Chapter 1 & 3

The Role of Statistics&

Graphical Methods for Describing Data

Statisticsthe science of collecting, analyzing, and drawing conclusions from data

Suppose we wanted to know the average GPA of high school graduates in the nation this year.

We could collect data from all high schools in the nation. What term would be

used to describe “all high school graduates”?

PopulationThe entire collection of

individuals or objects about which information is desired

A census is performed to gather about the entire population

What do you call it when you collect data about the

entire population?

Suppose we wanted to know the average GPA of high school graduates in the nation this year.

We could collect data from all high schools in the nation.

Why might we not want to use a census here?

If we didn’t perform a census, what would we do?

SampleA subset of the population,

selected for study in some prescribed manner

What would a sample of all high school graduates across the nation look like?

A list created by randomly selecting the GPAs of all high school graduates from each state.

Suppose we wanted to know the average GPA of high school graduates in the nation this year.

We could collect data from a sample of high schools in the nation.

Once we have collected the data, what would we do with it?

Descriptive statistics the methods of organizing &

summarizing data

• Create a graph

If the sample of high school GPAs contained 10,000 numbers, how could the data be described or summarized?

• State the range of GPAs• Calculate the average GPA

Suppose we wanted to know the average GPA of high school graduates in the nation this year.

We could collect data from a sample of high schools in the nation.Could we use the data from this sample to answer our question?

Inferential statistics involves making generalizations

from a sample to a populationBased on the sample, if the average GPA for high school graduates was 3.0, what generalization could be made?

The average national GPA for this year’s high school graduate is approximately 3.0.

Could someone claim that the average GPA for PISD graduates is 3.0?

No. Generalizations based on the results of a sample can only be made back to the population from which the sample came from.

Be sure to sample from the population of interest!!

Variable any characteristic whose value may change from one individual to another

Is this a variable . . .The number of wrecks per week

at the intersection outside?

Dataobservations on single variable or simultaneously on two or more variables

For this variable . . .The number of wrecks per week at the

intersection outside . . . What could observations be?

Types of variables

Categorical variablesor qualitativeidentifies basic

differentiating characteristics of the population

Numerical variablesor quantitative observations or measurements

take on numerical valuesmakes sense to average these

valuestwo types - discrete & continuous

Discrete (numerical)

listable set of valuesusually counts of items

Continuous (numerical)

data can take on any values in the domain of the variable

usually measurements of something

Classifying variables by the number of variables in a data set

Suppose that the PE coach records the height of each student in his class.

Univariate - data that describes a single characteristic of the population

This is an example of a univariate data

Classifying variables by the number of variables in a data set

Suppose that the PE coach records the height and weight of each student in his class.

Bivariate - data that describes two characteristics of the population

This is an example of a bivariate data

Classifying variables by the number

of variables in a data setSuppose that the PE coach records the height, weight, number of sit-ups, and number of push-ups for each student in his class.

Multivariate - data that describes more than two characteristics (beyond the scope of this course)

This is an example of a multivariate data

Identify the following variables:1. the appraised value of homes in Niceville

2. the color of cars in the teacher’s lot

3. the number of calculators owned by students at your school

4. the zip code of an individual

5. the amount of time it takes students to drive to school

Discrete numerical

Discrete numerical

Continuous numerical

Categorical

Categorical

Is money a measurement or a count?

Graphs for categorical data

Bar Graph

Used for categorical data Bars do not touch Categorical variable is typically on the horizontal

axis Frequency or relative frequency is on the vertical

axis To describe – comment on which occurred the

most often or least often May make a double bar graph or segmented bar

graph for bivariate categorical data sets

Relative frequency = frequency / total

Pie (Circle) graph

Used for categorical data To make:

– Proportion 360°

– Using a protractor, mark off each part

To describe – comment on which occurred the most often or least often

Using class survey data:

graph favorite ice cream

graph birth month

Graphs for numerical data

Dotplot

Used with numerical data (either discrete or continuous)

Made by putting dots (or X’s) on a number line

Can make comparative dotplots by using the same axis for multiple groups

Distribution Activity . . .

Types (shapes)of Distributions

Symmetricalrefers to data in which both sides are

(more or less) the same when the graph is folded vertically down the middle

bell-shaped is a special type–has a center mound with two

sloping tails

Uniformrefers to data in which every

class has equal or approximately equal frequency

Skewed (left or right)refers to data in which one

side (tail) is longer than the other side

the direction of skewness is on the side of the longer tail

Bimodal (multi-modal)refers to data in which two

(or more) classes have the largest frequency & are separated by at least one other class

How to describe a numerical,

univariate graph

What strikes you as the most distinctive difference among the distributions of exam scores in classes A, B, & C ?

1. Centerdiscuss where the middle of

the data fallsthree types of central

tendency–mean, median, & mode

What strikes you as the most distinctive difference among the distributions of scores in

classes D, E, & F?

2. Spreaddiscuss how spread out the data

isrefers to the variability of the

data–Range, standard deviation, IQR

What strikes you as the most distinctive difference among the distributions of exam scores in classes G, H, & I ?

3. Shaperefers to the overall shape of

the distributionsymmetrical, uniform,

skewed, or bimodal

What strikes you as the most distinctive difference among the distributions of exam scores in class K ?

4. Unusual occurrencesoutliers - value that lies away

from the rest of the datagapsclustersanything else unusual

5. In contextYou must write your answer

in reference to the specifics in the problem, using correct statistical vocabulary and using complete sentences!

More graphs for numerical data

Stemplots (stem & leaf plots)

Used with univariate, numerical data Must have key so that we know how to read

numbers Can split stems when you have long list of

leaves Can have a comparative stemplot with two

groups

Would a stemplot be a good graph for the number of pieces of gun chewed per day by

AP Stat students? Why or why not?

Would a stemplot be a good graph for the number of pairs of shoes owned by AP Stat

students? Why or why not?

Example:

The following data are price per ounce for various brands of dandruff shampoo at a local grocery store.

0.32 0.21 0.29 0.54 0.17 0.28 0.36 0.23

Can you make a stemplot with this data?

Example: Tobacco use in G-rated Movies

Total tobacco exposure time (in seconds) for Disney movies:223 176 548 37 158 51 299 37 11 165 74 9 2 6 23 206 9

Total tobacco exposure time (in seconds) for other studios’ movies:205 162 6 1 117 5 91 155 24 55 17

Make a comparative stemplot.

Histograms

Used with numerical data Bars touch on histograms Two types

– Discrete• Bars are centered over discrete values

– Continuous• Bars cover a class (interval) of values

For comparative histograms – use two separate graphs with the same scale on the horizontal axis

Would a histogram be a good graph for the fastest speed driven by AP Stat students?

Why or why not?

Would a histogram be a good graph for the number of pieces of gum chewed per day by

AP Stat students? Why or why not?

The two histograms below display the distribution of heights of gymnasts and the distribution of heights of female basketball players. Which is which? Why?

Heights – Figure A

Heights – Figure B

Suppose you found a pair of size 6 shoes left outside the locker room. Which team would you go to first to find the owner of the shoes? Why?

Suppose a tall woman (5 ft 11 in) tells you she is looking for her sister who is practicing in the gym. To which team would you send her? Why?

Cumulative Relative Frequency Plot(Ogive)

. . . is used to answer questions about percentiles. Percentiles are the percent of individuals that are

at or below a certain value. Quartiles are located every 25% of the data. The

first quartile (Q1) is the 25th percentile, while the third quartile (Q3) is the 75th percentile. What is the special name for Q2?

Interquartile Range (IQR) is the range of the middle half (50%) of the data.

IQR = Q3 – Q1