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Chebyshev’s Theorem and Its Implication

The Chebyshev’s theorem, also known as the Chebyshev’s inequality, is often related to the probability theory. The theorem presupposes that in the process of a probability distribution, almost every element is going to be very close to the expected mean. To be more exact, in case of having k values, only 1/k2 of their total number will be n times larger or smaller than the expected value of the predicted value (Brase and Brase “Getting Started” 5).

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To understand the meaning of the statement provided above, the concept of the expected value should be defined. Also belonging to the realm of the probability theory, the given notion can be defined as a value that one is most likely to obtain when picking the elements randomly for an unlimited number of times.

In other words, Chebyshev’s theorem helps define the number of observations, which one would expect to find within a particular number of standard deviations. Chebyshev’s theorem is often shortened to the following formula: (1 – (1/k2)) (Brase and Brase “Getting Started” 8).

Claiming that Chebyshev’s theorem applies to everything from butterflies to the orbits of the planets, I would have a point, since the theorem is related to the basics of probability theory. Seeing how the latter can be applied to any object, Chebyshev’s theorem as its major part can be as well. For instance, it can be used when defining the probability of planets changing their orbs, a person seeing a butterfly every Friday in May, etc.

The concept of the least square criterion, which is also often used in the field of probability theory, a least-squared criterion can be defined as a criterion that allows for locating the specific part of a regression line, which suits the scattered plot of the data in the best way possible. The least-square criterion includes the sum of squares of a range of particular data (Brase and Brase “Organizing Data” 19). It should be noted that the least square criterion is applied when the simple regression model – or, to be more exact, a simple linear regression model – is used to consider a particular list of data (Brase and Brase “Organizing Data” 21).

The least-square criterion can be calculated for a specific data set with the help of the following formula:

SSE = ∑e2 = ∑(y – y’)2

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The concept of the line of the best fit, also known as a regression line, is often related to the least square criterion concept. According to the existing definition, the line of the best fit can be defined as the line of data, in which the minimal sum of squares for error can be located, Brase explains (Brase and Brase “Organizing Data” 24). In other words, the least square criterion helps browse through all of the existing combinations between the elements of the data set to define the one that includes the smallest sum of squares of errors. Thus, it allows for defining the most appropriate solution of those provided.

Works Cited

Brase, Henry Charles and Corrinne Pellillo Brase. “Getting Started.”Understanding Basic Statistics. 6th ed. Ed. Henry Charles Brase and Corrinne Pellillo Brase. Pacific Grove, CA: Brooks/Cole Publishing Company. 2013. 3–18. Print.

“Organizing Data.” Understanding Basic Statistics. 6th ed. Ed. Henry Charles Brase and Corrinne Pellillo Brase. Pacific Grove, CA: Brooks/Cole Publishing Company. 2013. 19–32. Print.

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