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3.5 Chebyshev’s Rule and the Empirical Rule Objectives: By the end of this section, I will be able to… Calculate percentages using Chebyshev’s Rule. Find percentages and data values using the Empirical Rule.

CHEBYSHEV’S THEOREM

Chebyshev's Theorem Let 𝑘 ≥ 1. 𝒌= 𝒙− 𝒙 𝒔 The Russian mathematician P.L. Chebyshev (1821-1894) proved a theorem which is valid for any distribution of data:   𝒌= 𝒙− 𝒙 𝒔 Let 𝑘 ≥ 1. Then the % of distribution that lies within "k" SDs of the 𝑥 is at least 1− 1 𝑘 2 100.

NORMAL DISTRIBUTION IQ scores, heights, weights are all examples of normal distributions.

Chebyshev’s Theorem 75 5 Why is this theorem used? It can be used for samples or populations. What is the mean? What is the standard deviation? -5 75 +5 75 5

Easy Problem 1) Professor Costag is analyzing the grades from his Statistics course. The average grade was an 85 with a standard deviation of 3 points. What scores represent the scores within 2 standard deviations of the mean? - 2 s.d 𝒙 85 + 2 s.d.         79 to 91

k = 3.33 A little tougher…   Using the previous example, find the percentage of students who will score between 75 and a 95. Remember the mean is 85 with s = 3. WE NEED TO FIND k. Clue word is “percentage” To find k use the following formula:           k = 3.33 3.33 3.33

A little tougher… Using the previous example, find the percentage of students who will score between 75 and a 95. Remember the mean is 85 with s = 3. Now use Chebyshev’s Theorem.   k = 3.33   90.999% or 91%

Empirical Rule Metaphorically, the Empirical Rule is a Porshe compared to Chebyshev’s Rule, which is a go-anywhere ATV. Basically if you have data that is skewed or an unknown relationship, use CHEBYSHEV’S. If your distribution is bell shaped (normal) use EMPIRICAL.

EMPIRICAL RULE If the data distribution is normal (bell shaped): 68% of the data values will fall within 1 standard deviation of the x . 95% will fall within 2 SDs of the x . 99.7% will fall within 3 SDs of the x . You still start using Chebyshev’s, it will just end up being one of these values.

THESE ARE YOUR ANSWERS!!!!! EMPIRICAL RULE If the data distribution is normal (bell shaped): 68% 95% 99.7% THESE ARE YOUR ANSWERS!!!!!

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Practice Problem with Empirical Rule Your GPAs are bell-shaped with a mean of 3.1 and a standard deviation of 0.7. Find the proportion of GPAs between 2.4 and 3.8. WE NEED TO FIND k.   To find k use the following formula: Since k = 1, 68% of the students are within 1 standard deviation of 3.1, the mean.     1 k = 1     1

Practice Complete the following two problems. SHOW ALL YOUR WORK! ChebyEmpirical