With Chebyshev’s inequality, you can quickly determine the percentage of values that are a given number of standard deviations from the mean. Chebyshev’s inequality allows you to work with data samples from any distribution. Furthermore, you can check proportions for different standard deviations subject to your choice of standard deviation being real and positive.
Chebyshev’s inequality statement can simplybe expressed as follows:
Where k is the required number of standard deviations from the mean
μ is the population mean
Chebyshev’s inequality is a fact that is valid for any sample data distribution. It gives the lower bound for which results are valid for a given number of standard deviations from the mean. Empirical rule is an approximation that is valid for data samples that portray a bell shape distribution that is associated with the normal curve. It only gives us approximations for data values that fall within one, two and three standard deviations from the mean.
The number of students who sat for a math test is 44. The mean score in the math test was 56 with a standard deviation of 8. How many students attained a score between 40 and 72?
From above we have the following:
Mean, μ = 56
Standard deviation, σ = 8
For X = 40, Z-Score = (40-56)/8=-2
For X = 72, Z-Score = (72-56)/8=2
Applying Chebyshev’s inequality, we get; 1-1/2^2 =3/4
Thus, it follows; 3/4 of 44=33
33 students attained a score that lies between 40 and 72.
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