# I with this STATISTICS problem that deals with Chebyshev's theorem.

• rdapaul
In summary, Chebyshev's theorem tells us that for any given data set, the percentage of values within k standard deviations from the mean is at least (1-1/k^2) * 100%. In this case, since 58.6 and 68.6 are equally distant from the mean of 63.6, we can use the formula to find the number of standard deviations that 68.6 is above the mean. Then, we can apply Chebyshev's theorem to determine the percentage of men whose heights fall between 58.6 in. and 68.6 in. at Suny Rockland.
rdapaul
the heights of adult men at Suny Rockland have mean heights of 63.6 and a standard deviation of 2.5. What does Chebyshev's theorem tell us about the percentage of men whose heights are between 58.6 in. and 68.6 in.?

Since 58.6 and 68.6 are equally distant from the mean, find the number of standard deviations 68.6 is above the mean and then apply Chebyshev's theorem.

rdapaul said:
the heights of adult men at Suny Rockland have mean heights of 63.6 and a standard deviation of 2.5. What does Chebyshev's theorem tell us about the percentage of men whose heights are between 58.6 in. and 68.6 in.?

In such type of questions, the best way to handle them is to ask yourself what formula you require to obtain the interval values. This formula is: lower value=mean-k(s.d.) or upper value=mean+k(s.d.). Once you solve for k in either the two equations, you may use the value obtained to get the percentage using Chebyshev's Theorem.

## 1. What is Chebyshev's theorem and how is it used in statistics?

Chebyshev's theorem states that for any data set, the proportion of values within k standard deviations of the mean is at least 1 - 1/k^2, where k is any positive number greater than 1. This theorem is used in statistics to determine the minimum percentage of data that falls within a certain range, regardless of the shape of the data distribution.

## 2. How does Chebyshev's theorem differ from the empirical rule?

Chebyshev's theorem provides a minimum percentage of data within a certain range, while the empirical rule gives an estimate of the percentage of data within that range based on the assumption of a normal distribution. Chebyshev's theorem is more conservative and can be applied to any data set, while the empirical rule is more specific to a normal distribution.

## 3. Can Chebyshev's theorem be used for all types of data distributions?

Yes, Chebyshev's theorem can be used for any type of data distribution, as it makes no assumptions about the shape of the data. It only requires the mean and standard deviation of the data set.

## 4. How is Chebyshev's theorem helpful in analyzing data?

Chebyshev's theorem provides a way to determine the minimum proportion of data within a certain range, which can be useful in identifying outliers or extreme values in a data set. It also allows for a more conservative estimate of data within a range compared to the empirical rule.

## 5. What are the limitations of Chebyshev's theorem?

Chebyshev's theorem only provides a minimum percentage of data within a range, so it does not give an exact estimate of the data. It also assumes that the data set is unimodal, meaning it has one peak or mode. Additionally, it may not be as useful for smaller data sets, as the minimum percentage may not accurately represent the data distribution.

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