1.2 Displaying and Describing Categorical & Quantitative Data

You should be able to:• Recognize when a variable is categorical or quantitative• Choose an appropriate display for a categorical variable and a

quantitative variable• Summarize the distribution with a bar, pie chart, stem-leaf plot,

histogram, dot plot, box plots• Know how to make a contingency table• Describe the distribution of categorical variables in terms of relative

frequencies• Be able to describe the distribution of quantitative variables in terms

of its shape, center and spread• Describe abnormalities or extraordinary features of distribution• Discuss outliers and how they deviate from the overall pattern

3 RULES of EXPLORATORY DATA ANALYSIS

1. MAKE A PICTURE – find patterns in difficult to see from a chart

2. MAKE A PICTURE – show important features in graph

3. MAKE A PICTURE- communicates your data to others

Concepts to know!

• Bar graph• Histogram• Dot plot• Stem leaf plot• Scatterplots• Boxplots

Which graph to use?

• Depends on type of data

• Depends on what you want to illustrate

Categorical Data

• The objects being studied are grouped into categories based on some qualitative trait.

• The resulting data are merely

labels or categories.

Categorical Data(Single Variable)

Eye Color BLUE BROWN GREEN

Frequency

(COUNTS)

20 50 5

Relative Frequency

20/75 =

.27

50/75=

.66

5/75=

.07

Pie Chart(Data is Counts or Percentages)

Eye Color

Blue , 20, 27%

Brown, 50, 66%

Green, 5, 7%

Blue Brown Green

Bar Graph(Shows distribution of data)

Eye Color

0

10

20

30

40

50

60

Blue Brown Green

Color

Fre

qu

en

cy

Blue

Brown

Green

Bar Graph

• Summarizes categorical data.• Horizontal axis represents categories, while

vertical axis represents either counts (“frequencies”) or percentages (“relative frequencies”).

• Used to illustrate the differences in percentages (or counts) between categories.

Contingency Table(How data is distributed

across multiple variables)

Class

Survival

First Second Third Crew Total

ALIVE 203 118 178 212 711

DEAD 122 167 528 673 1490

Total 325 285 706 885 2201

What can go wrong when working with

categorical data?

• Pay attention to the variables and what the percentages represent(9.4% of passengers who were in first class survived is different from 67% of survivors were first class passengers!!!)

• Make sure you have a reasonably large data set

(67% of the rats tested died and 1 lived)

Analogy

Bar chart is to categorical data as histogram is to ...

quantitative data.

Histogram

18 19 20 21 22 23 24 25 26 27

0

10

20

30

40

50

Age (in years)

Fre

quency (

Count)

Age of Spring 1998 Stat 250 Students

n=92 students

Histogram

• Divide measurement up into equal-sized categories (BIN WIDTH)

• Determine number (or percentage) of measurements falling into each category.

• Draw a bar for each category so bars’ heights represent number (or percent) falling into the categories.

• Label and title appropriately.

• http://www.stat.sc.edu/~west/javahtml/Histogram.html

Use common sense in determining number of

categories to use.

Between 6 & 15 intervals is preferable

(Trial-and-error works fine, too.)

Histogram

Too few 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

Too many categories

2 3 4

0

1

2

3

4

5

6

7

GPA

Fre

quen

cy (

Co

unt)

GPAs of Spring 1998 Stat 250 Students

n=92 students

Dot Plot

160150140130120110100908070Speed

Fastest Ever Driving Speed

Women126

Men100

226 Stat 100 Students, Fall '98

Dot Plot

• Summarizes quantitative data.• Horizontal axis represents measurement

scale.• Plot one dot for each data point.

Stem-and-Leaf PlotStem-and-leaf of Shoes N = 139 Leaf Unit = 1.0

12 0 223334444444 63 0 555555555555566666666677777778888888888888999999999 (33) 1 000000000000011112222233333333444 43 1 555555556667777888 25 2 0000000000023 12 2 5557 8 3 0023 4 3 4 4 00 2 4 2 5 0 1 5 1 6 1 6 1 7 1 7 5

Stem-and-Leaf Plot

• Summarizes quantitative data.• Each data point is broken down into a “stem”

and a “leaf.”• First, “stems” are aligned in a column.• Then, “leaves” are attached to the stems.

Box Plot

0

1

2

3

4

5

6

7

8

9

10

Hours

of sl

eep

Amount of sleep in past 24 hours

of Spring 1998 Stat 250 Students

Box Plot

• Summarizes quantitative data.• Vertical (or horizontal) axis represents

measurement scale.• Lines in box represent the 25th percentile (“first

quartile”), the 50th percentile (“median”), and the 75th percentile (“third quartile”), respectively.

5 Number Summary

• Minimum• Q1 (25th percentile)• Median (50th percentile)• Q3 (75th percentile)• Maximum

An aside...

• Roughly speaking:– The “25th percentile” is the number such

that 25% of the data points fall below the number.

– The “median” or “50th percentile” is the number such that half of the data points fall below the number.

– The “75th percentile” is the number such that 75% of the data points fall below the number.

Box Plot (cont’d)

• “Outliers” are drawn to the most extreme data points that are not more than 1.5 times the length of the box beyond either quartile(IQR).

IQR = Q3 - Q1

Outliers(upper) > Q3+1.5 * IQR

Outliers(lower)<Q1-1.5*IQR

Using Box Plots to Compare

female male

60

110

160

Gender

Fast

est

Speed (

mph)

Fastest Ever Driving Speed

226 Stat 100 Students, Fall 1998 Outliers

Strengths and Weaknesses of Graphs for Quantitative Data

• Histograms– Uses intervals– Good to judge the “shape” of a data– Not good for small data sets

• Stem-Leaf Plots– Good for sorting data (find the median)– Not good for large data sets

Strengths and Weaknesses of Graphs for Quantitative Data

• Dotplots– Uses individual data points– Good to show general descriptions of

center and variation– Not good for judging shape for large data sets

• Boxplots– Good for showing exact look at center,

spread and outliers– Not good for judging shape

Analogy

Contingency table is to categorical data with two

variables as

scatterplot is to ..

quantitative data with two

variables.

Scatter Plots

22 23 24 25 26 27 28 29 30 31

22

23

24

25

26

27

28

29

30

31

Left foot (in cm)

Rig

ht fo

ot (in c

m)

Foot sizes of Spring 1998 Stat 250 students

n=88 students

Scatter Plots

• Summarizes the relationship between two quantitative variables.

• Horizontal axis represents one variable and vertical axis represents second variable.

• Plot one point for each pair of measurements.

No relationship

52 57 62

22

23

24

25

26

27

28

29

30

31

32

Head circumference (in cm)

Left fore

arm

(in

cm

)Lengths of left forearms and head circumferences

of Spring 1998 Stat 250 Students

n=89 students

Summary

• Many possible types of graphs.• Use common sense in reading graphs.• When creating graphs, don’t summarize your

data too much or too little.• When creating graphs, label everything for

others. Remember you are trying to communicate something to others!

GRAPHICAL ANALYSIS

• CENTER

• SPREAD

Graphical Analysis

Center Center

(Location)(Location)

SpreadSpread

(Variation)(Variation)

ShapeShape

Interesting Features Identified by Graphs

• Center (Location)• Spread (Variability)• Shape • Individual Values• Compare Groups• Identify Outliers• LOOK FOR PATTERNS, CLUSTERS, GAPS!!!!!• LOOK FOR DEVIATIONS FROM THE GENERAL

PATTERN!!!!!

http://bcs.whfreeman.com/bps3e/

Shape of Graph

Shape of Graph

• Symmetry-Skewness• Modes(peaks)

SYMMETRY & SKEWNESS

• Skewness - a measure of symmetry, or more precisely, the lack of symmetry.

• A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Kurtosis

• Kurtosis - a measure of whether the data are peaked or flat relative to a normal distribution.

• High kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. L

• Low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Symmetric -- Mesokurtic

Symmetric -- Platykurtic

Symmetric -- Leptokurtic

Nonsymmetric – Skewed Positive -Skewed Right

Nonsymmetric – Skewed Negative -- Skewed Left

Symmetric -- Bimodal

•

Describing Distributions –

SOCS Rock!• Shape

• Outliers

• Center

• Spread

Graphical Analysis - Scale

375 400 425 450 475 500 525 350 400 450 500 550 600 650

SOL Scores – Class 1 SOL Scores – Class 2

Graphical Analysis - Scale

400 450 500 550 600 400 450 500 550 600

SOL Scores – Class 1 SOL Scores – Class 2

400 450 500 550 600 400 450 500 550 600

375 400 425 450 475 500 525 350 400 450 500 550 600 650

SOL Scores – Class 1 SOL Scores – Class 2

• Comparing Histograms