Chapter 15: Statistics

Section 15.1: Formulating Statistical Questions, Gathering Data, and Using

Samples

Statistical Questions• Statistical Questions: ones that can be answered by collecting and

analyzing data (pieces of information, can be numerical or categorical)

• Ex’s: (a) What is the height of each student in our class?(b) How many home runs has each player hit in a baseball team’s starting lineup?(c) What on campus restaurant is the favorite among UK students?(d) What is the average weight of watermelons sold at your local grocery store?

Types of Statistical Studies

• Observational Studies: observe characteristics or quantities without influencing these characteristics or quantities

• Experiments: try to determine factors that influence characteristics or quantities

Gathering Data

• The population is the full set of people or things that the study is designed to investigate.• Ex’s: (a) the students in our class, (b) the players in the starting

lineup, (c) the UK student body, (d) all of the watermelons• A sample of a population consists of some collection of members of

the population.• Samples need to be representative, i.e. the characteristics of the

sample reflect those of the population• Ideal representative sample: random sample- every member of the

population has an equal chance of being in the sample.

Gathering Data

• Samples need to be representative, i.e. the characteristics of the sample reflect those of the population• Ideal representative sample: random sample- every member of the

population has an equal chance of being in the sample.

• Ex: Asking random students in the Student Center at lunchtime about their favorite on campus restaurant is not random or representative.

Using Samples

• Ex: If a random sampling of 250 students showed that 60 students said Panda Express was their favorite on campus restaurant, about how many of UK’s 20,000 students would be expected to say that Panda Express is their favorite?

Section 15.2: Displaying and

Interpreting Data

Displaying Categorical Data

• Real graph: display using real objects in graph form

• Ex: using Starburst candy or wrappers to display how many pieces there are of each color

Displaying Categorical Data

• Pictograph: uses icons or pictures to display the data

• Ex: each small, colored rectangle represents a piece of candy for each flavor in one package of candy

Displaying Categorical Data

• Bar Graph: uses a single rectangle for each category to display the data

• Ex: the height of each bar represents the number of a piece of candy for each flavor

Displaying Categorical Data• Double Bar Graph: each category is subdivided into 2 smaller categories

• Ex: Displays median weekly earnings broken down by race and further subdivided by gender.

Displaying Categorical Data

• Pie Graph: uses a subdivided circle to show how data is partitioned into categories

• Ex: Shows the percentage of UK students that prefer that particular on campus restaurant

Displaying Numerical Data

• Dot Plot: a pictograph with categories being numbers or intervals and the icons being dots

• Ex: The following displays the home runs hit by hitters in a baseball lineup, where the home runs hit were 12, 17, 25, 32, 24, 17, 12, 12, 8

Displaying Numerical Data

• Ex: The following displays the home runs hit by hitters in a baseball lineup, where the home runs hit were 12, 17, 25, 32, 24, 17, 12, 12, 8

Displaying Numerical Data• Histogram: a bar graph with categories being numbers or intervals

• Ex: The number of students earning each letter grade on an exam is displayed.

Displaying Numerical Data

• Stem and Leaf Plot: 2 columns in which the stem (left column) and leaf (right column) together form the data• 2 | 0572 = data includes 20, 25, 27, and 22

• Ex: Displays the exam scores from the previous histogram5 266

6 93

7 187827

8 02053901

9 58263

Displaying Numerical Data

• Line Graphs: data points are plotted and adjacent points are connected by line segments, used for continuously varying data

• Ex: Displays the U.S. popu- lation each decade over the 20th century.

Displaying Numerical Data

• Scatterplot: collection of data points in a plane, shows how 2 kinds of data are related

• Ex: Compares homework and exam averages for my MA 201 students from last semester

40 50 60 70 80 90 100 1100

20

40

60

80

100

120

Homework Averages

Exam

Ave

rage

s

Reading Graphs

• 1. Reading the data: lift facts directly from the graphs

• 2. Reading between the data: use mathematical concepts and skills to compare or combine quantities and identify relationships between data

• 3. Reading beyond the data: predict or infer from the data

Section 15.3: The Center of Data: Mean,

Median, and Mode

Motivating Example

• Ex: If your 4th grade math class had the following scores on their adding fractions quiz, what score did the “average” student get?

7, 4, 9, 10, 7, 9, 9, 6

• A single-number summary of a set of numerical data is called a measure of center. Ex’s are mean, median, and mode.

The Mean

• Def: The mean (or arithmetic mean or average) of a list of numbers is calculated by adding all of the numbers and dividing that sum by the length of the list (the number of numbers).

• Ex 1: The mean of the quiz scores 7, 4, 9, 10, 7, 9, 9, 6 is …

Visualizations of the Mean

• Consider a dot plot of your numerical data as if the axis is a seesaw. The mean is the location of the fulcrum so that the seesaw is perfectly balanced.

• See Activity 15K for another visual approach to finding the mean.

The Median

• Def: The median of a list of numbers is the middle number of the list once it is ordered from smallest to largest.

• Ex 2: The median of the quiz scores 7, 4, 9, 10, 7, 9, 9, 6 is …

• If the length of the list is even, then take the average of the middle two numbers. If the length is odd, no average is needed.

See Activity 15O

The Mode

• Def: The mode of a list of numbers is the number that occur most frequently.

• Ex 3: The mode of the quiz scores 7, 4, 9, 10, 7, 9, 9, 6 is …

Revisiting Our Motivating Example

• What score did the average or typical student get if the quiz scores were 7, 4, 9, 10, 7, 9, 9, 6?

• 7.625, 8, or 9 are all reasonable answers

Categorical Data

• Def: The modal category is the category listed most frequently in your categorical data.

• Note that the modal category is not necessarily the “favorite” category in statistical studies like the favorite on campus restaurant question. Study voting theory for other possibilities.

Section 15.4: Data Distributions

Definition and Types

• Def: Data distribution: numerical data displayed in a dot plot or histogram.• Data distributions with a long tail that extends to the right (left) of the

majority of the data is skewed to the right (left).

Types of Data Distributions

• Data distributions which have two or more peaks bimodal.

Types of Data Distributions

• Data distributions which have or nearly have reflectional symmetry are symmetric.

Other statistical measures

• Def: The Pth percentile of a set of numerical data is the number such that P% of the data is that number.• Def: The 1st quartile is the 25th percentile and the 3rd quartile is the

75% percentile. • What is the 50th percentile called?

The median• Ex 4: What is the 1st quartile, 3rd quartile, and 90th percentile of the

following quiz scores? 3, 4, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10

Another Graphical Display

• Def: A box plot (box and whiskers plot) is a display of the lowest value, highest value, 1st and 3rd quartiles, and the median, as shown in the example below.

• Ex: Draw a box plot for the quiz scores from Ex 4: 3, 4, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10