Stats Tutorial - Instrumental Analysis and Calibration

Probability Distributions:

When we make a measurement, we expect to get the true value. This is known as the expected value E(x), or the population (true) mean μ. We rarely get this value, however, as there is always some degree of random error or fluctuation in the system. The result we get will probably not be the true value, but somewhere close to it. If we repeat the measurements enough times, we expect that the average will be close to the true value, with the actual results spread around it. The distribution of a few measurements might look something like the following histogram:

Histogram of replicate concentration determinations

Here the frequency is the number of times that a particular result occurs. From this plot, we see that the values are distributed relatively evenly around a point somewhere between 1.2 and 1.6, so the mean value of these measurements is probably around 1.4 or 1.5. This type of plot is called a probability distribution and, as you can see, it has a bell shape to it. In fact, this type of distribution is sometimes called a bell-curve, or more commonly, the Normal Distribution.

The Normal Distribution:

A normal distribution implies that if you take a large enough number of measurements of the same property for the same sample under the same conditions, the values will be distributed around the expected value, or mean, and that the frequency with which a particular result (i.e. value) ocurs will become lower the farther away the result is from the mean.

Put another way, a normal distribution is a probability curve where there is a high probability of an event (i.e. a particular value) occurring near the mean value, with a decreasing chance of an event occurring as we move away from the mean.

The normal distribution curve and equation look like this:

Note: to convert y values to probabilities, P(x), the data must be normalized to give unit area.

The important thing to know about the Normal Distribution is that the probability of getting a certain result decreases the farther that result is from the mean. The concept of the normal distribution will be important when we talk about 1- and 2-tailed tests, confidence levels, and statistical tests, such as the t-test and F-test.

There are many other types of distributions, but we will only consider the normal distribution here. This is because we assume that the measurements we perform in this course will be normally distributed about the mean, and that the random errors will also be normally distributed. Generally, this is a good assumption, though there are many situations where it does not apply.

Continue to Confidence Levels...