The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.
Learning Outcomes
- Explain the difference between descriptive and inferential statistics
- For a given scenario, identify the population, parameter, sample, statistic, variables, and data
Activity
In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data:
[latex]5[/latex]; [latex]5.5[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]6[/latex]; [latex]6.5[/latex]; [latex]6.5[/latex]; [latex]6.5[/latex]; [latex]6.5[/latex]; [latex]7[/latex]; [latex]7[/latex]; [latex]8[/latex]; [latex]8[/latex]; [latex]9[/latex]
The dot plot for this data would be as follows:
Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not?
Where do your data appear to cluster? How might you interpret the clustering?
The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics.
Recall: Sets of numbers
The set of natural numbers includes the numbers used for counting: [latex]\{1,2,3,\dots\}[/latex].
The set of whole numbers is the set of natural numbers plus zero: [latex]\{0,1,2,3,\dots\}[/latex].
The set of integers adds the negative natural numbers to the set of whole numbers: [latex]\{\dots,-3,-2,-1,0,1,2,3,\dots\}[/latex].
The set of rational numbers includes fractions written as [latex]\{\frac{m}{n}|m\text{ and }n\text{ are integers and }n\ne 0\}[/latex].
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\{h|h\text{ is not a rational number}\}[/latex].
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from “good” data. The formal methods are called inferential statistics. Statistical inference uses probability (the likelihood that an event will occur) to determine how confident we can be that our conclusions are correct.
Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.
Recall: Fractions
A fraction is written as part(s) divided by whole (total number of parts in whole). For example, [latex]\frac{1}{2}[/latex] means [latex]1[/latex] out of two parts.
A fraction is written [latex]\Large{\frac{a}{b}}[/latex], where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b\ne 0[/latex]. In a fraction, [latex]a[/latex] is called the numerator and [latex]b[/latex] is called the denominator.
Recall: Probability
To calculate the probability of an event, you find the number of times that event happens in a situation (favorable outcomes) and divide that by the total number of outcomes possible in that situation.
[latex]\mathrm{Probability} = \frac{\mathrm{number \ of \ favorable \ outcomes}}{\mathrm{total \ number \ of \ outcomes}}[/latex]
Probability is typically written in terms of a fraction or a decimal.
Probability
Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin [latex]4,000[/latex] times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is [latex]12[/latex] or [latex]0.5[/latex]. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin [latex]24,000[/latex] times with a result of [latex]12,012[/latex] heads, one of the authors tossed a coin [latex]2,000[/latex] times. The results were [latex]996[/latex] heads. The fraction [latex]\displaystyle\frac{{996}}{{2,000}}[/latex] is equal to [latex]0.498[/latex] which is very close to [latex]0.5[/latex], 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, of rain, or whether you will get an A in this 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. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.
Recall: Proportion and Probability
The words “proportion” and “probability” are often misunderstood. They are both calculated the same way; yet proportion is a measure of something we know to be true, and probability is a measure of uncertainty.
Key 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. 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 [latex]1,000[/latex]–[latex]2,000[/latex] 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 [latex]16[/latex] ounce can contains [latex]16[/latex] 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 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. The statistic is an estimate of a population parameter. A parameter is a number that is 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.
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] and [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 the number of points earned by one math student 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 Republican, Democrat, and Independent. [latex]Y[/latex] is a categorical variable. We could do some math with values of [latex]X[/latex] (calculate the average number of points earned, for example), but it makes no sense to do math with values of [latex]Y[/latex] (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 [latex]86[/latex], [latex]75[/latex], and [latex]92[/latex], you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be [latex]84.3[/latex] to one decimal place). If, in your math class, there are [latex]40[/latex] students and [latex]22[/latex] are male and [latex]18[/latex] are female, then the proportion of male students is [latex]\displaystyle\frac{{22}}{{40}}[/latex] and the proportion of female 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 is “arithmetic mean,” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”
Example
Determine what the key terms refer to in 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 survey [latex]100[/latex] first year students at the college. Three of those students spent [latex]$150[/latex], [latex]$200[/latex], and [latex]$225[/latex], respectively.
Try It
Determine what the key terms 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 [latex]100[/latex] families with children in the school. Three of the families spent [latex]$65[/latex], [latex]$75[/latex], and [latex]$95[/latex], respectively.
Example
Determine what the key terms refer to in the following study.
A study was conducted at a local college to analyze the average cumulative GPAs 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
a) all students who attended the college last year
b) the cumulative GPA of one student who graduated from the college last year
c) [latex]3.65[/latex], [latex]2.80[/latex], [latex]1.50[/latex], [latex]3.90[/latex]
d) a group of students who graduated from the college last year, randomly selected
e) the average cumulative GPA of students who graduated from the college last year
f) all students who graduated from the college last year
g) the average cumulative GPA of students in the study who graduated from the college last year
Example
Determine what the key terms 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) |
| [latex]35[/latex] miles/hour | Front Seat |
Cars with dummies in the front seats were crashed into a wall at a speed of [latex]35[/latex] 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 [latex]75[/latex] cars.
Example
Determine what the key terms 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 [latex]500[/latex] doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.
Activity
Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study:
You want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were [latex]1[/latex], [latex]0[/latex], [latex]1[/latex], [latex]3[/latex], and [latex]4[/latex] glasses of milk.
Watch the following video for a brief introduction to statistics.
References
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013).
