Looking at data: distributions- Displaying distributions with graphsIPS section 1.1

© 2006 W.H. Freeman and Company (authored by Brigitte Baldi, University of California-Irvine; adapted by Jim Brumbaugh-Smith, Manchester College)

ObjectivesDisplaying distributions with graphs

Recognize numerical vs. categorical data

Construct graphs representing a distribution of numerical data Histograms

Stemplots

Describe overall patterns in numerical data

Identify exceptions to overall patterns

Discuss pros and cons of histograms vs. stemplots

Terminology Individual (or “observation”)

Variable numerical (or “quantitative”)

categorical (or “qualitative)

Value

Frequency absolute

relative

Frequency table

Distribution

Terminology (cont’d) Graphs for quantitative data

Histogram

Stemplot (or “stem-and-leaf diagram”)

Boxplot (section 1.2)

Symmetric

Skewed right (or “positively skewed”)

Skewed left (or “negatively skewed”)

Peak (or “mode”)

Unimodal vs. Bimodal

Outlier

Variables

In a study, we collect data from individuals, more formally known as

observations. Observations can be people, animals, plants, or any

object or process of interest.

A variable is a characteristic that varies among individuals in a

population or in a sample (a subset of a population).

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 on these values. A distribution can also be

thought of as the pattern of variation seen in the data.

Two types of variables Variables can be either numerical (a/k/a quantitative) …

Something that can be counted or measured for each individual and then

added, subtracted, averaged, etc. across individuals in the population.

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

number of credit cards you own

… or categorical (a/k/a qualitative).

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

computed 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

How do you know if a variable is categorical or quantitative?Ask: What are the individuals in the sample? What is being recorded about those individuals? Is that a number (“quantitative”) or a statement (“categorical”)?

Individualsin 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 graphsEach category isrepresented by a bar.

Pie chartsPeculiarity: The slices must represent the parts of one whole.

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 Alzheimer’s 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.

Top 10 causes of deaths in the United States 2001

Bar graphsEach category is represented by one bar. The bar’s 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.

0100200300400500600700800

Counts

(x1000)

Bar graph sorted by rank Easy to analyze

Top 10 causes of deaths in the United States 2001

0100200300400500600700800

Cou

nts

(x10

00)

Sorted alphabetically Much less useful

Percent of people dying fromtop 10 causes of death in the United States in 2000

Pie chartsEach slice represents a piece of one whole. The size of a slice depends on what

percent of the whole this category represents.

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 intervention—UNICEF, 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 least—through taxes and

subsidies—to 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.

Histograms

Vertical bar chart where horizontal axis is a numerical scale corresponding to the data. Vertical axis represents frequency (how many) or relative frequency (what proportion) “Peaks” correspond to commonly occurring data values. “Valleys” and “tails” correspond to values which do not occur as frequently.

Most states have between 0 and 10 percent Hispanic residents

A very small number have between 25 and 45 percent.

Histograms

The range of values that a variable can take is divided into equal size intervals.

The histogram shows the number (i.e., frequency) of individual data points that fall in each interval.

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

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

many states (27) have the percent Hispanic residents in this range.

The last column represents all states with a percent Hispanic between 40%

and 44.99%. There is only one such state: New Mexico, at 42.1% Hispanics.

Creating a histogram

What “class size” should you use?

Use an appropriate number of classes − usually between 5 and 15 work well, depending on number of data values being represented.

Either too few or too many classes will obscure the pattern in the data.

Not so detailed that it is no longer summary

Avoid using many classes having frequency of only 0 or 1

Not overly summarized so that you lose all the information

rule of thumb: start with 5 to 10 classes

Look at the distribution and refine your classes.

(There isn’t a unique or “perfect” histogram.)

Guidelines for histograms

Label the horizontal axis with a consistent numerical scale. (Don’t leave any gaps in the scale or compress the scale toward the left or right side of the graph)

Use vertical bars of equal width.

Label the horizontal scale on the class boundaries.

An exception is when each bar corresponds to a single whole number. Then it is reasonable to label each bar at its midpoint.

Not summarized enough

Too summarized

Same data set

Interpreting histograms

When describing the distribution of a quantitative variable, we look for the

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

overall pattern 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

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 right 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 extends

much farther out than the right.

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 eventually

turn out nice and symmetrical.

Histogram of Drydays in 1995

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 two

states clearly not belonging

to the main trend.

Alaska and Florida have

unusual representation of

the elderly in their

population.

A large gap in the

distribution is typically a

sign of an outlier.

More on Outliers

If outlier is incorrect data (e.g., data entry error, false response) Correct it if possible. Discard if absolutely sure it is wrong. Discarding data might introduce a bias if uncorrectable errors

tend to be mainly high (or mainly low) values.

If outlier is correct data consider effect on analysis Which statistical technique is most appropriate? How are conclusions affected by the outlier?

Some Other Strategies: Refine population definition if unusual responses are not part

of intended study group. If sample size is quite small collect more data to see if any

gaps fill in.

Stemplots (or “stem-and-leaf” diagrams)How to make a stemplot:

1) Truncate (or “trim”) the data to appropriate level of

accuracy.

2) Separate each observation into a stem, consisting of

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

the remaining final digit. Stems may have as many

digits as needed, but each leaf contains a single digit.

3) Write the stems in a column with the smallest value

at the top, and draw a vertical line at the right of this

column.

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

increasing order out from the stem. Leaves should

be aligned in vertical columns. (Why?)

STEM LEAVES

State PercentAlabama 1.5Alaska 4.1Arizona 25.3Arkansas 2.8California 32.4Colorado 17.1Connecticut 9.4Delaware 4.8Florida 16.8Georgia 5.3Hawaii 7.2Idaho 7.9Illinois 10.7Indiana 3.5Iowa 2.8Kansas 7Kentucky 1.5Louisiana 2.4Maine 0.7Maryland 4.3Massachusetts 6.8Michigan 3.3Minnesota 2.9Mississippi 1.3Missouri 2.1Montana 2Nebraska 5.5Nevada 19.7NewHampshire 1.7NewJ ersey 13.3NewMexico 42.1NewYork 15.1NorthCarolina 4.7NorthDakota 1.2Ohio 1.9Oklahoma 5.2Oregon 8Pennsylvania 3.2RhodeIsland 8.7SouthCarolina 2.4SouthDakota 1.4Tennessee 2Texas 32Utah 9Vermont 0.9Virginia 4.7Washington 7.2WestVirginia 0.7Wisconsin 3.6Wyoming 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

State PercentMaine 0.7WestVirginia 0.7Vermont 0.9NorthDakota 1.2Mississippi 1.3SouthDakota 1.4Alabama 1.5Kentucky 1.5NewHampshire 1.7Ohio 1.9Montana 2Tennessee 2Missouri 2.1Louisiana 2.4SouthCarolina 2.4Arkansas 2.8Iowa 2.8Minnesota 2.9Pennsylvania 3.2Michigan 3.3Indiana 3.5Wisconsin 3.6Alaska 4.1Maryland 4.3NorthCarolina 4.7Virginia 4.7Delaware 4.8Oklahoma 5.2Georgia 5.3Nebraska 5.5Wyoming 6.4Massachusetts 6.8Kansas 7Hawaii 7.2Washington 7.2Idaho 7.9Oregon 8RhodeIsland 8.7Utah 9Connecticut 9.4Illinois 10.7NewJ ersey 13.3NewYork 15.1Florida 16.8Colorado 17.1Nevada 19.7Arizona 25.3Texas 32California 32.4NewMexico 42.1

Stemplots are quick and dirty histograms that can easily be done by

hand, therefore very convenient for back of the envelope calculations.

However, they are rarely found in scientific publications.

Stemplots versus histograms

1) Advantages of Stemplots

Quick to do by hand (no frequency table needed)

Maintains all the numerical data

Good for comparing two distributions (using back-to-back plots)

2) Disadvantages

Not as crisp visually (compared to a graphics-quality histogram)

Most people like numerical scale on horizontal axis.

Stemplots versus histograms

1) Stem-splitting

Use to increase the number of stems

Create one stem for leaves 0-4

Second stem for leaves 5-9

2) Back-to-back Stemplots

Use to compare two data sets measured on same scale (e.g., male vs.

female heights).

Use single stem with leaves on opposite sides for different data sets.

Variations on stemplots

Ways to chart quantitative 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 and stemplots

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

understand the pattern of variability in the data.

Other graphs to reflect numerical summaries (see Chapter 1.2)

Line graphs: time plots

A trend is a rise or fall that persist 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.

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 interest—here

“retail price of fresh oranges”—

goes on the vertical, y axis.

1918 influenza epidemicDate # Cases # Deaths

week 1 36 0week 2 531 0week 3 4233 130week 4 8682 552week 5 7164 738week 6 2229 414week 7 600 198week 8 164 90week 9 57 56week 10 722 50week 11 1517 71week 12 1828 137week 13 1539 178week 14 2416 194week 15 3148 290week 16 3465 310week 17 1440 149

0100020003000400050006000700080009000

10000

0100200300400500600700800

0100020003000400050006000700080009000

10000

# c

ase

s d

iag

no

sed

0

100

200

300

400

500

600

700

800

# d

ea

ths

rep

ort

ed

# Cases # 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.

Death rates from cancer (US, 1945-95)

0

50

100

150

200

250

1940 1950 1960 1970 1980 1990 2000

Years

Death

rate

(per

thousand)

Death rates from cancer (US, 1945-95)

0

50

100

150

200

250

1940 1960 1980 2000

Years

Dea

th r

ate

(per

thou

sand

)

Death rates from cancer (US, 1945-95)

0

50

100

150

200

250

1940 1960 1980 2000

Years

Death

rate

(per

thousand)

A picture is worth a thousand words,

BUT

There is nothing like hard numbers.

Look at the scales.

Scales matterHow you stretch the axes and choose your scales can give a different impression.

Death rates from cancer (US, 1945-95)

120

140

160

180

200

220

1940 1960 1980 2000

Years

Death

rate

(pe

r th

ousan

d)

Why does it matter?

What's wrong with these graphs?

Careful reading reveals that:

1. The ranking graph covers an 11-year period, the tuition graph 35 years, yet they are shown comparatively on the cover and without a horizontal time scale.

2. Ranking and tuition have very different units, yet both graphs are placed on the same page without a vertical axis to show the units.

3. The impression of a recent sharp “drop” in the ranking graph actually shows that Cornell’s rank has IMPROVED from 15th to 6th ...

Cornell’s tuition over time

Cornell’s ranking over time