Stats Tutorial - Probability Distributions:

Introduction

Basics of Excel™

Basic Statistics

Introduction

Stats in Anal Chem

Mean and Variance

Error and Residuals

Probability

Confidence Levels

Degrees of Freedom

Linear Regression

Data Evaluation & Comparison


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When we make a measurement, we expect to get the true value. This is known as the expected value E(x), or the population, or true, mean μ. However, this is not normally the case, since there is always some degree of random error or fluctuation in the system. As result, we probably will not get 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 this:

where the frequency is the number of times that a result occurs. From this plot, we can 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.

A normal distribution implies that if you take a large number of measurements of the same system, the values with be distributed around the expected value, or mean, and that the frequency of a result 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 occurring near the mean value, with a decreasing chance of an event occurring as we move away from the mean.

The normal distribution curve looks like this:


What is important 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. However, we will not consider this here.

© 2006 Dr. David C. Stone & Jon Ellis, Chemistry, University of Toronto
Last updated: September 25th, 2006