dispersion of the observations around the mean. It takes the sum of the single observation and the mean’s (average’s) difference elevated to the square and divides it by the number of the observations.

For the next step, we will need a standard deviation. Which is simply a square root of the variance.

Empirical Rule

If we have a normally distributed dataset we can apply the empirical rule (sometimes called also 3 sigma rule, because of the Greek sigma sign which defines the standard deviation) to know the probability of the occurrence of the observations to lie within a certain range of the dataset.

Empirical Rule Says:
68% of the observations lie within 1 standard deviation range from the mean.
95% of the observations lie within a 2 standard deviation range from the mean.
99.7% of the observations lie within a 3 standard deviation range from the mean.

Since our problem doesn’t explicitly says us the distribution is normal. We cannot be confident with the empirical rule. In this case, we can use the probability theorem of the Chebyshev.

Chebyshev Theorem

If the dataset is not normally distributed it doesn’t mean that we cannot have an approximation of the distribution of the observations within the dataset.
The Chebyshev Formula states:

Where k denotes how many standard deviations we have the distance between the ranges of the interest.
In our case. The standard deviation is going to be 2.24 (Square root of 5)

The equation to find how many standard equations are needed to capture the area before 15' is
15=7+k(2.24)
k(2.24)=8
k=3.57

3.57 standard deviations from the mean are needed to arrive at 15'. Now we have all inputs to calculate the percentage in the range between +- 3.57 from the mean.

1– 1/3.5⁷² = 0.92 * 100
and in order to find the maximum number of the students going beyond 15' is 0.07*60. = ~4 students.

Conclusion

Thank you for your attention. Hope you have enjoyed this article. For any queries or critics feel free to leave a comment.

How To Approximate the Results of Your Sample Set (Empirical Rule vs. Chebyshev’s Formula) was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and responding to this story.

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