1.2: 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.

Collaborative Exercise

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:

5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9

The dot plot for this data would be as follows:

Figure \(\PageIndex{1}\)

• 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.

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 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.

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.

Definition: Population

Population: Is the complete collection of measurements of data that are being considered. Typically, such a set is too large to directly measure, so we use inference techniques.

Definition: Sample

Sample: Is the sub-collection of members selected from the 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.

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.

Definition: Parameter

Parameter: A numerical measurement describing some characteristic of a population.

Definition: Statistic

Statistic: A numerical measurement describing some characteristic of a sample.

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 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 \(\frac{22}{40}\) and the proportion of women students is \(\frac{18}{40}\). 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."

Example \(\PageIndex{1}\)

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.

Answer

• 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 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 ABC College.
• The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225. 

Example \(\PageIndex{2}\)

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.

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 group of students who graduated from the college last year, randomly selected
5. the average cumulative GPA of 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

Answer

1. f; 2. g; 3. e; 4. d; 5. b; 6. c

Example \(\PageIndex{3}\)

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)
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 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 head injury, or no, did not.

Example \(\PageIndex{4}\)

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 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.

Answer

• 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.

 

Test Knowledge\(\PageIndex{1}\)

 

Exercise \(\PageIndex{1}\)

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.

Answer

• 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.

 

Collaborative Exercise

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 1, 0, 1, 3, and 4 glasses of milk.

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.

All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. 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.

All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in such numbers as \(\frac{\pi}{6}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{4}\), 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.

Sample of Quantitative Discrete Data

The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books (three, four, two, and one) are the quantitative discrete data.

Exercise \(\PageIndex{1}\)

The data are the number of machines in a gym. You sample 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

quantitative discrete data

Sample of Quantitative Continuous Data

The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured.

Exercise \(\PageIndex{2}\)

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

quantitative continuous data

Exercise \(\PageIndex{3}\)

You go to the supermarket and purchase three cans of soup (19 ounces) tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces Cherry Garcia ice cream and two pounds (32 ounces chocolate chip cookies).

Name data sets that are quantitative discrete, quantitative continuous, and qualitative.

Solution

One Possible Solution:

• The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data because you count them.
• The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible.
• Types of soups, nuts, vegetables and desserts are qualitative data because they are categorical.

Try to identify additional data sets in this example.

Sample of qualitative data

The data are the colors of backpacks. Again, you sample the same 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. The colors red, black, black, green, and gray are qualitative data.

Exercise \(\PageIndex{4}\)

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

qualitative data

Collaborative Exercise \(\PageIndex{1}\)

Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words "the number of."

1. the number of pairs of shoes you own
2. the type of car you drive
3. where you go on vacation
4. the distance it is from your home to the nearest grocery store
5. the number of classes you take per school year.
6. the tuition for your classes
7. the type of calculator you use
8. movie ratings
9. political party preferences
10. weights of sumo wrestlers
11. amount of money (in dollars) won playing poker
12. number of correct answers on a quiz
13. peoples’ attitudes toward the government
14. IQ scores (This may cause some discussion.)

Answer

Items a, e, f, k, and l are quantitative discrete; items d, j, and n are quantitative continuous; items b, c, g, h, i, and m are qualitative.

Exercise \(\PageIndex{5}\)

Determine the correct data type (quantitative or qualitative) for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete.

Answer

quantitative discrete

Exercise \(\PageIndex{6}\)

A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure \(\PageIndex{1}\). What type of data does this graph show?

Figure \(\PageIndex{1}\)

Answer

This pie chart shows the students in each year, which is qualitative data.

Exercise \(\PageIndex{7}\)

The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25.

Figure \(\PageIndex{2}\)

What type of data does this graph show?

Answer

A histogram is used to display quantitative data: the numbers of credit hours completed. Because students can complete only a whole number of hours (no fractions of hours allowed), this data is quantitative discrete.

 

Test Knowledge\(\PageIndex{2}\)

 

Qualitative Data Discussion

Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.

Table \(\PageIndex{1}\): Fall Term 2007 (Census day)

De Anza College				Foothill College
	Number	Percent			Number	Percent
Full-time	9,200	40.9%		Full-time	4,059	28.6%
Part-time	13,296	59.1%		Part-time	10,124	71.4%
Total	22,496	100%		Total	14,183	100%

Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.

• In a pie chart, categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
• In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
• A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).

Look at Figures \(\PageIndex{3}\) and \(\PageIndex{4}\) and determine which graph (pie or bar) you think displays the comparisons better.

Figure \(\PageIndex{3}\): Pie Charts

It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on what we are using the data for.

Figure \(\PageIndex{4}\): Bar chart

Percentages That Add to More (or Less) Than 100%

Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.

Table \(\PageIndex{2}\): De Anza College Spring 2010

Characteristic/Category	Percent
Full-Time Students	40.9%
Students who intend to transfer to a 4-year educational institution	48.6%
Students under age 25	61.0%
TOTAL	150.5%

Figure \(\PageIndex{2}\): Bar chart of data in Table \(\PageIndex{2}\).

Omitting Categories/Missing Data

The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.

Table \(\PageIndex{2}\): Ethnicity of Students at De Anza College Fall Term 2007 (Census Day)

	Frequency	Percent
Asian	8,794	36.1%
Black	1,412	5.8%
Filipino	1,298	5.3%
Hispanic	4,180	17.1%
Native American	146	0.6%
Pacific Islander	236	1.0%
White	5,978	24.5%
TOTAL	22,044 out of 24,382	90.4% out of 100%

Figure \(\PageIndex{3}\): Enrollment of De Anza College (Spring 2010)

The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.

This particular bar graph in Figure \(\PageIndex{4}\) can be difficult to understand visually. The graph in Figure \(\PageIndex{5}\) is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.

Figure \(\PageIndex{4}\): Bar Graph with Other/Unknown Category

Figure \(\PageIndex{5}\): Pareto Chart With Bars Sorted by Size

Pie Charts: No Missing Data

The following pie charts have the “Other/Unknown” category included (since the percentages must add to 100%). The chart in Figure \(\PageIndex{6}\) is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure \(\PageIndex{6}\).

Figure \(\PageIndex{6}\).

References

1. The Data and Story Library, https://dasl.datadescription.com/ (accessed May 1, 2013).
2. Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).
3. Data from www.bookofodds.com/Relationsh...-the-President
4. Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), ssh.dukejournals.org/content/36/1/23.abstract (accessed May 1, 2013).
5. “The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).
6. “Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/ga...9362004.aspx#4 (accessed May 1, 2013).
7. LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/f...hts.html#focus (accessed May 1, 2013).
8. Data from San Jose Mercury News

Practice

Use the following information to answer the next five exercises. 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. Two researchers each follow a different set of 40 patients with AIDS from the start of treatment until their deaths. The following data (in months) are collected.

Researcher A:

3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34

Researcher B:

3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29

Determine what the key terms refer to in the example for Researcher A.

Exercise \(\PageIndex{2}\)

population

Answer

AIDS patients.

Exercise \(\PageIndex{3}\)

sample

Exercise \(\PageIndex{4}\)

parameter

Answer

The average length of time (in months) AIDS patients live after treatment.

Exercise \(\PageIndex{5}\)

statistic

Exercise \(\PageIndex{6}\)

variable

Answer

\(X =\) the length of time (in months) AIDS patients live after treatment

Review

Data are individual items of information that come from a population or sample. Data may be classified as qualitative, quantitative continuous, or quantitative discrete.

Footnotes

1. lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).
2. Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).
3. Frequently Asked Questions, Pew Research Center for the People & the Press, www.people-press.org/methodol...wer-your-polls (accessed May 1, 2013).

Glossary

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.

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

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

 

Qualitative Data

The result of categorizing or describing attributes of a population.

 

Quantitative Data

Consists of numbers representing counts or measurements.