Download Know what Chebyshev says about every data set, no matter how

Know what Chebyshev says about every data set, no matter how skewed
or extreme the data might be
“The portion of the data that likes within 𝑘 standard distributions of the mean is at
1
least 1 – 2 .”
𝑘
 Chebyshev uses the letter 𝑘 to stand for “how many standard deviations
away from the mean?” But that’s really the same meaning as the letter 𝑧,
again, “how many standard deviations away from the mean?”
Four kinds of Chebyshev problems
1) Sometimes it’s easy, they just tell you the k value
Example: “In any data set, no matter what shape the data distribution has, at least
_________ % of the data lives within 1.6 standard deviations of the mean.”
 Just plug in the 𝑘 value.
 Convert to a percent.
 Round to the nearest tenth of a percent.
“In any data set, no matter what shape the data distribution has, at least
_________ % of the data lives within 1.8 standard deviations of the mean.”
2) Sometimes you need both 𝒌 (or 𝒛) and 𝒙 values
Example: “Suppose that a certain data set has a mean of 112 and a standard
deviation of 36. According to Chebyshev, we are guaranteed that at least ______%
of the data values must be between 29.2 and 194.8.”
 For the 𝑥 value 29.2, find the corresponding 𝑧 score: _______
 For the 𝑥 value 194.8, find the corresponding 𝑧 score: _______
 Observe – this problem was rigged so that the low and high were the same
distance from the mean.
 z is a number of standard distributions away from the mean. So is 𝑘. Plug
into the Chebyshev Theorem formula and convert to a percent.
Answer: “For that particular data set, at least ______% of the data values are
between 29.2 and 194.8, no matter what crazy distribution shape or extreme
outlier values there may be. And up to ______% lies outside that interval.”
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3) Sometimes Chebyshev runs backwards and you have to find k
Example: “In any data set, regardless of the distribution shape, we are guaranteed
that 95% of the data lies within how many standard deviations of the mean?”
 Plug in the 0.95 as the result of the Chebyshev Formula.
 Work backwards to solve for 𝑘.
o Clear the fractions: multiply each term by 𝑘 2 .
o Combine like terms: 𝑘 2 terms on one side = number term on other
side. Divide by the coefficient to finish isolating the 𝑘 2 .
o Use the Square Root Property.
Answer: “In any data set, regardless of the distribution shape, we are guaranteed
that at least 95% of the data lies within __________ standard deviations of the
mean.”
And: “Up to ____% could be more than _________ standard deviations away from
the mean.”
4) Sometimes Chebyshev runs backwards and you need to answer with x
values
Example: “Suppose a certain data set has a mean of 100 and a standard deviation
of 10 and some unknown distribution shape. It could be heavily skewed and/or it
could have some outrageous outliers. Based on Chebyshev’s Theorem, we are
guaranteed that 68% of the data values lie between what two values?
 Plug in 0.68 as the result of the Chebyshev Formula.
 Work backwards to solve for 𝑘. (Clear Fractions, Combine Like Terms,
Square Root Property).
 Convert those 𝑘 (that is, 𝑧) values into 𝑥 values.
Answer: “For this particular data set, we are guaranteed that 68% of the data lies
between the values _____ and ______.”
And: “Up to ______% of the data could be lower than _____ or higher than
_____.”
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