Find the minimum proportion of data within k standard deviations of the mean
For k = 2: at least 75% of data is within 2σ ·
For k = 3: at least 88.9% of data is within 3σPowered by Chebyshev's inequality · Statistics Tools
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When you’re analyzing data, one of the first questions you ask is: “How spread out is this?” The standard deviation tells you the spread, but it doesn’t directly tell you how much data falls within a given range unless you assume a specific distribution. Chebyshev’s theorem solves this problem differently: it guarantees a minimum proportion of data within k standard deviations of the mean for any dataset, regardless of its shape. This makes it incredibly useful for understanding data when you don’t know the underlying distribution.
This Chebyshev’s theorem calculator applies the inequality to your specific numbers. Enter the mean, standard deviation, and k value, the tool computes the minimum proportion of data within the interval, displays the interval bounds, and shows a visual chart. Whether you’re a statistics student learning about bounds, a data analyst exploring data spread, or a researcher applying the theorem to real data, this calculator delivers accurate results instantly. All calculations run locally in your browser, keeping your data private.
Enter the mean of your dataset using the first input field, this is the average value.
Enter the standard deviation of your dataset, a measure of spread (must be greater than zero).
Adjust the k value using the slider or type a number directly, k represents the number of standard deviations from the mean.
Review the minimum proportion of data within k standard deviations displayed in the highlighted result box.
Check the interval [μ − kσ, μ + kσ] and the lower and upper bounds.
Examine the visual chart showing the interval shaded relative to the data distribution.
The calculator applies Chebyshev’s inequality, which provides a universal lower bound on the proportion of data within k standard deviations of the mean for any distribution.
Formula: Proportion ≥ 1 − 1/k²
For any k > 1, at least (1 − 1/k²) × 100% of the data lies within k standard deviations of the mean. For k = 2, this gives at least 75% of data within 2σ. For k = 3, at least 88.9% within 3σ.
Formula: Interval = [μ − kσ, μ + kσ]
The interval is centered at the mean and extends k standard deviations in both directions.
The theorem is conservative, the actual proportion is often much higher than the guaranteed minimum. The bound is tight for some distributions but not for others. The calculator also computes the complementary proportion, the maximum proportion of data that can lie outside the interval.
Consider a dataset with a mean of 50 and a standard deviation of 10. We want to know the minimum proportion of data within 2 standard deviations of the mean.
Step 1: Identify the known values
μ = 50
σ = 10
k = 2
Step 2: Apply Chebyshev’s inequality
Proportion ≥ 1 − 1/2² = 1 − 1/4 = 0.75 = 75%
Step 3: Calculate the interval
Lower bound = 50 − 2 × 10 = 30
Upper bound = 50 + 2 × 10 = 70
Interval = [30, 70]
Step 4: Determine the outside proportion
Outside = 1 − 0.75 = 0.25 = 25%
Interpretation: At least 75% of the data lies between 30 and 70. At most 25% of the data can lie outside this interval. This is a guaranteed bound — the actual proportion within this range could be much higher (e.g., 95% in a normal distribution), but it cannot be lower than 75%.
Chebyshev’s theorem states that for any dataset, at least (1 − 1/k²) of the data lies within k standard deviations of the mean, for any k > 1. It provides a universal lower bound that applies to all distributions.
The formula is: Proportion ≥ 1 − 1/k². For example, for k = 2, at least 75% of data lies within 2 standard deviations of the mean.
The empirical rule applies only to normal distributions and gives approximate proportions (68%, 95%, 99.7%). Chebyshev’s theorem applies to any distribution and gives guaranteed minimum proportions. The theorem is more conservative but universally applicable.
For k = 1, the theorem gives 1 − 1/1² = 0%. This means the theorem guarantees nothing for data within 1 standard deviation, in some distributions, very little data may fall within 1σ. For k > 1, the bound becomes meaningful.
This interval represents the range from k standard deviations below the mean to k standard deviations above the mean. Chebyshev’s theorem guarantees that at least (1 − 1/k²) of the data falls within this interval.
Yes, the theorem gives a lower bound, meaning the actual proportion is always at least the calculated value. For many distributions, especially normal distributions, the actual proportion is much higher.
The outside proportion is the complement of the proportion within the interval. It represents the maximum possible proportion of data that can fall outside the interval. For k = 2, the outside proportion is at most 25%.
Choose k based on how many standard deviations you want to consider. Common choices are k = 2 (at least 75% coverage) and k = 3 (at least 88.9% coverage). For a broader interval, choose a larger k.
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