The probability of a result *x* in an experiment consisting of a
large number of equally probable independent trials *n* is
approximated by the *normal probability density function*:

where μ, the *mean value*, is *n*/2 and
σ,
the *standard deviation*, is a measure of the breadth of the
curve which, for experiments with two equally probable outcomes
of each trial is:

Since the mean value and standard deviation depend upon the number
of trials in the experiment, comparison between experiments
with differing numbers of trials is facilitated by
*standardising* the result: transforming it
to a distribution with mean value zero and standard deviation
of 1. A normally distributed experimental result *x*
is thus standardised by subtracting the mean and dividing
by the standard deviation of the experiment:

This *z-value* or *z score* expresses the
divergence of the experimental result *x* from the
most probable result μ as a number of standard
deviations σ
The larger the value of *z*, the less probable the
experimental result is due to chance. The probability
can be calculated from the *cumulative standard
normal distribution*:

Which gives the probability *P* that an experimental
result with a *z* value less than or equal to that
observed is due to chance. Subtracting *P* from one:

gives *Q*, the probability that the observed *z*
score is due to chance.

Unfortunately, there is no closed form solution for *P*, so it
must be evaluated numerically. This page allows you to calculate the
probability of chance occurrence of a given *z*, or to
calculate *z* given a probability *Q*. All
calculations are performed with six decimal places of accuracy; the
maximum *z* accepted is 6.

Note that the probability calculated from the *z*
is an approximation which is valid only for large
values of *n*, and is therefore only meaningful when
calculated for experiments with a large number of
individual trials.

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One of the most common probability calculations is determining, given
the measured *z* value from an experiment or set of
experiments, the probability of the result being due to chance. Enter
the *z* value in the box below, press the **Return** key or
the **Calculate** button, and the probability will appear in the Q
box.

To determine the *z* score indicating a probability *Q*
of non-chance occurrence for an experiment,
enter *Q* in the box below and
press the **Return** key or the **Calculate** button.

by John Walker