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$$\mathbf{1}-\frac{\mathbf{1}}{k^{2}}$$ For example, at least $1-1 / 2^{2}=3 / 4$ of any set of numbers lie within 2 standard deviations of the mean. Similarly, for any probability distribution, the probability that a number will lie within $k$ standard deviations of the mean is at least $1-1 / k^{2}$ . For example, if the mean is 100 and the standard deviation is 10, the probability that a number will lie within 2 standard deviations of 100, or between 80 and 120, is at least Use Chebyshev’s theorem to find the fraction of all the numbers of a data set that must lie within the following numbers of standard deviations from the mean. $$5$$

Name: Problem 15, Chapter 9: Finite Mathematics and Calculus with Applications
Uploaded: 2020-08-23T05:51:21-08:00
Duration: 4 min 5 s
Channel: Sonam Khatri

   $$\mathbf{1}-\frac{\mathbf{1}}{k^{2}}$$
For example, at least $1-1 / 2^{2}=3 / 4$ of any set of numbers lie within 2 standard deviations of the mean. Similarly, for any probability distribution, the probability that a number will lie within $k$ standard deviations of the mean is at least $1-1 / k^{2}$ .
For example, if the mean is 100 and the standard deviation is 10, the probability that a number will lie within 2 standard deviations of 100, or between 80 and 120, is at least Use Chebyshev’s theorem to find the fraction of all the numbers of a data set that must lie within the following numbers of standard deviations from the mean.

$$5$$

Finite Mathematics and Calculus with Applications

Margaret L. Lial,… 9th Edition

Chapter 9, Problem 15

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Solved by Expert Sonam Khatri

08/23/2020

Step 1

We can use Chebyshev's theorem for this, which states that the proportion of any distribution that lies within k standard deviations of the mean is at least $1-\frac{1}{k^{2}}$. Show more…

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$$\mathbf{1}-\frac{\mathbf{1}}{k^{2}}$$ For example, at least $1-1 / 2^{2}=3 / 4$ of any set of numbers lie within 2 standard deviations of the mean. Similarly, for any probability distribution, the probability that a number will lie within $k$ standard deviations of the mean is at least $1-1 / k^{2}$ . For example, if the mean is 100 and the standard deviation is 10, the probability that a number will lie within 2 standard deviations of 100, or between 80 and 120, is at least Use Chebyshev’s theorem to find the fraction of all the numbers of a data set that must lie within the following numbers of standard deviations from the mean. $$5$$

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Key Concepts

Chebyshev's Theorem

Chebyshev's theorem provides a lower bound on the proportion of values in any data set or probability distribution that lie within a specified number of standard deviations from the mean. The theorem states that at least 1 - 1/k² of the data will fall within k standard deviations, making it a useful tool for analyzing data regardless of distribution shape.

Standard Deviation

Standard deviation is a measure of the amount of dispersion or variability in a set of data values. It quantifies how much the values in a data set deviate from the mean, thus indicating the spread of the data.

Mean

The mean is the arithmetic average of a data set and serves as a measure of central tendency. It is calculated by adding all the values together and dividing the sum by the number of values, providing a central point around which the data is distributed.

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