Warm-Up: Believe It or Not?

A student claims that they have flipped a fair coin 200 times and only had 84 times the heads side of the coin showed up.

Do you believe this student or not, discuss with your neighbor why or why not.

Chapters 2 - 4

The Role of Statistics&

Graphical Methods for Describing Data

In order to learn Statistics, we need to learn the language of statistics first.

We’ll be learning a lot of new vocabulary today – through examples and activities

Statistics the science of collecting, analyzing, and drawing conclusions from data

Suppose we wanted to know something about the GPAs of high school graduates in the nation this year.

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

Suppose we wanted to know something about the GPAs 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”?

Population The entire group of individuals or

objects we want information about A census attempts to contact every

individual in the entire population

What do you call it when you collect data about the entire population?

Suppose we wanted to know something about the GPAs of high school graduates in the nation this year.

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

Suppose we wanted to know something about the GPAs of high school graduates in the nation this year.

We could collect data from all high schools in the nation.If we didn’t perform a

census, what would we do?

Why might we not want to use a census here?

Sample A part of the population that we

actually examine in order to gather information

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 something about the GPAs of high school graduates in the nation this year.

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

Suppose we wanted to know something about the GPAs 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 something about the GPAs 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 population

Be sure to sample from the population of interest!!

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 FISD 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.

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

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

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

at the intersection outside?

Dataobservations on a single variable or simultaneously on two or more variables

Dataobservations on a single variable or simultaneously on two or more variables

For this variable . . .The number of wrecks per week at the intersection outside . . . what could the

observations be?

VariabilityThe range of possible data values

The goal of statistics is to understand the nature of variability in a population

VariabilityThe range of possible data values

The goal of statistics is to understand the nature of variability in a population

Can you think of a population that has no variability?

Populations with no variability are rare and boring (of little statistical interest).

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

Variability

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) you see is looking for her sister who is practicing in the gym. To which team would you send her? Why?

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) you see is looking for her sister who is practicing in the gym. To which team would you send her? Why?

What aspects of the graphs helped you answer these questions?

Types of variables

Categorical variables (qualitative) Variables where the possible

values are set of categories

Numerical variables or quantitative Variables where the values are

numbers (are numerical) (makes sense to average these

values) two types - discrete & continuous

Numerical: Discrete Values are isolated points on

a number line usually counts of items

Numerical: Continuous Set of possible values form an

entire interval on the number line

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 Faraway

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

Continuous numerical

Discrete numerical

Continuous numerical

Categorical

Categorical

Warm-Up: Classifying variables

Write an example of a variable on the index card provided (try to come up with something we have not discussed in class already). Please include your name.

When done, fold your index card in half and place in the bowl in the back of the room.

We will classify these before completing notes on display types.

Graphs for categorical data

Bar Graph Used for categorical data Bars do not touch Categorical variable is

typically on the horizontal axis

Best used to describe or 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

Comparative Bar Charts

Use relative frequency If observations are the same for all groups (50

boys and 50 girls), you could use the frequency Vertical scale the same

Pie Chart (circle graph) Used for categorical data To make:

Proportion X 360° Using a protractor, mark

off each part Best used to describe or

comment on which occurred the most often or least often

Using class survey data, make bar graphs for:

birth month

gender & handedness

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

Dotplot

To compare the weights of the males and females we put the dotplots on top of each other, using the same scales.

Using class survey data make dot plots of:

# AP classes

# siblings

Types (shapes)of Distributions

1) Symmetrical refers 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 typehas a center mound with two

sloping tails

2) Uniform refers to data in which every

class has equal or approximately equal frequency

3) 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

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

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

Warm-Up: Example 1 (From Your Notes) Looking at Example 1 (about sports-related

injuries), complete the columns titled “Tally” and “Frequency”.

Graphical Methods for

Describing Data

Frequency DistributionsThe relative frequency for a particular

category is the fraction or proportion of the time that the category appears in the data set. It is calculated as

frequencyrelative frequency number of observations in the data set

When the table includes relative frequencies, it is sometimes referred to as a relative frequency distribution.

Category Tally Frequency Relative freq. Cum rel. freq

Sprain

Contusion

Fracture

Strain

Laceration

Chronic

Dislocation

Concussion

Dental

13

12

14

6

3

2

3

2

1

13/56.2321

12/56.2143

.25

.1071

.0534

.0357

.0534

.0357

.0179

Example1

.2321

.4464

.6964

.8035

.8569

.8926

.946

.9817

1.00

Bar Chart – Example (frequency)

02468

10121416

Injury

Freq

uenc

y

Bar Chart – (Relative Frequency)

0

0.05

0.1

0.15

0.2

0.25

0.3

Injury

Rela

tive

Freq

uenc

y

Pie ChartRelative freq.

23%

21%

25%

11%

5%

4%

5%

4%

2%

Sprain

Contusion

Fracture

Strain

Laceration

Chronic

Dislocation

Concussion

Dental

Class data: fastest speed you have ever driven.

Take that speed and your gender and put it on a sticky note.

In a moment, you will place you sticky note next to the digit(s) that represent tenths on the

white board. For example, you drove 102 – put it next to 10

You drove 87 – put it by the 8

Fastest speed driven3456789

10111213

8│7 is 87 mph

Stemplots (stem & leaf plots)

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

numbers

Stemplots (stem & leaf plots)

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

numbers

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

Stat students? Why or why not?

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

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 pairs of shoes owned by AP Stat

students? Why or why not?

Example 2:

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?

Do examples on separate note paper

Example 3: 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. Greed

How to describe a graph

DotplotsStem & leaf plots

HistogramsBoxplots

1. Centerdiscuss where the middle of

the data fallsthree types of central

tendency–mean, median, & mode

2. Spreaddiscuss how spread out the data

isrefers to the variability of the

data–Range, standard deviation, IQR

3. Type of distributionrefers to the overall shape of

the distributionsymmetrical, uniform,

skewed, or bimodal

4. Unusual occurrencesoutliers - value that lies

away from the rest of the data

gapsclustersanything 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!

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

Example 4Height (inches)

Tally Height (Inches)

Tally

62 1 69 1063 3 70 1064 6 71 965 7 72 666 9 73 367 12 74 168 13 75 1

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

Ex. 4

Example 5 Notes

Relative Frequencies, Cumulative Frequencies, & Ogives

Review: Histograms Patterns

Center: for now, the value that divides the observations roughly in half

Spread (variability): the extent of the data from smallest to largest value

Shape: overall appearance of distribution Outlier (Unusual): an individual

observation that falls outside the overall pattern

Let’s practice….

Can you describe the shape of the following distributions?

symmetricalFr

eque

ncy

slightly skewed left (negative skew)

Freq

uenc

y

strongly skewed right (positive skew)

Freq

uenc

y

bimodalFr

eque

ncy

skewed left with outlierFr

eque

ncy

bimodalFr

eque

ncy

strongly skewed leftFr

eque

ncy

Guess Age

Guess the age of the person in the picture (do not discuss this with your partner, just write down your guess.)

After everyone has guessed, I will give you the actual ages.

Graph a scatter plot of Guess Age vs. Actual Age (Be sure to label the axes.)

A

B

C

D

E

F and G

H

COMPARE

A 27 B 38 C 8 D 115 E 22 F she’s 80, G he’s 90 H 19

Generate a scatterplot

What would you expect from these scatterplots?– If your guesses were accurate?– If you under estimated?– Over estimated?

Scatterplots

A scatterplot shows the relationship between two quantitative variables measured on the same individuals.

The explanatory variable, if there is one, is graphed on the x-axis.

Scatterplots reveal the direction, form, and strength.

Patterns

Direction: variables are either positively associated or negatively associated

Form: linear is preferred, but curves and clusters are significant

Strength: determined by how close the points in the scatterplot are linear

Note: A strong association does NOT indicate cause and effect!

Scatterplot

Plot the data.

Describe the relationship between the SAT Math and SAT Verbal scores

Math

Verbal

690 510720 610590 550760 690660 700660 630650 700710 730710 540800 720620 780

Time Plot (aka time-series plot)

This a plot of each observation against the time at which it was measured– Stock prices, sales figures,

other socio-economic data– Invaluable for identifying

trends– Y-variable, x-time when

observation made– Used to plot trends or

cycles

Life Expectancy

60

62

64

66

68

70

72

74

76

78

80

1940 1950 1960 1970 1980 1990

Life Expectancy

0

10

20

30

40

50

60

70

80

90

100

1940 1950 1960 1970 1980 1990