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Lecture

# Chekyshev’s rule, normal distribution, how to calculate probabilities

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Colorado State University

Statistics

STAT 301

Brett Hunter

Fall

Description

7 September
Chekyshev’s Rule
For any population distribution, we know that the percentage of observations within k
standard deviations of the mean is at least
1
100 ( 1 - k2 ) %
For any k ≥ 1
Example
We have an exam where mean = 83 and standard deviation = 7.
Between what two values do at least 75% of the exam scores fall?
1
2
100 ( 1 - k ) = 75
K = 4
K = 2
At least 75% of the exam scores fall within 2 standard deviations of the mean.
Between μ – kσ and μ + kσ
83 – 2(7) = 69
83 + 2(7) = 97
So at least 75% of the exam scores are between a 69 and a 97.
Normal Distribution (Gaussian)
2
We denote a random variable X that follows a normal distribution as X ~N (μ, σ )
The normal distribution is defined by the density curve
1 −1 (x−μ)
F(x) = 2 (2σ2
√ 2πσ

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