1.1 Definitions of Statistics and Key Terms

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.

Example:

  1. 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.

  2. 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.

  3. City of Houston wants to know if the annual household income in the city is higher than national average. The statisticians collect data from 1500 families.

  4. An automobile manufacturer wanted to know if more than 50% of the US drivers own at least a domestic car. This company surveyed 10,000 drivers over US.

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.

Example:

  1. If we assume that X is equal to the number of points earned by one math student at the end of a term, then X is a numerical variable.
  2. If we let Y be a person’s 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 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.

NOTE (We will learn the meaning of mean in next chapter! )

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.


Try It

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.

 


Example 2

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 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively.

Solution:

Population: All families with children at Knoll Academy
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.



Example 3 

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.

Solution:


Try It

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.

 


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).


Concept Review

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.


Glossary

Average
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
Data
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
Parameter
a number that is used to represent a population characteristic and that generally cannot be determined easily
Population
all individuals, objects, or measurements whose properties are being studied
Probability
a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
Proportion
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
Sample
a subset of the population studied
Statistic
a numerical characteristic of the sample; a statistic estimates the corresponding population parameter.
Variable
a characteristic of interest for each person or object in a population