statistics is the science of: collecting information...

What is statistics?

Statistics is the science of:

Collecting information

Organizing and summarizing the information collected

Analyzing the information collected in order to draw conclusions

Two types of Statistics

Descriptive Statistics

Organizing and summarizing the

information collected.

Inferential Statistics

Draws conclusion from the information

collected.

Chapter 1

Exploring Data

Lesson 1-1, Displaying

Distributions with Graphs

Bar Graphs and Pie Charts

Data

Individuals

are objects described by a set of data. Individuals may be people, animals or things.

Variable

is any characteristic of an individual. A variable can take different values for different individuals

Types of Variables

Categorical variable

allows for classification of individuals

based on some attribute or

characteristics.

Quantitative variable

provides numerical measures of

individuals.

Example, , #1.2

Data from a medical study contain values of many

variables for each of the people who where subjects

of the study. Which of the following variables are

categorical and which are quantitative?

Example, Page 7, #1.2

a) Gender (female or male)

categorical

b) Age (years)

Quantitative

c) Race (Asian, black, white or other)

categorical

d. Smoker (yes or no)

categorical

e) Systolic blood pressure (millimeters of mercury)

Quantitative

f) Level of calcium in blood (micrograms per milliliter)

Quantitative

Distribution

Distribution Tells us what values the variable takes

and how often it takes each value

Displaying Distributions

Categorical Variables

Bar Graphs

Pie Charts

Quantitative Variables

Dotplots

Stemplots

Histograms

Example – , #1.6

In 1997 there were 92,353 deaths from accidents in the

United States. Among these were 42.340 deaths from

Motor vehicle accidents, 11,858 from falls, 10,163 from

poisoning, 4051 from drowning, and 3601 from fires.

A) Find the percent of accidental deaths from each of

these causes, rounded to the nearest percent. What

percent of accidental deaths were due to other causes?

Example – Page 11, #1.6

Accidents Number Percentage

Motor Vehicle 42,340

Falls 11,858

Poisoning 10,163

Drowning 4051

Fires 3601

Other Causes 20,340

Total 92,353

42,34045.8 46%

92,353

13%

11%

4%

4%

22%

100%


STAT


B) Make a well-labeled bar graph of the distribution of

causes of accidental deaths. Be sure to include an

“other causes” bar.


Causes of Accidental Deaths

Pe

rce

nta

ge

of A

ccid

en

tal D

ea

ths US Accidental Death – 1997

50

40

10

20

30

MV Falls Drown FiresPoison OC


C) Would it also be correct to use a pie chart to display

these data? If so, construct the pie chart. If not

explain why not.

Yes, since categories represent parts of a whole.


Accidents Number Percentage

MV 42,340

Falls 11,858

Poisoning 10,163

Drowning 4051

Fires 3601

OC 20,340

Total 92,353

46%

13%

11%

4%

4%

22%

100%

Pie Chart

0.46 360

165.6 166

47

40

14

14

79

360°


US Accidental Deaths - 1997

46%

13%

11%

4%

4%

22% Motor Vehicle

Falls

Poisoning

Drowning

Fires

Other Causes

Lesson 1-1, Displaying


Dot Plots and Stem Leaf Plots

Overall Pattern of Distribution

(Quantitative Variables)

Center Divides the data in half

Spread Smallest to largest values

Shape Skewness of the data

Outlier Data that falls outside of the pattern


Are you driving a gas guzzler? Table 1.3 displays the highway

gas mileage for 32 model year 2000 midsize cars.

A). Make a dot plot of these data.


21 23 25 27 29 31 33

Highway Gas Mileage

21 23 25


B) Describe the shape, center, and spread of the distribution

of gas mileages. Are there any potential outliers?

The shape of the distribution is skewed to the left, with a

major peak at 28 and a minor peak at 24. The spread is

relatively narrow (21 to 32 mpg). The two observations at

21 and the observation at 32 appear to outliers. The center

is 28 mpg.


In 1978 the English scientist Henry Cavendish measured

the density of the earth by careful work with a torsion balance.

The variable recorded was the density of the earth as a

multiple of the density water. Here are Cavendish’s 29

measurements:

5.50 5.61 4.88 5.07 5.26 5.55 5.36 5.29 5.58 5.65

5.57 5.53 5.62 5.29 5.44 5.34 5.79 5.10 5.27 5.39

5.42 5.47 5.63 5.34 5.46 5.30 5.75 5.68 5.85


5.50 5.61 4.88 5.07 5.26 5.55 5.36 5.29 5.58 5.65

5.57 5.53 5.62 5.29 5.44 5.34 5.79 5.10 5.27 5.39

5.42 5.47 5.63 5.34 5.46 5.30 5.75 5.68 5.85

Present these measurements graphically in a stemplot.

Discuss the shape, center, and spread of the distribution.

Are there any outliers? What is estimate of the density of

the earth based on these measurements?


48 8

49

50 7

51 0

52 6 7 9 9

53 0 4 4 6 9

54 2 4 6 7

55 0 3 5 7 8

56 1 2 3 5 8

57 5 9

58 5

48|8 = 4.88%

The shape of the distribution is

roughly symmetric with one

possible outlier at 4.88 that is

somewhat low. The spread

between 4.88 to 5.85. The

center of the distribution if

between 5.4 and 5.5. Based on

the plot, we would estimate the

Earth’s density to be about

halfway between 5.4 and 5.5.

Density of the Earth

Lesson 1-1 Displaying


Histograms and Relative

Frequency Graphs

Histogram and categories

18 23 28

0

10

20

30

40

50

60

Age (in years)

Fre

quency (

Count)

Age of Spring 1998 Stat 250 Students

n=92 students

2 3 4

0

1

2

3

4

5

6

7

GPAF

req

uency (

Co

unt)

GPAs of Spring 1998 Stat 250 Students

n=92 students

too few categories too many categories

Example – Histogram

Suppose you are considering investing in a Roth IRA.

You collect the data table, which represent the three-year

rate of return (in percent) for 40 small capitalization growth

mutual funds.

27.4 12.7 22.6 32.1 18.2 23.7 18.4 14.7

16.7 28.5 29.6 47.7 32.0 14.7 21.3 37.0

10.8 22.2 11.6 10.9 25.5 12.8 27.0 19.2

24.1 18.4 45.9 18.4 23.7 31.1 19.6 18.5

35.9 17.4 16.6 23.3 38.1 21.9 18.5 29.1


STAT


A) Construct a histogram to display these data. Record

your class intervals and counts

Step 1 – Find the class intervals

Locate the smallest number (10.8) and the largest

number (47.7)

Lower class limit will be 10.0 with a class width of 5


3-yr Rate of Return Frequency

Total

10.00

15.0

14.9

20.0

25.0

30.0

35.0

40.0

45.0

19.9

24.9

29.9

34.9

39.9

44.9

49.9

7

11

8

6

3

3

0

2

40


Stat Plot

2nd Y=

Step – 2 Graph it using the TI

Window


Graph Trace

Example - Histogram

10 15 20 25 30 35 40 45 50

4

8

12

Rate of Return

Fre

qu

ency

3 – Year Rate of Return of Mutual Funds


B) Describe the distribution of 3 – Year Rate of Return.

The shape of the distribution is

skewed to the right with the

center at class 15.0% – 19.9%.

There is one outlier in class the

45.0% – 49.9%. The spread is

between 10% to 50%.

Shape of a Distribution

Uniform (symmetric)

Bell-shaped (Symmetric)

Skewed Right

Skewed Left

Uniform Distribution

Symmetric – Bell Shaped

Skewed Right

Skewed left

Example – Relative Cumulative Frequency

Suppose you are considering investing in a Roth IRA.

You collect the data table, which represent the three-year

rate of return (in percent) for 40 small capitalization growth

mutual funds.

27.4 12.7 22.6 32.1 18.2 23.7 18.4 14.7

16.7 28.5 29.6 47.7 32.0 14.7 21.3 37.0

10.8 22.2 11.6 10.9 25.5 12.8 27.0 19.2

24.1 18.4 45.9 18.4 23.7 31.1 19.6 18.5

35.9 17.4 16.6 23.3 38.1 21.9 18.5 29.1


Class Freq Relative

Frequency

Cumulative

Frequency

Relative cumulative

Frequency

10.0 – 14.9 7

15.0 – 19.9 11

20.0 – 24.9 8

25.0 – 29.9 6

30.0 – 34.9 3

35.0 – 39.9 3

40.0 – 44.9 0

45.0 – 49.9 2

Total 40 1

70.175

40

0.20

0.275

0.15

0.075

0.075

0

0.05

7

7 111 8

18 28 6

32

35

38

38

40

0.175

0.2750.175 0.45

0.20.4 655 0.

0.8

0.875

0.95

0.95

1


Class Freq Rel Freq Cum Freq Rel Cum Freq

20.0 – 24.9 8 0.2 26 0.65

45.0 – 49.9 2 0.05 40 1

26 of the 40 mutual funds had a 3 year rate of

return of 24.9% or less

65% of the mutual funds had 3 year rate of return of

24.9% or less

A mutual fund with a 3 year rate of return of 45% or

higher is out performing 95% of its peers.


L3 – Upper Class Limits

L4 – Relative Cumulative Frequency


3 Year Rate of Return for Small Capitalization

Mutal Funds

0

0.2

0.40.6

0.8

1

1.2

10 14.9 19.9 24.9 29.9 34.9 39.9 44.9 49.9

Rate of Return

Cu

mu

lati

ve

Rela

tive F

req

uen

cy

Lesson 1-2 Describing

Distributions with Numbers

Measuring the center

Mean

1 2 ... nx x x

Xn

To find the sample mean add up all of the observations and

divided by the number of observations.

x

Xn

Is affected by unusual values called outliers.

Median

Another name for the 50th percentile

Is not affected by unusual values called outliers

The median is the midpoint of a distribution, such

that half the observation are smaller and the other

half are larger.

Center and Distribution

Mean < Median

Skewed Left

Mean = Median

Symmetric

Mean > Median

Skewed Right

Measuring the Spread

Range

Quartiles

Boxplots

Standard Deviation

Variance

Range

The range is the difference between the largest

and smallest observation.

max minR x x

Quartiles

1Q 2Q

Quartiles divides the observation into fourths, or four equal

parts.

Smallest

Data Value

Largest

Data Value3Q

25% of

the data

25% of

the data

25% of

the data

25% of

the data

Interquartile Range (IQR)

The interquartile range (IQR) is the distance between

the first and third quartiles

3 1IQR Q Q

Outliers

3 1.5( )Q IQR

Upper Cutoff

Lower Cutoff

1 1.5( )Q IQR

Five Number Summary

Smallest observation (minimum)

Quartile 1

Quartile 2 (median)

Quartile 3

Largest observation (maximum)


The Survey of Study Habits and Attitudes (SSHA) is a

Psychological test that evaluates college students’

Motivation, study habits and attitudes toward school.

A private college gives the SSHA to a sample of 18 of

Its incoming first-year women students. There scores are

154 109 137 115 152 140 154 178 101

103 126 126 137 165 165 129 200 148


A) Make a stemplot of these data. The overall shape of the

distribution is irregular, as often happens when only a few

observations are available. Are there any potential

outliers? About where is the center of the distribution (the

score with half the scores above it and half below)?

What is the spread of the scores (ignoring any outliers)?

STATEDIT1:edit


10 1 3 9

11 5

12 6 6 9

13 7 7

14 0 8

15 2 4 4

16 5 5

17 8

18

19

20 0

200 is a potential outlier. The center

Is approximately 140. The spread

(excluding 200) is 178 – 101 = 77.


154 109 137 115 152 140 154 178 101

103 126 126 137 165 165 129 200 148


B) Find the mean.

C) Find the median of these scores. Which larger: the

median or the mean? Explain why.

141.058x

138.5Median

The mean is larger than the median because the outlier

at 200, which pulls the mean towards the long right

tail of the distribution.


Here are the scores on the Survey of Study Habits and

Attitudes (SSHA) for 18 first-year college women:

and for 20 first-year college men:

A) Make side-by side boxplots to compare the distribution.

154 109 137 115 152 140 154 178 101 103 126 126 137 165 165 129 200 148

108 140 114 91 180 115 126 92 169 146 109 132 75 88 113 151 70 115 187 104


Men

Wo

men

0 40 80 120 160 200

SSHA SCORES Box Plot


B) Compute the numerical summaries for these two

distributions.

Min Q1 Median Q3 Max

Women 141.06 101 126 138.5 154 200

Men 121.25 70 98 114.5 143 187

x


C) Write a paragraph comparing SSHA scores for men and

women.

All the displays and descriptions reveal that women

generally score higher than men. The men’s scores

(IQR = 45) are more spread out than the women’s

(even if we don’t ignore the outlier). The shapes of the

distributions are reasonable similar, with each

displaying right skewness.

Describing Distributions with

Numbers

Standard Deviation and

Variance

Standard Deviation

The standard deviation (s) measures the average distance

of observations from their mean.


The level of various substances in the blood influence

our health. Here are measurements of the level of

phosphate in the blood of a patient, in milligrams

of phosphate per deciliter of blood, made on 6

consecutive visits to a clinic.

5.6 5.2 4.6 4.9 5.7 6.4


5.6 5.2 4.6 4.9 5.7 6.4

A. Find the mean.

5.6 5.2 4.6 4.9 5.7 6.4 32.45.4

6 6x


Observation Deviations Square Deviations

5.6

5.2

4.6

4.9

5.7

6.4

ixix x

2

ix x

5.6 5.4 0.2

5.2 5.4 0.2

4.6 5.4 0.8

4.9 5.4 0.5

5.7 5.4 0.3

6.4 5.4 1

0


5.04.5 5.5 6.56.0

5.4x 4.6x 6.4x

0.8 1


Observation Deviations Square Deviations

5.6

5.2

4.6

4.9

5.7

6.4

ixix x

2

ix x

5.6 5.4 0.2

5.2 5.4 0.2

4.6 5.4 0.8

4.9 5.4 0.5

5.7 5.4 0.3

6.4 5.4 1

0SUM

2(0.2) 0.04

0.04

0.64

0.25

0.09

1

2.06SUM


22 1

1is x x

n

B) Find the standard deviation (s) from its definition.

1 1

2.06 2.06 0.4126 1 5

0.412 0.64187 0.6419 2s s


C) Use your TI-83 to find and Do the result agree with

part B.

x .s

STAT

Standard Deviation

Standard deviation (s) is the square root of the

variance (s² )

Units are the original units

Measures spread about the mean and should only

be used when the mean is chosen as the center

If s = 0 then there is no spread. Observations are

the same value

As s gets larger the observations are more spread

out.

Highly affected by outliers. Best for symmetric data

Variance

Variance (s²) measures the average

squared deviation of observations from the

mean

Units are squared

Highly affected by outliers.

How to Choose?

Skewed Distribution or Outliers

Five number summary

Symmetric Distribution or No Outliers

Mean

Standard Deviation

Homework

HW, page 52, #1.41, 1.43

Read pages 53 – 61

Linear Transformation

A linear transformation changes the original variable

x into the new variable xnew given by an equation of the

form

newx a bx

Adding the constant a shifts all values of x upward or

downward by the same amount.

Multiplying by the positive constant b changes the size

of the unit of measurement.


Maria measures the lengths of 5 cockroaches that

she finds at school. Here are her results in inches

1.4 2.2 1.1 1.6 1.2

A. Find the mean and standard deviation.


1.4 2.2 1.1 1.6 1.2


B) Maria’s science teacher is furious to discover that

she has measured the cockroaches lengths in inches

rather than centimeters. (There are 2.54 cm in 1 inch).

Find the mean and standard deviation of the 5

cockroaches in centimeters.

1.5

1.5(2.54)

3.81

x

cm

0.436

0.436(2.54)

1.017

s

cm


C) Considering the 5 cockroaches that Maria found

as a small sample from the population of all

cockroaches at her school, what would you

estimate as the average length of the population

of cockroaches? How sure of your estimate

are you?

The average cockroach length can be estimate

as the mean length of the 5 sampled cockroaches

of 1.5 inches. This is a questionable estimate,

because the sample is so small.


A change of units that multiplies each unit by b, such

as change from inches x to centimeters

xnew, multiplies our usual measures of spread by b. This

is true of the IQR and standard deviation. What happens

to the variance when we change units this way?

0 2.54newx x

Variance is changed by a factor of 2.54² = 6.4516

Homework

HW, Page 56, #1.45

HW, Page 63, #1.55

1-2 Describing Distributions

with Numbers.

Comparing Distributions


The table below gives the distribution grades earned

by students taking the Calculus AB and Statistics

exam in 2000.

5 4 3 2 1

16.8% 23.2% 23.5% 19.6% 16.8%

9.8% 21.5% 22.4% 20.5% 25.8%

Calculus

Statistics

A. Make a graphical display to compare the AP exam

grades for Calculus AB and Statistics.

2000 AP Exam

0.0

5.0

10.0

15.0

20.0

25.0

30.0

1 2 3 4 5

Grade on Exam

% o

f stu

den

ts E

arn

ing

Gra

de

Calculus AB

Statistics



B) Write a few sentences comparing the two

distributions of exam grade. Do you know which

now know which exam is easier? Why or why not?

The distributions are very similar for grades 2, 3, and 4.

The major difference occurs for grades 1 and 5. With a

larger proportion of Statistics students receiving a grade

of 1 and a smaller proportion of Statistics student receiving

a grade of 5.

This suggest that the Statistics exam is harder in the

sense that students are more likely to get a poor grade

on the Statistics Exam than on the Calculus AB exam.

Example – Page 63, 1.54

x

The mean and standard deviation s measure the center

and spread but are not a complete description of a

distribution. Data sets with different shapes can have

the same mean and standard deviation. To demonstrate

this fact, use your calculator to find and s for the

following to small data sets. Then make a stem plot

of each and comment on the shape of each distribution

x

Data A 9.14 8.14 8.74 8.77 9.26 8.10 6.13 3.10 9.13 7.26 4.74

Data B 6.58 5.76 7.71 8.84 8.47 7.04 5.25 5.56 7.91 6.89 12.50


Set A Set B


3 1

4 7

5

6 1

7 2

8 1 1 7 7

9 1 1 2

Set A

3|1 = 3.1

5 2 5 7

6 5 8

7 0 7 9

8 4 8

9

10

11

12 5

Set B


The means and standard are basically the same. Set A is

skewed to the left, while Set B has a higher outlier.

Homework

HW, Page 59, #1.47, #1.49

HW, Page 62, #1.51, 1,57

statistics is the science of: collecting information...

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