Chebyshev’s Theorem
Chebyshev’s theorem is used to find the proportion of observations you would expect to find within two standard deviations from the mean.
Chebyshev’s Interval refers to the intervals you want to find when using the theorem. For example, your interval might be from -2 to 2 standard deviations from the mean.
For normal distributions, about 68% of results will fall between +1 and -1 standard deviations from the mean. About 95% will fall between +2 and -2 standard deviations. The Theorem allows you to use this idea for any distribution, even if that distribution isn’t normal. The theorem states:
You can only use the formula to get results for standard deviations more than 1; It can’t be used to find results for smaller values like 0.1 or 0.9. Technically, you could use it and get some kind of a result, but those results wouldn’t be valid.
Example problem: a left-skewed distribution has a mean of 4.99 and a standard deviation of 3.13. Use the theorem to find the proportion of observations you would expect to find within two standard deviations from the mean:
Step 1: Square the number of standard deviations:
22 = 4.
Step 2: Divide 1 by your answer to Step 1:
1 / 4 = 0.25.
Step 3: Subtract Step 2 from 1:
1 – 0.25 = 0.75.
At least 75% of the observations fall between -2 and +2 standard deviations from the mean.
That’s:
mean – 2 standard deviations
4.99 – 3.13(2) = -1.27
mean + 2 standard deviations
4.99 + 3.13(2) = 11.25
Or between -1.27 and 11.25
Microsoft Excel has a wide variety of built-in functions and formulas that can help you with statistics. However, it does not have a built-in formula for Chebyshev’s Theorem. In order to calculate the theorem in Excel, you’ll need to add the formula yourself. If you want to use it just once or twice, you can type the formula into a cell. However, if you intend on using the formula several times, you can add a custom function (=CHEBYSHEV) to Microsoft Excel.
Step 1: Type the following formula into cell A1: =1-(1/b1^2).
Step 2: Type the number of standard deviations you want to evaluate in cell B1.
Step 3: Press “Enter.” Excel will return the percentage of results you can expect to find within that number of standard deviations in cell B1.
Step 1: Open the Visual Basic editor in Excel. To open the Visual Basic editor, click the “Developer” tab and then click “Visual Basic.”
Step 2: Click “Insert” and then click “New Module.”
Step 3: Type the following code into the blank window:
Function Chebyshev(stddev)
If stddev >= 0 Then
Chebyshev = (1 – (1 / stddev ^ 2))
Else: Chebyshev = 0
End If
End Function
.
Typing the code for a Chebyshev custom function in Microsoft Excel.
Step 4: Close the visual basic window and return to the worksheet. The custom function is now ready to use: type “=chebyshev(x)” into a blank cell, where “x” is the number of standard deviations. Excel will calculate Chebyshev’s theorem and return the result in the same cell.
Pafutny Lvovich Chebyshev (1821-1894)was a Russian mathematician. His friend, mathematician and.webpted linguist Irenée-Jules Bienaymé translated many of Chebyshev’s works into French.
In 1867 Chebyshev published a paper On mean values which first mentioned the inequality to give a generalized law of large numbers. However, the inequality actually first appeared fourteen years earlier in Bienaymés Considérations à l’appui de la découverte de Laplace. The editor who discovered Chebyshev’s use of Bienaymé’s inequality (without mention of the original author) said:
It is a pity that their common interest in the Inequality somehow “slipped through the cracks” in the early contacts between Bienaymé and Chebyshev. Possibly the Inequality was regarded by Bienaymé as a minor result compared with his main themes of linear least squares and Laplacian defence. Chebyshev’s recognition of its significance and its clear statement has, at any rate, always been a defensive point in his favour stressed by some historiographers. From The University of St. Andrews.
Chebyshev’s theorem is often spelled many different ways>you’ll find it spelled as Chebychev’s theorem, Chebyschev’s theorem and even Tschebyscheff’s theorem. That’s mostly due to the fact that his original name was Russian, which uses a different alphabet (cyrillic). “Chebyshev” is just the word, taken as it sounds and translated into an English approximation.
Fun fact: There is a crater on the moon named after him: Crater Chebyschev.
Note: Technically, Chebyshev’s Inequality is defined by a different formula than Chebyshev’s Theorem. That said, it’s become common usage to confuse the two terms; A quick Google search for “Chebyshev’s Inequality” will bring up a dozen sites using the formula (1 – (1 / k2)). If you’re in a beginning stats class, pretty much the only form of Chebyshev’s formula you’ll be dealing with is that one.
For disambiguation, here’s the “other” inequality, which is mainly used to prove the Law of Large Numbers and other academic exercises. This is not the inequality used in elementary statistics. For that one, see Chebyshev’s Theorem above.
Chebyshev’s Inequality gives an upper bound to the probability that the absolute deviation of a random variable from the mean will exceed a stated amount. The formula reads as follows:
Applications of Chebyshev’s Inequality
The formula was used with calculus to develop the weak version of the law of large numbers. This law states that as a sample set increases in size, the closer it should be to its real mean (i.e. the one you would expect to see in a population). A simple example is that when rolling a six-sided die, the probable average is 3.5. A sample size of 5 rolls may result in drastically different results. Roll the die 20 times; The average should begin approaching 3.5. As you add more and more rolls, the average should continue to near 3.5 until reaching it. Or, it becomes so close that they are pretty much equal.
Another application is in finding the difference between the mean and median of a set of numbers. Using a one-sided version of Chebyshev’s Inequality theorem, also known as Cantelli’s theorem, you can prove the absolute value of the difference between the median and the mean will always be less than or equal to the standard deviation. This is handy in figuring out if a median you derived is plausible.
Chebyshev’s inequality doesn’t provide good accuracy for lower bounds if the sample size is small. For large random samples, it’s much more useful. That said, as there aren’t any restrictions at all on the shape of the underlying probability distribution, it tends to be very weak. Therefore, it’s not used much at all outside of academia.