Statistiques et probabilités

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Chebyshev's Theorem Calculator.

Find the minimum proportion within k standard deviations.

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k standard deviations

minimumProportion

0,750000

minimumPercent

75,000000

Results update automatically as you type.

Use Cases

Quality Control

Determine the minimum percentage of products within a specified number of standard deviations from the mean, without assuming a normal distribution.

Example: If k=3, at least 88.89% of items are within 3 standard deviations.

Data Analysis

Quickly assess the spread of data in any dataset, especially when the distribution is unknown or skewed.

Example: For k=2, you know at least 75% of data lies within 2 standard deviations.

Frequently Asked Questions

What does Chebyshev's Theorem calculate?: It calculates the minimum proportion of data that must lie within k standard deviations from the mean, for any distribution. The formula is 1 - 1/k², expressed as a percentage.
Can I use this calculator for any data set?: Yes, Chebyshev's Theorem applies to any distribution, regardless of shape. It provides a conservative lower bound, so the actual proportion within k standard deviations is at least the calculated value.
What is the minimum value for k?: k must be greater than 1. For k=1, the theorem gives 0%, which is trivial. For k=2, it guarantees at least 75% of data within 2 standard deviations.

Tips & Common Mistakes

Tips

Enter a value greater than 1 for k. The larger the k, the higher the minimum proportion.
Use this calculator as a quick reference for statistical analysis or to check if your data meets certain spread criteria.
Remember that Chebyshev's Theorem gives a lower bound; the actual proportion may be higher, especially for bell-shaped distributions.

Common Mistakes to Avoid

Entering k=1 or less, which yields a meaningless result (0% or negative).
Assuming the result is the exact proportion; it is only a minimum guarantee.
Using this theorem for small samples without considering sample size limitations.

Last updated: August 13, 2026

Use Cases

Quality Control

Determine the minimum percentage of products within a specified number of standard deviations from the mean, without assuming a normal distribution.

Example: If k=3, at least 88.89% of items are within 3 standard deviations.

Data Analysis

Quickly assess the spread of data in any dataset, especially when the distribution is unknown or skewed.

Example: For k=2, you know at least 75% of data lies within 2 standard deviations.

Frequently Asked Questions

What does Chebyshev's Theorem calculate?

It calculates the minimum proportion of data that must lie within k standard deviations from the mean, for any distribution. The formula is 1 - 1/k², expressed as a percentage.

Can I use this calculator for any data set?

Yes, Chebyshev's Theorem applies to any distribution, regardless of shape. It provides a conservative lower bound, so the actual proportion within k standard deviations is at least the calculated value.

What is the minimum value for k?

k must be greater than 1. For k=1, the theorem gives 0%, which is trivial. For k=2, it guarantees at least 75% of data within 2 standard deviations.

Tips & Common Mistakes

Tips

Enter a value greater than 1 for k. The larger the k, the higher the minimum proportion.
Use this calculator as a quick reference for statistical analysis or to check if your data meets certain spread criteria.
Remember that Chebyshev's Theorem gives a lower bound; the actual proportion may be higher, especially for bell-shaped distributions.

Common Mistakes to Avoid

Entering k=1 or less, which yields a meaningless result (0% or negative).
Assuming the result is the exact proportion; it is only a minimum guarantee.
Using this theorem for small samples without considering sample size limitations.

Use Cases

Quality Control

Determine the minimum percentage of products within a specified number of standard deviations from the mean, without assuming a normal distribution.

Example: If k=3, at least 88.89% of items are within 3 standard deviations.

Data Analysis

Quickly assess the spread of data in any dataset, especially when the distribution is unknown or skewed.

Example: For k=2, you know at least 75% of data lies within 2 standard deviations.

Frequently Asked Questions

What does Chebyshev's Theorem calculate?

It calculates the minimum proportion of data that must lie within k standard deviations from the mean, for any distribution. The formula is 1 - 1/k², expressed as a percentage.

Can I use this calculator for any data set?

What is the minimum value for k?

k must be greater than 1. For k=1, the theorem gives 0%, which is trivial. For k=2, it guarantees at least 75% of data within 2 standard deviations.

Tips & Common Mistakes

Tips

Enter a value greater than 1 for k. The larger the k, the higher the minimum proportion.
Use this calculator as a quick reference for statistical analysis or to check if your data meets certain spread criteria.
Remember that Chebyshev's Theorem gives a lower bound; the actual proportion may be higher, especially for bell-shaped distributions.

Common Mistakes to Avoid

Entering k=1 or less, which yields a meaningless result (0% or negative).
Assuming the result is the exact proportion; it is only a minimum guarantee.
Using this theorem for small samples without considering sample size limitations.