Any statistical method for determining a mean (and confidence limit) from a set of observations is based on a probability density function. This function describes the distribution of observations for a hypothetical, infinite set of observations called a population. The Gaussian probability density function (normal distribution) has the familiar bell-shaped form shown in Figure 11.1a. The meaning of the probability density function f(z) is that the proportion of observations within an interval of incremental width dz centered on z is f(z)dz.

The Gaussian probability density function is given by:

| (11.1) |

where

x is the variable measured, μ is the true mean, and σ is the standard deviation. The parameter μ determines the value of x about which the distribution is centered, while σ determines the width of the distribution about the true mean. By performing the required integrals (computing area under curve f(z)), it can be shown that 68% of the readings in a normal distribution are within σ of μ, while 95% are within 1.96σ of μ.

The usual situation is that one has made a finite number of measurements of a variable x. In the literature of statistics, this set of measurements is referred to as a sample. Let us say that we made 1000 measurements of some parameter, say bed thickness (in cm) in a particular sedimentary formation. We plot these in histogram form in Figure 11.1b.

By using the methods of Gaussian statistics, one is supposing that the observed sample
has been drawn from a population of observations that is normally distributed. The true
mean and standard deviation of the population are, of course, unknown. But the following
methods allow estimation of these quantities from the observed sample. A normal
distribution can be characterized by two parameters, the mean (μ) and the variance σ^{2}. How
to estimate the parameters of the underlying distribution is the art of statistics. We all know
that the arithmetic mean of a batch of data drawn from a normal distribution is calculated
by:

The mean estimated from the data shown in Figure 11.1b is 10.09. If we had measured an infinite number of bed thicknesses, we would have gotten the bell curve shown as the dashed line and calculated a mean of 10.

The “spread” in the data is characterized by the variance σ^{2}. Variance for normal
distributions can be estimated by the statistic s^{2}:

| (11.2) |

In order to get the units right on the spread about the mean (cm – not cm^{2}), we have to
take the square root of s^{2}. The statistic s gives an estimate of the standard deviation σ and
is the bounds around the mean that includes 63% of the values. The 95% confidence bounds
are given by 1.96s (this is what a “2-σ error” is), and should include 95% of the observations.
The bell curve shown in Figure 11.1b has a σ (standard deviation) of 3, while the s is
2.97.

If you repeat the bed measuring experiment a few times, you will never get exactly the
same measurements in the different trials. The mean and standard deviations measured for
each trial then are “sample” means and standard deviations. If you plotted up all those
sample means, you would get another normal distribution whose mean should be
pretty close to the true mean, but with a much more narrow standard deviation. In
Figure 11.1c we plot a histogram of means from 100 such trials of 1000 measurements
each drawn from the same distribution of μ = 10,σ = 3. In general, we expect the
standard deviation of the means (or standard error of the mean, s_{m}) to be related to s
by

What if we were to plot up a histogram of the estimated variances as in Figure 11.1c?
Are these also normally distributed? The answer is no, because variance is a squared
parameter relative to the original units. In fact, the distribution of variance estimates from
normal distibutions is expected to be chi-squared (χ^{2}). The width of the χ^{2} distribution is
also governed by how many measurements were made. The so-called number of degrees of
freedom (ν) is given by the number of measurements made minus the number of
measurements required to make the estimate, so ν for our case is N - 1. Therefore we expect
the variance estimates to follow a χ^{2} distribution with N - 1 degrees of freedom of
χ_{ν}^{2}.

The estimated standard error of the mean, s_{m}, provides a confidence limit for the
calculated mean. Of all the possible samples that can be drawn from a particular normal
distribution, 95% have means, , within 2s_{m} of . (Only 5% of possible samples have means
that lie farther than 2s_{m} from .) Thus the 95% confidence limit on the calculated
mean, , is 2s_{m}, and we are 95% certain that the true mean of the population from
which the sample was drawn lies within 2s_{m} of . The estimated standard error of
the mean, s_{m} decreases 1/. Larger samples provide more precise estimations
of the true mean; this is reflected in the smaller confidence limit with increasing
N.

We often wish to consider ratios of variances derived from normal distributions (for
example to decide if the data are more scattered in one data set relative to another). In order
to do this, we must know what ratio would be expected from data sets drawn from the same
distributions. Ratios of such variances follow a so-called F distribution with ν_{1} and ν_{2}
degrees of freedom for the two data sets. This is denoted F[ν_{1},ν_{2}]. Thus if the ratio F, given
by:

A related test to the F test is Student’s t-test. This test compares differences in normal
data sets and provides a means for judging their significance. Given two sets of
measurements of bed thickness, for example in two different sections, the t test addresses the
likelihood that the difference between the two means is significant at a given level of
probability. If the estimated means and standard deviations of the two sets of N_{1} and N_{2}
measurements are _{1},σ_{1} and _{2},σ_{2} respectively, the t statistic can be calculated
by: