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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%
—
Display and chart settings
Chart bounds set the x-axis range. The shaded interval and reference line must lie within that range.
Distribution chart
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 | |
| \(x_p =\) | \(\exp\bigl(\mu + \sigma\,\Phi^{-1}(p)\bigr)\) |