Important Terms You Need to Know in Probability and Statistics
INTRODUCTION
Probability and statistics are concerned with events which occur by chance. Examples
include occurrence of accidents, errors of measurements, production of defective and
nondefective items from a production line, and various games of chance, such as
drawing a card from a well-mixed deck, flipping a coin, or throwing a symmetrical
six-sided die.
In each case of probability and statistics we may have some knowledge of the likelihood of various possible results, but we cannot predict with any certainty the outcome of any particular
trial.
Probability and statistics are used throughout engineering. In electrical engineering, signals and noise are analyzed by means of probability theory. Civil, mechanical, and industrial engineers use statistics and probability to test and account for variations in materials and goods. Chemical engineers use probability and statistics to assess experimental data and control and improve chemical processes.
It is essential for today’s engineer as part or function of probability and statistics to master these tools.
Important Terms You Need to Know in Probability and Statistics
(a) Probability is an area of study which involves predicting the relative likelihood
of various outcomes. It is a mathematical area which has developed over the past three or four centuries. One of the early uses was to calculate the odds of various gambling games.
Its usefulness for describing errors of scientific and engineering measurements was soon realized. Engineers study probability for its many practical uses, ranging from quality control and quality assurance to communication theory in electrical engineering. Engineering measurements are often analyzed using statistics, as we shall see
later in this book, and a good knowledge of probability is needed in order to understand statistics.
(b) Statistics is a word with a variety of meanings. To the man in the street it most
often means simply a collection of numbers, such as the number of people living in a country or city, a stock exchange index, or the rate of inflation.
These all come under the heading of descriptive statistics, in which items are counted or measured and the results are combined in various ways to give useful results. That type of statistics certainly has its uses in engineering, and we will deal with it later, but another type of statistics will engage our attention in this book to a much greater extent. That is inferential statistics or statistical inference. For example, it is often not practical to measure all the items produced by a process.
Instead, we very frequently take a sample and measure the relevant quantity on each member of the sample. We infer something about all the items of interest from our knowledge of the sample.
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A particular characteristic of all the items we are interested in constitutes a population. Measurements of the diameter of all possible bolts as they come off a production process would make up a particular population. A sample is a chosen part of the population in question, say the measured diameters of twelve bolts chosen to be representative of all the bolts made under certain conditions. We need to know how reliable is the information inferred about the population on the basis of our measurements of the sample. Perhaps we
can say that “nineteen times out of twenty” the error will be less than a certain stated limit.
(c) Chance is a necessary part of any process to be described by probability or statistics. Sometimes that element of chance is due partly or even perhaps entirely to our lack of knowledge of the details of the process.
For example, if we had complete knowledge of the composition of every part of the raw
materials used to make bolts, and of the physical processes and conditions in their manufacture, in principle we could predict the diameter of each bolt.
But in practice we generally lack that complete knowledge, so the diameter of the next bolt to be produced is an unknown quantity described by a random variation. Under these conditions the distribution of diameters can be described by probability and statistics. If we want to improve the quality of those bolts and to make them more uniform, we will have to look into the causes of the variation and make changes in the raw materials or the production process. But even after that, there will very likely be a random variation
in diameter that can be described statistically.
Relations which involve chance are called probabilistic or stochastic relations. These are contrasted with deterministic relations, in which there is no element of chance. For example, Ohm’s Law and Newton’s Second Law involve no element of chance, so they are deterministic. However, measurements based on either of these laws do involve elements of chance, so relations between the measured quantities are probabilistic.
(d) Another term which requires some discussion is randomness. A random action cannot be predicted and so is due to chance. A random sample is one in which every member of the population has an equal likelihood of appearing.
Just which items appear in the sample is determined completely by chance. If some items are more likely to appear in the sample than others, then the sample is not random.
