AP StatisticsOverview

What is Statistics?Statistics is the science of learning from data.

Ex. Take a sample of 50 seniors and record the number of AP classes they are taking. Use this to make a prediction, or educated

guess, about how many AP classes ALL seniors are taking.

Parameter – summary measurement (ex: p, µ) that describes the population

Statistic – summary measurement (ex: ) that describes the sample

AP Statistics – At a Glance

I. Exploring Data (Chapters 1 – 4) A. Create Distributions (graph of data)

B. Describe / Compare Distributions

II. Observational Studies and Experiments (Ch 5)

III. Anticipating Patterns (Chapters 6 – 9)

IV. Statistical Inference (Chapters 10 – 15)

The key to AP Stats:THINK—SHOW—TELL

Think first! Know where you’re headed and why. It will save you a lot of work.

Show is what most people think Statistics is about. The mechanics of calculating statistics and making displays is important, but not the most important part of Statistics.

Tell what you’ve learned. Until you’ve explained your results so that someone else can understand your conclusions, the job is not done.STAY FOCUSED!

WHO is being described? How many?Individuals are the objects described by a set of data. These individuals go by different names depending on the situation.

Respondents Individuals who answer a survey.

Subjects/Participants

People on who we experiment.

Experimental Units

Animals, plants, Web sites, and other inanimate subjects on which we experiment.

WHAT are the variables? Units?

Categorical Group of category names w/no order

Eye Color (brown, blue, green)

Quantitative Numerical values Weight (117lbs 170oz)

Variables – characteristics recorded about each individual

Univariate Data One Variable Final Exam Scores

Bivariate Data 2 Paired Variables Homework % vs. Final Exam Scores

Discrete Numbers have specific values

# of desks, money

Continuous Estimated numbers Time, height, age

CHAPTER 1Exploring

Data

Summarize Categorical Data using a

Bar Chart or Pie ChartAP Scores

1 2 3 4 5AP SCORES

30282624222018161412108642

FR

EQ

UE

NC

Y

Dotplot for Univariate Quantitative Data

0 20 40 60 80 100 120 140 160Temperature

paneldat Dot Plot

Stemplot for Quantitative DataAges of Death of U.S. First Ladies

3 | 4, 6

4 | 3

5 | 2, 4, 5, 7, 8

6 | 0, 0, 1, 2, 4, 4, 4, 5, 6, 9

7 | 0, 1, 3, 4, 6, 7, 8, 8

8 | 1, 1, 2, 3, 3, 6, 7, 8, 9, 9

9 | 7

StemLeaf

3 | 4 indicates 34 years old

Leaf – single digitDo not skip stemsLeafs – smallest to largest

Split Stemplot

1 | 7

1 | 8, 9, 9, 9, 9, 9

2 | 0, 0, 0, 0, 1, 1, 1, 1, 1, 1

2 | 2, 2, 2, 3, 3

2 | 4, 5

2 |

2 | 8

3 | 0, 1

Stem is split for every 2 leaves— (0, 1), (2, 3), (4, 5), (6, 7), and (8, 9)

Age of 27 students randomly selected from Stat 303 at A&M

Split Stemplot

1 |

1 | 7, 8, 9, 9, 9, 9, 9

2 | 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4

2 | 5, 8

3 | 0, 1

3 | Stem is split for every 5 leaves—(0 thru 4) AND ( 5 thru 9)

Age of 27 students randomly selected from Stat 303 at A&M

Back-to-back Stemplot

Babe Ruth Roger Maris

| 0 | 8

| 1 | 3, 4, 6

5, 2 | 2 | 3, 6, 8

5, 4 | 3 | 3, 9

9, 7, 6, 6, 6, 1, 1 | 4

9, 4, 4 | 5 |

0 | 6 | 1

Number of home runs in a season

• Frequency - # of times something occurs• Cumulative Frequency – keep adding• Relative Frequency – percents• Cumulative Relative Frequency – add

percents (AKA ogive) See graphs on page 62

Letter Grade

Frequency Cumulative Frequency

Relative Frequency

Cumulative Relative

Frequency

A

B

C

D

F

Co

un

t

20

40

60

80

100

120

age0 20 40 60 80 100

TX_betweenHoustonDallas Histogram

Histogram—Univariate Quantitative data

Univariate Variable Age

Frequency Count • Classes should be equal width• Reasonable width• Reasonable starting point• Roughly 7 bars• Bars should touch• This is not a bar graph!

HistogramsDiscrete vs. Continuous

ContinuousDiscrete

Location—pth Percentile

The pth percentile of a distribution (set of data) is the value such that p percent of the observations fall at or below it.

Suppose your Math SAT score is at the 80th percentile of all Math SAT scores. This means your score was higher than 80% of all other test takers.

5 Number SummaryMinimum, Q1, Median, Q3, Maximum

Q1 (Quartile 1) is the 25th percentile of ordered data or median of lower half of ordered data

Median (Q2) is 50th percentile of ordered data

Q3 (Quartile 3) is the 75th percentile of ordered data or median of upper half of ordered data

Range = Maximum – minimum

IQR = Interquartile Range (Q3 – Q1) middle 50%

Calculating OUTLIERS

“1.5IQR above Q3 or below Q1”

IQR(Interquartile Range) = Q3 – Q1

Any point that falls outside the interval calculated by

Q1- 1.5(IQR) and Q3 + 1.5(IQR)

is considered an outlier.

121, 132, 134, 154, 164, 175, 188, 192, 201, 203, 203

3, 4, 4, 5, 10, 12, 13, 24

Calculate the 5 Number Summary

Calculate the 5 Number Summary and Check for Outliers

Boxplot - Using 5 Number Summary

ThouComputers0 2000 4000 6000 8000

ComputerDensity Box Plot

5# Summary of Computers: 250, 1000, 2950, 5400, 8600

min

250

Q1

1000

median

2950

Q3

5400

Max

8600

Boxplot and Modified Boxplot

Age_Wife15 20 25 30 35 40 45 50 55 60 65

HusbandsAndWives Box Plot

Ht_Husband1550 1650 1750 1850 1950

HusbandsAndWives Box Plot

Modified – show outliers

25% of data in each section

se

x

FM

age0 10 20 30 40 50 60 70 80 90

TX_betweenHoustonDallas Box Plot

Comparative Parallel (Side by Side) Boxplots

Outliers

Mean or

Median?

Robust (Resistant) Statistic

Median is resistant to extreme values (outliers) in data set.

Mean is NOT robust against extreme values. Mean is pulled away from the center of the distribution toward the extreme value (“tails of graph”).

Of the 2 segments, where is the Mean with respect to the Median?

Remember the mean is pulled toward extreme values.

Where’s the Mean with respect to the Median?

Describing Spread: Standard DeviationRoughly speaking, standard deviation is the average distance values fall from the mean (center of graph).

Population and SampleStandard Deviation

2 population variance s2 sample variance

n

xi

2

1

2

n

xxs i

What is Variance???

What is Variance?

Variance = (Standard deviation)2

Calculated Standard Deviationis a measure of Variation in data

Sample Data Set Mean Standard Deviation

100, 100, 100, 100, 100 100 0

90, 90, 100, 110, 110 100 10

30, 90, 100, 110, 170 100 50

90, 90, 100, 110, 320 142 99.85

LET’S CUSS!

Center

Unusual Features

Spread

Shape

To describe a distribution:LET’S CUSS!

Center Mean, Median

Unusual Features Gaps, Outliers, Clusters

Spread Standard Deviation, Range, IQR

Shape Normal, Symmetric, Skewed Right (left)

CENTERMean(, ) —add up data values and divide by

number of data values

Median—list data values in order, locate middle data value

Data Set: 19, 20, 20, 21, 22

Mean is 04.205

2221202019

x

Median is 20 since it is the middle number of the ranked (ordered) data values.

x

Cluster---Gaps---Potential OutliersC

ou

nt

51015202530354045

Age_Husb_at_Marriage20 30 40 50 60

HusbandsAndWives Histogram

UNUSUAL FEATURES

SHAPE

Normal – bell-shaped Skewed Right

The shape can also be skewed left or symmetric or uniform.

“Tail” points to right

SPREAD

The spread can be described using: Standard Deviation (about 10) or

Range (80 – 150 or 70) orIQR (about 100 – 130)

Summary Features of Quantitative Variables

Center – Location

Unusual Features – Outliers, Gaps, Clusters

Spread – Variability

Shape – Distribution Pattern

How to Choose Measures of Center and Spread?

NON - SKEWED DISTRIBUTIONS – use mean and standard deviation

SKEWED DISTRIBUTIONS – use 5# Summary

Comparing Distributions

• CUSS • COMPARE in CONTEXT• GENERAL CONCLUSION

Linear Transformationsusing the height of all LHS Seniors (in inches)

What happens to center and spread if everyone is put in 3 inch heels (add 3 inches)?

What happens to the center and spread if we change everyone height to feet (divide by 12)?

Summary of Linear Transformations

Multiplying each observation by a positive number b multiplies both measures of center (mean and median) and measures of spread (IQR and standard deviation) by b.

Adding the same number a (either positive, negative, or zero) to each observation adds a to measures of center and to quartiles but does not change measures of spread.

NOTE: The shape NEVER changes!