1. Gathering and Organizing Data
Math in Our World
Section 12.1
Gathering and Organizing Data

2. Learning Objectives Identify binomial experiments.
Define data and statistics.
Explain the difference between a population and a sample.
Describe four basic methods of sampling.
Construct a frequency distribution for a data set.
Draw a stem and leaf plot for a data set.

3. Statistics
Data are measurements or observations that are gathered for an event under study.
Statistics is the branch of mathematics that involves collecting, organizing, summarizing, and presenting data and drawing general conclusions from that data.

4. Populations and Samples
When statistical studies are performed, we usually begin by identifying the population
for the study.
A population consists of all subjects under study. (i.e. all colleges in the United States)
More often than not, it’s not realistic to gather data from every member of a population.
A sample is a representative subgroup or subset of a population.

5. Sampling Methods We will study four basic sampling methods:
1. In order to obtain a random sample, each subject of the population must have an equal chance of being selected.
2. A systematic sample is taken by numbering each member of the population and then selecting every kth member, where k is a natural number. When using systematic sampling, it’s important that the starting number is selected at random.

6. Sampling Methods We will study four basic sampling methods:
3. When a population is divided into groups where the members of each group have similar characteristics and members from each group are chosen at random, the result is called a stratified sample.
4. When an existing group of subjects that represent the population is used for a sample, it is called a cluster sample.

7. EXAMPLE 1 Choosing a Sample
A student in an education class is given an assignment to find out how late typical students at his campus stay up to study. He decides to stop by the union before his 9 A.M. class and ask everyone sitting at a table how late they were up studying the night before. (a) What method of sampling is he using? (b) Do you think he’s likely to get a representative sample?

8. EXAMPLE 1 Choosing a Sample
SOLUTION
(a) Since he is choosing all students in a particular place at a particular time, he has chosen a cluster sample. (b) The sample is unlikely to be representative. Since he’s polling people early in the morning, those that tend to stay up very late studying are less likely to be included in the sample.

9. Descriptive vs. Inferential
There are two main branches of statistics: descriptive and inferential.
Statistical techniques that are used to describe data are called descriptive statistics.
For example, a researcher may wish to determine the average age of the full-time students enrolled in your college and the percentage who own automobiles.

10. Descriptive vs. Inferential
There are two main branches of statistics: descriptive and inferential.
Statistical techniques used to make inferences are called inferential statistics.
For example, every month the Bureau of Labor and Statistics estimates the number of people in the US who are unemployed. Since it’s would be impossible to survey everyone, they use a sample of adults to see what percent are unemployed. In this case, the information obtained from a sample is used to estimate a population measure.

11. Descriptive vs. Inferential
Another area of inferential statistics is called hypothesis testing. A researcher tries to test a hypothesis to see if there is enough evidence to support it.
A third aspect of inferential statistics is determining whether or not a relationship exists between two or more variables. This area of statistics is called correlation and regression.

12. Frequency Distributions
The data collected for a statistical study are called raw data. In order to describe situations and draw conclusions, the researcher must organize the data in a meaningful way.
Two methods that we will use are frequency distributions and stem and leaf plots.
The first type of frequency distributions that we will investigate is the categorical frequency distribution. This is used when the data are categorical rather than numerical.

13. EXAMPLE 2 Constructing a Frequency Distribution
Twenty-five volunteers for a medical research study were given a blood test to obtain their blood types. The data follow. Construct a frequency distribution for the data.

14. EXAMPLE 2 Constructing a Frequency Distribution
SOLUTION
Step 1 Make a table with all categories represented.
Type Tally Frequency A B O AB
Step 2 Tally the data using the second column.
Step 3 Count the tallies and place the numbers in the third column.

15. Frequency Distributions
Another type of frequency distribution that can be constructed uses numerical data and is called a grouped frequency distribution. In a grouped frequency distribution, the numerical data are divided into classes.

16. Frequency Distributions
When deciding on classes, here are some useful guidelines:
1. Try to keep the number of classes between 5 and 15.
2. Make sure the classes do not overlap.
3. Don’t leave out any numbers between the lowest and highest, even if nothing falls into a particular class.
4. Make sure the range of numbers included in a class is the same for each one.

17. EXAMPLE 3 Constructing a Frequency Distribution
These data represent the record high temperatures for each of the 50 states in degrees Fahrenheit. Construct a grouped frequency distribution for the data.
112
100
127
120
134
105
110
109
118
117
116
114
122
107
115
106
108
121
113
119
111
104
Source: The World Almanac Book of Facts

18. EXAMPLE 3 Constructing a Frequency Distribution
SOLUTION
Step 1 Subtract the lowest value from the highest value: 134 – 100 = 34.
Step 2 If we use a range of 5 degrees, that will give us seven classes, since the entire range (34 degrees) divided by 5 is 6.8.
112
100
127
120
134
105
110
109
118
117
116
114
122
107
115
106
108
121
113
119
111
104

19. EXAMPLE 3 Constructing a Frequency Distribution
SOLUTION
Class
Tally
Frequency
100
105
110
115
120
125
130
Step 3 Start with the lowest value and add 5 to get the lower class limits: 100, 105, 110, 115, 120, 125, 130.
112
100
127
120
134
105
110
109
118
117
116
114
122
107
115
106
108
121
113
119
111
104

20. EXAMPLE 3 Constructing a Frequency Distribution
SOLUTION
Class
Tally
Frequency
100
105
110
115
120
125
130
104
109
114
119
124
129
- 134
Step 4 Set up the classes by subtracting one from each lower class limit except the first lower class limit.
112
100
127
120
134
105
110
109
118
117
116
114
122
107
115
106
108
121
113
119
111
104

21. EXAMPLE 3 Constructing a Frequency Distribution
SOLUTION 2 8 18 13 7 1
104
109
114
119
124
129
- 134
Class
Tally
Frequency
100
105
110
115
120
125
130
Step 5 Tally the data and record the frequencies.
112
100
127
120
134
105
110
109
118
117
116
114
122
107
115
106
108
121
113
119
111
104

22. Stem and Leaf Plots
Another way to organize data is to use a stem and leaf plot (sometimes called a stem plot).
Each data value or number is separated into two parts. For a two-digit number such as 53, the tens digit, 5, is called the stem, and the ones digit, 3, is called its leaf.
For the number 72, the stem is 7, and the leaf is 2.
For a three-digit number, say 138, the first two digits, 13, are used as the stem, and the third digit, 8, is used as the leaf.

23. EXAMPLE 4 Drawing a Stem and Leaf Plot
The data below show the number of games won by the Chicago Cubs in each of the 21 seasons from 1988–2008, with the exception of 1994, which was a short season because of a player strike. Draw a stem and leaf plot for the data
SOLUTION
Notice that the first digit ranges from 6 to 9, so we set up a table with stems 6, 7, 8, 9.

24. EXAMPLE 4 Drawing a Stem and Leaf Plot
SOLUTION 7 3
Stems
Leaves 6 7 8 9 6 7 5 8 9 6 3 8 7 5 9 8 4
Now we go through the data one value at a time, putting the appropriate leaf next to the matching stem, starting with the 60s.
Now the 70s and so on.