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 renowned Russian mathematician Pafnuty Chebyshev, this theorem provides a conservative estimate for 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:
- P represents probability
- X is a random variable
- μ (mu) denotes the mean
- σ (sigma) signifies the standard deviation
- k is 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 within 2 standard deviations of 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 least 88.89% of the data falls within 3 standard deviations of 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) = 70
- Upper 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: 7.5
- Standard deviation: 1.2
We can assert that at least 75% of the scores fall between:
- Lower bound: 7.5 – (2 * 1.2) = 5.1
- Upper bound: 7.5 + (2 * 1.2) = 9.9
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