Estimate spread using a generalized interval rule. Enter mean, deviation, and k. Get bounds, proportions, counts, exports, and graph easily.
| Mean | Standard Deviation | k | Total Observations | Lower Bound | Upper Bound | Minimum Percentage |
|---|---|---|---|---|---|---|
| 50 | 8 | 2 | 1000 | 34 | 66 | 75% |
| 72 | 6 | 3 | 500 | 54 | 90 | 88.8889% |
| 120 | 15 | 2.5 | 200 | 82.5 | 157.5 | 84% |
The generalized empirical rule gives a guaranteed minimum proportion of observations within k standard deviations of the mean for any distribution with finite variance.
Interval: μ ± kσ
Lower Bound: μ - kσ
Upper Bound: μ + kσ
Minimum Proportion: 1 - 1/k²
Minimum Percentage: (1 - 1/k²) × 100
Minimum Count: Total Observations × (1 - 1/k²)
This rule is often linked to Chebyshev's inequality. It works when k is greater than 1.
The generalized empirical rule calculator estimates the minimum share of values expected within a selected number of standard deviations from the mean. It is useful for bounded interpretation when the data distribution is unknown or not perfectly normal.
This page lets you enter a mean, standard deviation, k value, and total observations. It then computes the lower and upper interval bounds, the guaranteed minimum proportion inside the interval, the equivalent percentage, and the minimum count.
The graph helps you see the selected interval around the mean. The included table shows sample cases, while the export tools make it easier to save results for reports, assignments, or quick reference.
Because the result is based on a general inequality, it provides a conservative estimate rather than an exact observed percentage. That makes it valuable when you want a safe lower bound for spread analysis.
It estimates the minimum proportion of data that must lie within k standard deviations of the mean. It gives a guaranteed lower bound, not an exact percentage.
No. The 68-95-99.7 rule is for normal distributions. The generalized empirical rule works for broader distributions and gives more conservative minimum percentages.
The formula 1 - 1/k² becomes meaningful for interval guarantees when k exceeds 1. Smaller values do not provide the standard generalized bound used here.
This calculator still applies because the rule does not require normality. It only assumes the dataset has a defined mean and finite variance.
Proportion is the decimal form of the result. Percentage is the same value multiplied by 100. Both express the same minimum coverage.
The calculator multiplies the minimum proportion by the total number of observations. This gives the minimum number of values expected inside the interval.
Yes. The calculator accepts decimal values for mean, standard deviation, k, and observations. Decimal observations are allowed for planning or estimated datasets.
No. It is a theoretical lower bound based on the selected inputs. Your actual dataset may have a larger percentage within the interval.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.