4.1: Introduction to Statistics
- Recognize and differentiate between key terms.
- Identify different types of data
You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess."
Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.
Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad."
The science of statistics deals with the collection, analysis, interpretation, and presentation of data . We see and use data in our everyday lives.
Key Terms in Statistics
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 . 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.
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 1,000–2,000 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 to determine if a 16 ounce can contains 16 ounces of carbonated drink.
A parameter is a number that represents a property of the population . Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter. 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 number of points earned by students in that one math class at the end of the term is an example of a statistic. A statistic can be used to estimate the unknown population parameters.
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 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 \(X\) equal the number of points earned by one math student at the end of a term, then \(X\) is a numerical variable. If we let \(Y\) be a person's political party affiliation, then some examples of \(Y\) include Republican, Democrat, and Independent; \(Y\) is a categorical variable. We could do some math with values of \(X\) (calculate the average number of points earned, for example), but it makes no sense to do math with values of \(Y\) (calculating an average party affiliation makes no sense).
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 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 rounded to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is \(\frac{22}{40}\) and the proportion of women students is \(\frac{18}{40}\). Mean and proportion are discussed in more detail in later chapters.
Determine what the key terms (population, sample, parameter, statistics, variable, data) refer to in the following study. We want to know the average (mean) amount of money first-year college students spend at MSJC on school supplies that do not include books. We randomly survey 100 first-year students at the college. Three of those students spent $150, $200, and $225, respectively.
Answer
- The population is all first-year students attending AMSJC this term.
- The sample could be all students enrolled in one section of a beginning statistics course at MSJC (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 MSJC this term.
- The statistic is the average (mean) amount of money spent (excluding books) by first-year college students in the sample.
- The variable could be the amount of money spent (excluding books) by one first-year student. Let \(X\) = the amount of money spent (excluding books) by one first-year student attending MSJC.
- The data are the dollar amounts spent by the first-year students. Examples of the data are $150, $200, and $225.
Determine what the key terms (population, sample, parameter, statistics, variable, data) refer to in 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.
- Answer
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- 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 \(X\) = 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.
Determine what the key terms (population, sample, parameter, statistics, variable, data) refer to in 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 | Location of “drive” (i.e. dummies) |
|---|---|
| 35 miles/hour | Front Seat |
Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles 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.
Answer
- 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 the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.
- The variable \(X\) = the number of driver dummies (if they had been real people) who would have suffered head injuries.
- The data are either: yes, had a head injury, or no, did not.
Determine what the key terms (population, sample, parameter, statistics, variable, data) refer to in 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.
- Answer
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- 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 \(X\) = 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.
Types of Data
Data may come from a population or from a sample. Small letters like \(x\) or \(y\) generally are used to represent data values. Most data can be put into the following categories:
- Qualitative
- Quantitative
Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous . Quantitative discrete data takes on only certain numerical values or in other words it would be possible to list all possible values. For example, if you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three, thus the number of daily calls is discrete data. All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring lengths in yards might result in such numbers as \(10.2\), \(7.9\), \(102.6\), \(92.7\), and so on. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data.
Consider the number of books students carry in their backpacks in a sample of five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. What type of data is this?
Solution
The numbers of books (three, four, two, and one) are quantitative discrete data .
Consider the number of machines in a gym in a sample of five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?
- Answer
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The numbers of machines (three, four, two, and one) are quantitative discrete data.
You go to the supermarket and purchase three cans of soup (19.4 ounces) tomato bisque, 14.1 ounces lentil, and 19.6 ounces Italian wedding, two packages of nuts (walnuts and peanuts), four different kinds of vegetables (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces Cherry Garcia ice cream and two pounds (32 ounces chocolate chip cookies). What type of data are the weights of the soups?
Solution
The weights of the soups (19.4 ounces, 14.1 ounces, 19.6 ounces) are quantitative continuous data because you measure weights as precisely as possible.
The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?
- Answer
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quantitative continuous data
Consider the colors of backpacks of a sample of five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. What type of data is this?
Solution
The colors red, black, black, green, and gray are qualitative data .
The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?
- Answer
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qualitative data