Lecture PPT Section1-1

8/2/2019 Lecture PPT Section1-1

1/22

8/16/20

Variables

In a study, we collect informationdatafrom individuals. Individuals

can be people, animals, plants, or any object of interest.

A variable is any characteristic of an individual. A variable varies among

individuals.

Example: age, height, blood pressure, ethnicity, leaf length, first language

The distribution of a variable tells us what values the variable takes and

how often it takes these values.

Two types of variables Variables can be eitherquantitative

Something that takes numerical values for which arithmetic operations,

such as adding and averaging, make sense.

Example: How tall you are, your age, your blood cholesterol level, the

number of credit cards you own.

orcategorical.

Something that falls into one of several categories. What can be counted

is the count or proportion of individuals in each category.

Example: Your blood type (A, B, AB, O), your hair color, your ethnicity,

whether you paid income tax last tax year or not.


2/22

8/16/20

How do you know if a variable is categorical or quantitative?

Ask:

What are the n individuals/units in the sample (of size n)? What is being recorded about those n individuals/units?

Is that a number ( quantitative) or a statement ( categorical)?

Individuals

in sample

DIAGNOSIS AGE AT DEATH

Patient A Heart disease 56

Patient B Stroke 70

Patient C Stroke 75

Patient D Lung cancer 60

Patient E Heart disease 80

Patient F Accident 73

Patient G Diabetes 69

QuantitativeEach individual is

attributed a

numerical value.

CategoricalEach individual is

assigned to one of

several categories.

Ways to chart categorical dataBecause the variable is categorical, the data in the graph can be

ordered any way we want (alphabetical, by increasing value, by year,

by personal preference, etc.)

Bar graphs

Each category is

represented by

a bar.

Pie charts

The slices must

represent the parts of one whole.


3/22

8/16/20

Example: Top 10 causes of death in the United States 2001

Rank Causes of death Counts% of top

10s

% of total

deaths

1 Heart disease 700,142 37% 29%

2 Cancer 553,768 29% 23%

3 Cerebrovascular 163,538 9% 7%

4 Chronic respiratory 123,013 6% 5%

5 Accidents 101,537 5% 4%

6 Diabetes mellitus 71,372 4% 3%

7 Flu and pneumonia 62,034 3% 3%

8 Alzheimers disease 53,852 3% 2%

9 Kidney disorders 39,480 2% 2%

10 Septicemia 32,238 2% 1%

All other causes 629,967 26%

For each individual who died in the United States in 2001, we record what was

the cause of death. The table above is a summary of that information.

0

100

200

300

400

500

600

700

800

Heartdise

ases

Cancers

Cerebrovascular

Chronicrespira

tory

Accid

ents

Diabetes

mellitu

s

Flu&pneumonia

Alzheimer'sdisease

Kidney

diso

rders

Septice

mia

Counts(x1000)

Top 10 causes of deaths in the United States 2001

Bar graphs

Each category is represented by one bar. The bars height shows the count (or

sometimes the percentage) for that particular category.

The number of individuals

who died of an accident in

2001 is approximately

100,000.


4/22

8/16/20

0

100

200

300

400

500

600

700

800

Heartdise

ases

Cancers

Cerebrovascular

Chronicrespira

tory

Accidents

Diabetes

mellitu

s

Flu&pneumonia

Alzheimer'sdisease

Kidney

diso

rders

Septice

mia

Counts(x1000)

Bar graph sorted by rank

Easy to analyze

Top 10 causes of deaths in the United States 2001

0

100

200

300

400

500

600

700

800

Accidents

Alzheimer'sdisease

Cancers

Cerebrovascular

Chronicrespira

tory

Diabetes

mellitu

s

Flu&pneumonia

Heartdise

ases

Kidney

diso

rders

Septice

mia

Counts(x1000) Sorted alphabetically

Much less useful

Percent of people dying from

top 10 causes of death in the United States in 2000

Pie charts

Each slice represents a piece of one whole. The size of a slice depends on whatpercent of the whole this category represents.


5/22

8/16/20

Percent of deaths from top 10 causes

Percent of

deaths from

all causes

Make sure your

labels match

the data.

Make sure

all percents

add up to 100.

Child poverty before and after government

interventionUNICEF, 1996

What does this chart tell you?

The United States has the highest rate of child

poverty among developed nations (22% of under 18).

Its government does the leastthrough taxes and

subsidiesto remedy the problem (size of orange

bars and percent difference between orange/blue

bars).

Could you transform this bar graph to fit in 1 pie

chart? In two pie charts? Why?

The poverty line is defined as 50% of national median income.


6/22

8/16/20

Graphing Data with Excel Bar Graph

Graphing Data with Excel Pie Chart


7/22

8/16/20

Tire model for 2969 accidents that

involved Firestone tires Exercise:Display this

set of data

graphically

using a bar

graph and a

pie chart.

Exercise Solution Bar Graph


8/22

8/16/20

Exercise Solution Pie Chart

Using the TI-83/84 Bar Graph


9/22

8/16/20

Using the TI-83/84

Using the TI-83/84


10/22

8/16/20

Ways to chart quantitative data

Histograms and stemplots

These are summary graphs for a single variable. They are very useful to

understand the pattern of variability in the data.

Line graphs: time plots

Use when there is a meaningful sequence, like time. The line connecting

the points helps emphasize any change over time.

Histograms

The range of values that a

variable can take is divided

into equal size intervals.

The histogram shows the

number of individual data

points that fall in each

interval.

The first column represents all states with a Hispanic percent in their

population between 0% and 4.99%. The height of the column shows how

many states (27) have a percent in this range.

The last column represents all states with a Hispanic percent in their

population between 40% and 44.99%. There is only one such state: New

Mexico, at 42.1% Hispanics.


11/22

8/16/20

IQ Scores - Example

81 101 109 114 123 13182 102 110 115 124 133

89 102 110 116 124 134

90 102 110 117 124 134

94 103 112 117 125 136

96 105 112 117 126 137

97 106 113 118 127 139

100 108 113 118 127 139

101 109 114 122 128 142

101 109 114 122 130 145

IQ Scores Example - HistogramInterval Frequency

85 2

85 - 95 3

95 - 105 11

105 - 115 16

115 - 125 13

125 - 135 9

> 135 6


12/22

8/16/20

How to Create a Histogram - Excel

1. Select Histogram & click OK

2. Input data range

3. Check Labels (if applicable)

4. Check Chart Output

5. Select New Worksheet Ply (default)

6. Click OK

How to Create a Histogram - ExcelExcel creates a frequency table by

dividing the data range into intervals

of equal width automatically and charts the data.

Intervals:

[ x 81], [81< x 90.14286], [90.14286 < x 99.28571], , [ x > 135.8571]


13/22

8/16/20

How to Create a Histogram Excel

Excel creates a bar graph (with gaps between intervals). Right-click onany of the bars and select Format Data Point and move the Gap

Width slider to No Gap (0%)

How to Create a Histogram ExcelYou can instruct Excel how to divide the data into equal intervals by

entering your own bin ranges to a column and then entering it into the

Bin Range in the Histogram dialog window.


14/22

8/16/20

How to Create a Histogram TI -83/84



15/22

8/16/20


Stem plots

How to make a stemplot:

1) Separate each observation into a stem, consisting of

all but the final (rightmost) digit, and a leaf, which is

that remaining final digit. Stems may have as many

digits as needed, but each leaf contains only a single

digit.

2) Write the stems in a vertical column with the smallest

value at the top, and draw a vertical line at the right

of this column.

3) Write each leaf in the row to the right of its stem, in

increasing order out from the stem.

STEM LEAVES


16/22

8/16/20

State Percent

Ala ba m a 1.5

Ala sk a 4.1

Ar izo na 25 .3

Ar ka ns as 2.8

Ca lifornia 32.4

Co lo rado 17.1

Connectic ut 9.4

De la w are 4.8

Florida 16.8

Georgia 5.3Haw ai i 7.2

Idaho 7.9

Illino is 10.7

Ind ian a 3.5

Iow a 2.8

Ka nsas 7

Ke ntuck y 1.5

Lou is iana 2.4

M aine 0.7

M aryland 4.3

M as sac hu se tt 6.8

M ic higan 3.3

M innes ota 2.9

M is s is s ipp i 1.3

M is souri 2.1

M ontana 2

Nebrask a 5.5

Nevada 19.7

New Ha mp sh ir 1.7

New Jersey 13.3

New Mexico 42.1

New York 15.1

N orth Ca ro lin a 4 .7

NorthDako ta 1.2

Ohio 1.9

Okla hom a 5.2

Oregon 8

Pe nns ylvan ia 3.2

Rhode Is land 8.7S ou th Ca ro lin a 2 .4

So uthDa ko ta 1.4

Tennes see 2

Texas 32

Utah 9

Ve rm on t 0.9

Virginia 4.7

W ash ing ton 7.2

W e stV irgin ia 0.7

W iscons in 3.6

W yo m ing 6.4

Percent of Hispanic residents

in each of the 50 states

Step 2:

Assign the

values to

stems and

leaves

Step 1:

Sort the

data

S tate Pe rc ent

M a in e 0 .7

W e stV irg in ia 0 .7

V erm o n t 0 .9

N orthD ak o ta 1 .2

M is s is s ip p i 1 .3

S ou th D ako ta 1 .4

Al ab am a 1. 5

K en tu c ky 1 .5

N ew H am psh ir 1 .7

O h io 1 .9M o n ta n a 2

Tenne s s ee 2

M is sou ri 2 .1

L ou is ian a 2 .4

S ou th Ca ro lin a 2 .4

Ar ka ns as 2. 8

Io w a 2 .8

M inn e so ta 2 .9

P en ns ylva nia 3 .2

M ic h igan 3 .3

In d iana 3 .5

W is c ons in 3 .6

Al as ka 4. 1

M a ryla nd 4 .3

N orth Ca ro lin a 4 .7

V irg in ia 4 .7

D e law a re 4 .8

O k lah o m a 5 .2

G eo rg ia 5 .3

N ebras k a 5 .5

W yom ing 6 .4

M a ss ach us et t 6 .8

K an s as 7

H aw a ii 7 .2

W a sh ing ton 7 .2

Id aho 7 .9

O re go n 8

R hod e Is la nd 8 .7

U tah 9C onn ec tic u t 9 .4

Ill ino is 10 .7

N ew J ers e y 13 .3

N ew Yo rk 15 .1

F lo rida 16 .8

C o lo rado 1 7 .1

N evada 19 .7

Ar izo na 25 .3

Texas 32

C a lifo rn ia 3 2 .4

N ew M exic o 4 2 .1

Stem Plot

To compare two related distributions, a back-to-back stem plot with

common stems is useful.

Stem plots do not work well for large datasets.

When the observed values have too many digits, trim the numbers

before making a stem plot.

When plotting a moderate number of observations, you can split

each stem.


17/22

8/16/20

Stemplots are quick and dirty histograms that can easily be done byhand, and therefore are very convenient for back of the envelope

calculations. However, they are rarely found in scientific or laymen

publications.

Stemplots versus histograms

Interpreting histogramsWhen describing the distribution of a quantitative variable, we look for the

overall pattern and for striking deviations from that pattern. We can describe

the overallpattern of a histogram by its shape, center, and spread.

Histogram with a line connecting

each column too detailed

Histogram with a smoothed curve

highlighting the overall pattern of

the distribution


18/22

8/16/20

Most common distribution shapes

A distribution is symmetric if the right and left

sides of the histogram are approximately mirror

images of each other.

Symmetric

distribution

Complex,

multimodal

distribution

Not all distributions have a simple overall shape,

especially when there are few observations.

Skewed

distribution

A distribution is skewed to the right if the right

side of the histogram (side with larger values)

extends much farther out than the left side. It is

skewed to the left if the left side of the histogram

extends much farther out than the right side.

Alaska Florida

Outliers

An important kind of deviation is an outlier. Outliers are observations

that lie outside the overall pattern of a distribution. Always look for

outliers and try to explain them.

The overall pattern is fairly

symmetrical except for 2

states that clearly do not

belong to the main trend.

Alaska and Florida have

unusual representation of

the elderly in theirpopulation.

A large gap in the

distribution is typically a

sign of an outlier.


19/22

8/16/20

2

How to create a histogram

It is an iterative process try and try again.

What bin size should you use?

Not too many bins with either 0 or 1 counts

Not overly summarized that you loose all the information

Not so detailed that it is no longer summary

rule of thumb: start with 5 to 10 bins

Look at the distribution and refine your bins

(There isnt a unique or perfect solution)

Not

summarized

enough

Too summarized

Same data set


20/22

8/16/20

2

IMPORTANT NOTE:

Your data are the way they are.

Do not try to force them into a

particular shape.

It is a common misconception

that if you have a large enough

data set, the data will eventuallyturn out nice and symmetrical.

Histogram of Drydays in 1995

Line graphs: time plots

A trend is a rise or fall that

persists over time, despite

small irregularities.

In a time plot, time always goes on the horizontal, x axis.

We describe time series by looking for an overall pattern and for striking

deviations from that pattern. In a time series:

A pattern that repeats itself

at regular intervals of time is

called seasonal variation.


21/22

8/16/20

2

Retail price of fresh oranges

over time

This time plot shows a regular pattern of yearly variations. These are seasonal

variations in fresh orange pricing most likely due to similar seasonal variations in

the production of fresh oranges.

There is also an overall upward trend in pricing over time. It could simply be

reflecting inflation trends or a more fundamental change in this industry.

Time is on the horizontal, x axis.

The variable of interesthere

retail price of fresh oranges

goes on the vertical, y axis.

1918 influenza epidemic

Date # Cases # Deaths

week 1 36 0

week 2 531 0

week 3 4233 130

week 4 8682 552

week 5 7164 738

week 6 2229 414

week 7 600 198

week 8 164 90

week 9 57 56

week 10 722 50

week 11 1517 71week 12 1828 137

week 13 1539 178

week 14 2416 194week 15 3148 290week 16 3465 310

week 17 1440 149

01000

2000300040005000

600070008000

900010000

we

ek1

we

ek3

we

ek5

we

ek7

we

ek9

wee

k11

wee

k13

wee

k15

wee

k17

#casesdiagnosed

0

10 0

20 0

30 0

40 0

50 0

60 0

70 0

80 0

#

deathsreported

# C as es # Deaths

A time plot can be used to compare two or more

data sets covering the same time period.

The pattern over time for the number of flu diagnoses closely resembles that for the

number of deaths from the flu, indicating that about 8% to 10% of the people

diagnosed that year died shortly afterward, from complications of the flu.


22/22

8/16/20

Death rates from cancer (US, 1945-95)

0

50

100

150

200

250

1 940 1950 1960 1970 1980 1990 20 00

Years

Deathrate(per

thousand)


0

50

100

150

200

250

1940 1960 1980 2000

Years

Deathrate

(perthousa

nd)


0

50

100

150

200

250

1940 1960 1980 2000

Years

Deathrate(perthousand)

A picture is worth a

thousand words,

BUT

There is nothing like

hard numbers.

Look at the scales.

Scales matter

How you stretch the axes and choose your

scales can give a different impression.


120

140

160

180

200

220

1940 1960 1980 2000

Years

Deathrate(per

thousand)

Lesson SummaryKey Concepts Individuals

Variables

o Categorical Variables

o Quantitative Variables

Graphs for Categorical Variables

o Bar graphs

o Pie charts

Graphs for Quantitative Variables

o Stemplots

o Histograms

o Timeplots

Examining distributions

o Overall pattern (Shape, center,

spread)

o Deviations from the pattern (Outliers)

o Trends

o Seasonal Variation

Skills Learned

Displaying data graphically

o Bar graphs

o Pie charts

o Stemplots

o Histograms

o Timeplots

Interpreting graphs

Describing distributions

Date post:	05-Apr-2018
Category:	Documents
Upload:	engr-shahbaz-siddiqui
View:	223 times
Download:	0 times

Lecture PPT Section1-1

Documents