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1.1 Definitions of Statistics, Probability, and Key Terms

LEARNING OBJECTIVES

  • Recognize and differentiate between key terms used in statistics.

The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives. The organization and summation of data is called descriptive statistics. Two ways to summarize data are by graphing, such as a histogram or box plot, and by using numbers, such as average and standard deviation. After we have studied probability and probability distributions, we will use formal methods for drawing conclusions from “good” data. These formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.

Effective interpretation of data, or inference, is based on good procedures for producing data and thoughtful examination of the data. Although there are numerous mathematical formulas for analyzing data, the goal of statistics is to gain an understanding of the data, and not to simply perform calculations using the formulas. These days, we use computers to perform the calculations. The understanding and interpretation of the data comes from us. If we can thoroughly grasp the basics of statistics, we can be more confident in the decisions you make in life.

Probability

Probability is a mathematical tool used to study randomness and the chance, or likelihood, of an event occurring. For example, if we toss a fair coin four times, the outcomes may not necessarily be two heads and two tails. However, if we toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 50%. Even though the outcomes from a small number of repetitions are uncertain, there is a regular pattern to the outcomes when there is a large number of repetitions.

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, of rain, or whether a student will get an A in a particular course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client’s investments. We might use probability to decide whether or not to buy a lottery ticket. In the study of statistics, we use the power of mathematics through probability calculations to analyze and interpret the data.

Key Terms

In statistics, we generally want to study a population. A population is a collection of persons, things, or objects under study. Because populations tend to be very large, it is too expensive and too time-consuming to study the entire population. Instead of studying the population, we study a sample taken from the population. The idea of sampling is to select a portion, or subset, of the larger population and study that sample to gain information about the population.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. Consider the following examples:

  • If we wished to compute the overall grade point average at a 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 federal elections, opinion polls typically sample between 1,000 and 2,000 people. The opinion poll is supposed to represent the views of the people in the entire country.
  • Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can actually contains 16 ounces of a 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 one math class to be a sample of the population of all math classes, then the average grade earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a number that is a property of the population. Because we considered all math classes to be the population, then the average grade earned per student over all the math classes 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 [latex]X[/latex] or [latex]Y[/latex], is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let [latex]X[/latex] equal one student’s grade in a math class at the end of a term, then [latex]X[/latex] is a numerical variable. If we let [latex]Y[/latex] be a person’s party affiliation, then some examples of [latex]Y[/latex] include Conservative, Liberal, and New Democrat. In this case, [latex]Y[/latex] is a categorical variable. We could do some math with values of [latex]X[/latex], such as calculate the average grade, but it makes no sense to do math with values of [latex]Y[/latex]. 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 in statistics are mean and proportion. If we take three exams written by a single student in a math class and obtain scores of 86, 75, and 92, we would calculate the student’s mean score by adding the three exam scores and dividing by three, in this case a mean of 84.3. If a class has 40 students and 22 are men and 18 are women, then the proportion of men students is [latex]\displaystyle{\frac{22}{40}}[/latex] and the proportion of women students is [latex]\displaystyle{\frac{18}{40}}[/latex]. Mean and proportion are discussed in more detail in later chapters.

NOTE

The words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term for mean  is “arithmetic mean,” and “average” is technically a centre of location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”

EXAMPLE

Determine the key terms for the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first-year students at the college. Three of those students spent $150, $200, and $225, respectively.

Solution

  • The population is all first year students attending ABC College this term.
  • The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).
  • The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term (the population mean).
  • The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample (the sample mean).
  • The variable could be the amount of money spent (excluding books) by one first year student. Let [latex]X[/latex] be the amount of money spent (excluding books) by one first year student attending ABC College.
  • The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225.

EXAMPLE

Determine the key terms refer for the following study. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies. Here is the criterion they used:

Speed at which cars crashed 56 kilometres/hour
Location of crash test dummies Front seat

Cars with dummies in the front seats were crashed into a wall at a speed of 56 kilometres per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars.

Solution

  • The population is all cars containing dummies in the front seat.
  • The sample is the 75 cars, selected by a simple random sample.
  • The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population.
  • The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.
  • The variable [latex]X[/latex] = the number of driver dummies (if they had been real people) who would have suffered head injuries.
  • The data are either: yes, had head injury, or no, did not.

EXAMPLE

Determine the key terms for the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.

Solution

  • The population is all medical doctors listed in the professional directory.
  • The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.
  • The sample is the 500 doctors selected at random from the professional directory.
  • The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.
  • The variable [latex]X[/latex] = the number of medical doctors who have been involved in one or more malpractice suits.
  • The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not.

TRY IT

Determine the key terms for the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively.

 

Click to see Solution
  • The population is all families with children attending Knoll Academy.
  • The sample is a random selection of 100 families with children attending Knoll Academy.
  • The parameter is the average (mean) amount of money spent on school uniforms by families with children at Knoll Academy.
  • The statistic is the average (mean) amount of money spent on school uniforms by families in the sample.
  • The variable is the amount of money spent by one family. Let [latex]X[/latex] be the amount of money spent on school uniforms by one family with children attending Knoll Academy.
  • The data are the dollar amounts spent by the families. Examples of the data are $65, $75, and $95.

TRY IT

Determine the key terms for the following study. A study was conducted at a local college to analyze the average cumulative GPA’s of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below.

  1. Population ____
  2. Statistic ____
  3. Parameter ____
  4. Sample ____
  5. Variable ____
  6. Data ____
  1. all students who attended the college last year.
  2. the cumulative GPA of one student who graduated from the college last year.
  3. 3.65, 2.80, 1.50, 3.90.
  4. a randomly selected group of students who graduated from the college last year.
  5. the average cumulative GPA of all students who graduated from the college last year.
  6. all students who graduated from the college last year.
  7. the average cumulative GPA of students in the study who graduated from the college last year.
Click to see Solution
  1. f
  2. g
  3. e
  4. d
  5. b
  6. c

Exercises

  1. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once they start the treatment. A researcher selects [latex]40[/latex] patients with AIDS from the start of treatment until their deaths. Identify the population, the sample, the parameter, the sample and the variable for this study.
    Click to see Answer
    • Population: all AIDS patients
    • Sample: the 40 AIDS patients in the study.
    • Parameter: the average length of time (in months) AIDS patients live after treatment.
    • Sample: the average length of time (in months) patients in the study live after treatment.
    • Variable: the length of time an individual AIDS patient lives after treatment.

     

  2. For each of the following eight exercises, identify the key terms: the population, the sample, the parameter, the statistic, and the variable.
    1. A fitness centre is interested in the mean amount of time a client exercises in the centre each week.
    2. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.
    3. A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.
    4. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.
    5. A politician is interested in the proportion of voters in his district who think he is doing a good job.
    6. A marriage counsellor is interested in the proportion of clients she counsels who stay married.
    7. Political pollsters may be interested in the proportion of people who will vote for a particular cause.
    8. A marketing company is interested in the proportion of people who will buy a particular product.
    Click to see Answer
      • Population: all clients at the fitness centre.
      • Sample: a group or subset of the clients at the fitness centre.
      • Parameter: the mean amount of time all clients at the fitness centre exercise each week.
      • Statistic: the mean amount of time the clients in the sample exercise each week.
      • Variable: the amount of time an individual client at the fitness centre exercises each week.
      • Population: all children who take ski or snowboard lessons.
      • Sample: a group or subset of children who take ski or snowboard lessons.
      • Parameter: the mean age of all children who take ski or snowboard lessons.
      • Statistic: the mean age of the children in the sample.
      • Variable: the age of an individual child who takes ski or snowboard lessons.
      • Population: all of the cardiologist’s patients who have had heart attacks.
      • Sample: a group or subset of cardiologist’s patients who have had heart attacks.
      • Parameter: the mean recovery time of all patients who have had heart attacks.
      • Statistic: the mean recovery time of the patients in the sample.
      • Variable: the recovery time of an individual patient who has had a heart attack.
      • Population: all clients at the insurance company.
      • Sample: a group or subset of clients at the insurance company.
      • Parameter: the mean health cost of all clients at the insurance company.
      • Statistic: the mean health cost of the clients in the sample.
      • Variable: the health cost of an individual client at the insurance company.
      • Population: all voters in the politician’s district.
      • Sample: a group or subset of voters in the politician’s district.
      • Parameter: the proportion of all voters in the district who think the politician is doing a good job.
      • Statistic: the proportion of the voters in the sample who think the politician is doing a good job.
      • Variable: the number of voters who think the politician is doing a good job.
      • Population: all clients of the counsellor.
      • Sample: a group or subset of the counsellor’s clients.
      • Parameter: the proportion of all clients who stayed married.
      • Statistic: the proportion of clients in the sample who stayed married.
      • Variable: the number of clients who stayed married.
      • Population: all voters.
      • Sample: a group or subset of voters.
      • Parameter: the proportion of all voters who vote for the cause.
      • Statistic: the proportion of voters in the sample who vote for the cause.
      • Variable: the number of voters who vote for the cause.
      • Population: all consumers.
      • Sample: a group or subset of consumers.
      • Parameter: the proportion of all consumers who purchase the product.
      • Statistic: the proportion of consumers in the sample who purchase the product.
      • Variable: the number of consumers who purchase the product.

     

  3. A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.
    1. What is the population she is interested in?
    2. Consider the following: [latex]X[/latex] = number of days a Lake Tahoe Community College math student is absent. In this case, [latex]X[/latex] is an example of what?
    3. The instructor’s sample produces a mean number of days absent of [latex]3.5[/latex] days. This value is an example of what?
    Click to see Answer
    1. All math students at Lake Tahoe Community College.
    2. A variable.
    3. A statistics.

     

“1.2 Definitions of Statistics, Probability, and Key Terms” and “1.6 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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Introduction to Statistics - Second Edition Copyright © 2025 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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