1.2 Statistical Thinking and Key Terms

What is Probability?

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring in the future. When expressed numerically, probability is a number between 0 and 1 (including 0 and 1), that gives the likelihood that a specific event will occur in the future. An event with a probability of 0 will not happen but an event with a probability of 1 will happen for sure. What about probabilities somewhere between 0 and 1?

Suppose you toss a fair coin four times, the outcome may not be exactly two heads and two tails. We cannot say for sure what the outcome will be. However, if you toss the same coin 4000 times, and look at the collective result, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is [latex]{\frac{1}{2}}[/latex] or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. There is a dependability of probability in the long run. After reading about the English statistician Karl Pearson, who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors of this book tossed a coin 2000 times. The results were 996 heads! The fraction [latex]{\frac{996}{2000}}[/latex] is equal to 0.498 which is very close to 0.5, the expected probability.

The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake this year, rain this week, or the result of a roll of a six-sided die, we use probabilities. Sometimes we base probability on experimentation and observed events. This is called empirical probability. Doctors use probability to determine the chance of a vaccination causing the disease that the vaccination is supposed to prevent. A stockbroker uses probability to determine the likely rate of return on a client’s investments. You might use probability to decide to buy a raffle ticket, or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

Key Statistical Terms

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample, which is a subset of the population. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. Data is a set of observations gained from the sample.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students’ grade point averages. In presidential elections, opinion poll samples of 1000 to 2000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples from a production run to determine if a 16 ounce can contains 16 ounces of carbonated drink.

From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider the students in our statistics class to be a sample of the population of all students at our school, then the average number hours our class spends working at a paid job is an example of a statistic. While a statistic is a property of a sample, a parameter is a numeric property of a population. The statistic is an estimate of a population parameter. Since we considered all the students at our school to be the population, then the average number of hours working at a paid job for all the students at our school is an example of a parameter.

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

A variable, notated by capital letters such as X and Y, is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values that are counted, such as the number of accidents in a week, or with measurement units, such as weight in pounds or time in hours. Categorical variables place the person or thing into a category. If we let X represent the number of defective bolts in a production run, then X is a numerical variable. If we let Y represent a person’s political party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable.

Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.

Two words that come up often daily life, but which have specific meanings in statistics, are “average” and “proportion.” We will go into more detail in later chapters, but for now, when someone uses the term “average” they likely are talking about the mean. For example, if you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 12 are left-handed and 28 are right-handed, then the proportion of left-handed students is [latex]{\frac{12}{40}}[/latex] or 0.3 and the proportion of right-handed students is [latex]{\frac{28}{40}}[/latex] or 0.7.