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.

Monday, 9/4/2006

Not summarized enough

Too summarized

Same data set0.0625 0.125

0.25 0.5 0.75

How to create a histogram It is an iterative process—try and try again. What bin size should you use?

Not overly summarized that you lose all the information

How to create a histogram It is an iterative process—try and try again. What bin size should you use?

Not so detailed that it is no longer summary

Not too many bins with either 0 or 1 counts

How to create a histogram

Rule of thumb: Start with 5 to10 bins.

Look at the distribution and refine your bins.

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

It is an iterative process—try and try again. What bin size should you use?

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.

Stemplots

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

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 or laymen publications.

Stemplots versus histograms

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

Line graphs: time plots

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 always goes on the

horizontal, or x, axis.

The variable of interest—

here “retail price of fresh

oranges”—goes on the

vertical, or y, axis.

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)

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

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 from the flu.

1918 influenza epidemic

0100020003000400050006000700080009000

10000

Wee

k 1

Wee

k 3

Wee

k 5

Wee

k 7

Wee

k 9

Wee

k 11

Wee

k 13

Wee

k 15

Wee

k 17

Num

ber

of c

ases

dia

gnos

ed

0

100

200

300400

500

600

700

800

Num

ber

of d

eath

s re

port

ed

# of Cases # of Deaths

Why does it matter?

What's wrong with these graphs?

Careful reading reveals that:

1) The ranking graph covers an 11-year period and 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