The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.
Important 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.
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.
Population: all math classes
Sample: One of the math classes
Parameter: Average number of points earned per student over all math classes
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.
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 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 2240 and the proportion of women students is 1840. Mean and proportion are discussed in more detail in later chapters.
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.”
Determine what the key terms refer to in 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.
all students who attended the college last year
all students who attended the college last year
a group of students who graduated from the college last year, randomly selected
3.65, 2.80, 1.50, 3.90
the cumulative GPA of one student who graduated from the college last year
the average cumulative GPA of students who graduated from the college last year
the average cumulative GPA of students in the study who graduated from the college last year
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 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.
Population: All the first year college students at ABC College
Sample: 100 first year students who are surveyed at the college.
Parameter: Average amount of money a first year college student spent on school supplies that do not include books.
Statistics: Average amount of money these 100 first year college student spent on school supplies that do not include books.
Variable: The amount of money a first year ABC College student spend on school supplies that do not include books.
Data: The amount that we collected from these 100 students, like $150, $200, and $225.
Sample: 100 families with children who are surveyed in the school.
Parameter: average (mean) amount of money spent on school uniforms by families with children at Knoll Academy.
Statistics: average (mean) amount of money spent on school uniforms by families in the sample.
Variable: the amount of money spent by one family
Data: The amount that we collected from these 100 families, like $65, $75, and $95.
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.
all cars containing dummies in the front seat. the 75 cars, selected by a simple random sample. the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population. proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample. the number of driver dummies (if they had been real people) who would have suffered head injuries. yes, had head injury, or no, did not.
all cars containing dummies in the front seat.
the 75 cars, selected by a simple random sample.
the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population.
proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.
the number of driver dummies (if they had been real people) who would have suffered head injuries.
yes, had head injury, or no, did not.
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.
all medical doctors listed in the professional directory.
the 500 doctors selected at random from the professional directory.
the proportion of medical doctors who have been involved in one or more malpractice suits in the population.
the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.
the number of medical doctors who have been involved in one or more malpractice suits.
Yes, was involved in one or more malpractice lawsuits; or no, was not.
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013).
The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text.
- also called mean; a number that describes the central tendency of the data
- Categorical Variable
- variables that take on values that are names or labels
- a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage)
- Numerical Variable
- variables that take on values that are indicated by numbers
- a number that is used to represent a population characteristic and that generally cannot be determined easily
- all individuals, objects, or measurements whose properties are being studied
- a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
- the number of successes divided by the total number in the sample
- Representative Sample
- a subset of the population that has the same characteristics as the population
- a subset of the population studied
- a numerical characteristic of the sample; a statistic estimates the corresponding population parameter.
- a characteristic of interest for each person or object in a population