Whereas the bias for formula_154 is negligible, a less biased estimator for formula_20 is obtained as for the normal distribution by replacing the denominator "n" by "n-1" in the equation for formula_156. When the individual values formula_157 are not available, but the sample's mean formula_158 and standard deviation "s" is, then the corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation formula_159 and variance formula_160 for formula_16 and formula_20:  The most efficient way to analyze log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate. A basic example is given by scatter intervals: For the normal distribution, the interval formula_164 contains approximately two thirds (68 %) of the probability (or of a large sample), and formula_165 contain 95 %. Therefore, for a log-normal distribution,  of the probability. Using estimated parameters, the approximately the same percentages of the data should be contained in these intervals. Using the principle, note that a confidence interval for formula_16 is formula_170, where formula_171 is the standard error and "q" is the 97.5 % quantile of a t distribution with "n-1" degrees of freedom. Back-transformation leads to a confidence interval for formula_34,    The log-normal distribution is important in the description of natural phenomena. In a prototype case, a justification runs as follows: Many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation). This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal. A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables.