Our **chebyshev’s theorem calculator** is a **powerful tool** that helps estimate the **proportion** of data points falling within a **specific range** of standard deviations from the mean.

Named after the

renownedRussian mathematician Pafnuty Chebyshev, this theorem provides aconservative estimatefor data distribution, regardless of the underlying shape.

## Chebyshev’s Theorem Calculator

k (std deviations) | Minimum % within interval | Example dataset (μ = 50, σ = 10) | Interval [μ – kσ, μ + kσ] |
---|---|---|---|

1 | 0% | [40, 60] | [50 – 110, 50 + 110] = [40, 60] |

1.5 | 55.56% | [35, 65] | [50 – 1.510, 50 + 1.510] = [35, 65] |

2 | 75% | [30, 70] | [50 – 210, 50 + 210] = [30, 70] |

2.5 | 84% | [25, 75] | [50 – 2.510, 50 + 2.510] = [25, 75] |

3 | 88.89% | [20, 80] | [50 – 310, 50 + 310] = [20, 80] |

3.5 | 91.84% | [15, 85] | [50 – 3.510, 50 + 3.510] = [15, 85] |

4 | 93.75% | [10, 90] | [50 – 410, 50 + 410] = [10, 90] |

4.5 | 95.11% | [5, 95] | [50 – 4.510, 50 + 4.510] = [5, 95] |

5 | 95.84% | [0, 100] | [50 – 510, 50 + 510] = [0, 100] |

5.5 | 96.44% | [-5, 105] | [50 – 5.510, 50 + 5.510] = [-5, 105] |

6 | 97.22% | [-10, 110] | [50 – 610, 50 + 610] = [-10, 110] |

6.5 | 97.78% | [-15, 115] | [50 – 6.510, 50 + 6.510] = [-15, 115] |

7 | 98.00% | [-20, 120] | [50 – 710, 50 + 710] = [-20, 120] |

7.5 | 98.56% | [-25, 125] | [50 – 7.510, 50 + 7.510] = [-25, 125] |

8 | 98.75% | [-30, 130] | [50 – 810, 50 + 810] = [-30, 130] |

## Chebyshev’s Theorem Formula

The core of Chebyshev’s Theorem is expressed through a **concise yet potent formula**:

**P(|X - μ| ≤ kσ) ≥ 1 - (1/k²)**

Where:

Prepresents probabilityXis a random variableμ(mu) denotes the meanσ(sigma) signifies the standard deviationkis the number of standard deviations from the mean

This formula allows us to calculate the **minimum proportion** of data points within **k standard deviations** of the mean.

When we want to know the proportion of data within **2 standard deviations**:

**P(|X - μ| ≤ 2σ) ≥ 1 - (1/2²) = 1 - (1/4) = 3/4 = 75%**

Thus, at least

75%of the data falls within2 standard deviationsof the mean, regardless of the distribution’s shape.

## How do you calculate Chebyshev’s theorem?

**Determine k**: Decide how many standard deviations from the mean you want to consider.**Apply the formula**: Substitute k into the equation 1 – (1/k²).**Interpret the result**: The outcome represents the**minimum proportion**of data within**k standard deviations**.

Suppose we want to find the **minimum proportion** of data within **3 standard deviations** of the mean.

k = 3- 1 – (1/k²) = 1 – (1/3²) = 1 – (1/9) ≈ 0.8889
Interpretation: At least88.89%of the data falls within3 standard deviationsof the mean.

## How to compute a 75% Chebyshev interval?

To find the Chebyshev interval that contains at least **75%** of the data:

**Set up the inequality**: 1 – (1/k²) ≥ 0.75**Solve for k**:- (1/k²) ≤ 0.25
- k² ≥ 4
- k ≥ 2

**Interpret**: The interval [μ – 2σ, μ + 2σ] contains at least**75%**of the data.

**For a concrete example, let’s say a dataset has a mean of 100 and a standard deviation of 15:**

Lower bound: 100 – (2 * 15) = 70Upper bound: 100 + (2 * 15) = 130

Therefore, at least **75%** of the data points lie between **70** and **130**.

## What is at least 75% according to the Chebyshev rule?

According to Chebyshev’s rule, at least **75%** of the data in **any distribution** falls within **2 standard deviations** of the mean. This is a **crucial insight** for analyzing datasets where the distribution shape is unknown or non-normal.

In a company’s employee satisfaction survey with scores ranging from **1 to 10**:

Mean score:7.5Standard deviation:1.2

**We can assert that at least 75% of the scores fall between:**

Lower bound: 7.5 – (2 * 1.2) = 5.1Upper bound: 7.5 + (2 * 1.2) = 9.9

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