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

Learning Objectives 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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.

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. 97 85 66 79 89 88 67 88 65 67 90 68 76 73 84 78 77 77 93 77 SOLUTION Notice that the first digit ranges from 6 to 9, so we set up a table with stems 6, 7, 8, 9.

EXAMPLE 4 Drawing a Stem and Leaf Plot SOLUTION 97 85 66 79 89 88 67 88 65 67 90 68 76 73 84 78 77 77 93 77 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.