Select up to two metrics for the selected distribution, or data on its own — a data set is fitted to a distribution of its own. Click a selected metric to deselect it.
Primary metric
No metric selected.
Secondary metric
No metric selected.
Computed results
μ
—
log-mean
σ
—
log-std dev
Mean
—
arithmetic
Median
—
= geom. mean
Mode—
Std dev—
Variance—
GSD—
CV—
Error factor—
Skewness—
Excess kurtosis—
Entropy—
Percentiles
5%
—
25%
—
50%
—
75%
—
95%
—
Distribution chart
Display and chart settings
Significant digits apply to the results and the comparison, the unit only titles the chart’s x-axis, and the rest apply to the chart. Chart bounds set the x-axis range, which covers every distribution shown, and the shaded band and reference line must lie within it. The band, the reference line and the raw data can each be switched off, which is worth doing once several distributions overlap. The log₁₀(x) view plots log₁₀ x, where each step of 1 is a factor of ten, a lognormal is a symmetric bell and distributions of different size can be read against each other — untick it for a linear x-axis.
Equations & formulas
| Log-space parameters | |
| \(\mu =\) | \(\ln\mu_g = \ln\bigl(\mu_a^2/\sqrt{\mu_a^2+\sigma_a^2}\bigr)\) |
| \(\sigma =\) | \(\ln\sigma_g = \sqrt{\ln\bigl(1+\sigma_a^2/\mu_a^2\bigr)}\) |
| Central tendency | |
| \(\mu_g =\) | \(e^\mu\quad\) (geometric mean = median) |
| \(\mu_a =\) | \(e^{\mu+\sigma^2/2}\) |
| mode \(=\) | \(e^{\mu-\sigma^2}\) |
| Spread | |
| \(\sigma_g =\) | \(e^\sigma\) |
| \(\sigma_a =\) | \(\mu_a\sqrt{e^{\sigma^2}-1}\) |
| \(\mathrm{CV} =\) | \(\sqrt{e^{\sigma^2}-1}\) |
| EF \(=\) | \(e^{1.645\,\sigma}\) (central 90% coverage) |
| Higher moments | |
| skewness \(=\) | \(\mathrm{CV}^3+3\,\mathrm{CV}\) |
| excess kurtosis \(=\) | \(\mathrm{CV}^8+6\,\mathrm{CV}^6+15\,\mathrm{CV}^4+16\,\mathrm{CV}^2\) |
| entropy \(=\) | \(\tfrac{1}{2}(1+\ln 2\pi\sigma^2)+\mu\) |
| Percentile inversion | |
| \(x_p =\) | \(\exp\bigl(\mu + \sigma\,\Phi^{-1}(p)\bigr)\) |