Lesson 1 – 1a from http://www.pendragoncove.info/statistics/ch1.htm

Displaying Distribution with Graphs

Knowledge Objectives• What is meant by exploratory data analysis

• What is meant by the distribution of a variable

• Differentiate between categorical variables and quantitative variables

• What is meant by the mode of a distribution

• What is meant by an outlier in a stemplot or histogram

Construction Objectives• Construct bar graphs and pie charts for a set of

categorical data

• Construct a stemplot for a set of quantitative data

• Construct a back-to-back stemplot to compare two related distributions

• Construct a stemplot using split stems

• Construct a histogram for a set of quantitative data, and discuss how changing the class width can change the impression of the data given by the histogram

Construction Objectives cont• Describe the overall pattern of a distribution by its

shape, center, and spread

• Recognize and identify symmetric and skewed distributions

• Construct and interpret an ogive (relative cumulative frequency graph) from a relative frequency table

• Construct a time plot for a set of data collected over time

Vocabulary• Roundoff error – errors associated with decimal inaccuracies• Pie chart – chart that emphasize each category’s relation to the

whole • Bargraph – displays the distribution of a categorical variable• Stemplot – includes actual numerical values in a plot that gives

a quick picture of the distribution• Back-to-back stemplot – two distributions plotted with a

common stem• Splitting stems – divides step into 0-4 and 5-9• Trimming – removes the last digit or digits before making a

stemplot• Histogram – breaks range of values into classes and displays

their frequencies• Frequency – counts of data in a class• Frequency table – table of frequencies

Vocabulary• Modes – major peaks in a distribution• Unimodal – a distribution whose shape with a single peak (mode)• Bimodal – a distribution whose shape has two peaks (modes)• Symmetric – if values smaller and larger of the center are mirror

images of each other• Skewed – if smaller or larger values from the center form a tail• Ogive – relative cumulative frequency graph• Time plot – plots a variable against time on the horizontal scale of

the plot• Seasonal variation – a regular rise and fall in a time plot

Categorical Data

• Categorical Variable:– Values are labels or categories– Distributions list the categories and either the

count or percent of individuals in each

• Displays: BarGraphs and PieCharts

Categorical Data Example

Body Part Frequency Relative Frequency

Back 12 0.4

Wrist 2 0.0667

Elbow 1 0.0333

Hip 2 0.0667

Shoulder 4 0.1333

Knee 5 0.1667

Hand 2 0.0667

Groin 1 0.0333

Neck 1 0.0333

Total 30 1.0000

Physical Therapist’s Rehabilitation Sample

Categorical Data

• Items are placed into one of several groups or categories (to be counted)

• Typical graphs of categorical data:– Pie Charts; emphasizes each category’s relation to the whole– Bar Charts; emphasizes each category’s relation with other

categories

0

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Back40%

Wrist7%

Elbow3%

Hip7%

Shoulder13%

Knee17%

Hand7%

Groin3%

Neck3% Pie ChartBar Chart

Charts for Both Data Types

00.05

0.10.15

0.20.25

0.30.35

0.40.45

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00.05

0.10.15

0.20.25

0.30.35

0.40.45

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Pareto ChartRelative Frequency Chart

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0.6

0.8

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Cumulative Frequency Chart

Example 1Construct a pie chart and a bar graph.

Radio Station Formats

Format Nr of Stations Percentage

Adult contemporary 1,556 11.2

Adult standards 1.196 8.6

Contemporary Hits 569 4.1

Country 2,066 14.9

News/Talk/Info 2,179 15.7

Oldies 1,060 7.7

Religious 2,014 14.6

Rock 869 6.3

Spanish Language 750 5.4

Other formats 1,579 11.4

Total 13,838 99.9

Why not 100%?

Example 1 Pie Chart

Example 1 Bar Graph

Quantitative Data

• Quantitative Variable:– Values are numeric - arithmetic computation

makes sense (average, etc.)– Distributions list the values and number of times

the variable takes on that value

• Displays:– Dotplots– Stemplots– Histograms– Boxplots

Dot Plot

• Small datasets with a small range (max-min) can be easily displayed using a dotplot– Draw and label a number line from min to max– Place one dot per observation above its value– Stack multiple observations evenly

• First type of graph under STATPLOT

34 values

ranging from 0 to 8

Stem Plots

• A stemplot gives a quick picture of the shape of a distribution while including the numerical values– Separate each observation into a stem and a leaf

eg. 14g -> 1|4 256 -> 25|6 32.9oz -> 32|9– Write stems in a vertical column and draw a

vertical line to the right of the column– Write each leaf to the right of its stem

• Note: – Stemplots do not work well for large data sets– Not available on calculator

Stem & Leaf Plots Review

Given the following values, draw a stem and leaf plot

20, 32, 45, 44, 26, 37, 51, 29, 34, 32, 25, 41, 56

Ages Occurrences------------------------------------------------------------------2 | 0, 6, 9, 5

|3 | 2, 3, 4, 2

|4 | 5, 4, 1

|5 | 1, 6

Splitting Stems

• Double the number of stems, writing 0-4 after the first and 5-9 after second.

Back-to-Back Stemplots

• Back-to-Back Stemplots: Compare datasets

Example1.4, pages 42-43Literacy Rates in Islamic Nations

Example 1

The ages (measured by last birthday) of the employees of Dewey, Cheatum and Howe are listed below.

a) Construct a stem graph of the ages

b) Construct a back-to-back comparing the offices

c) Construct a histogram of the ages

22 31 21 49 26 42

42 30 28 31 39 39

20 37 32 36 35 33

45 47 49 38 28 48

Office A

Office B

Example 1a: Stem and Leaf

2 0, 1, 2, 6, 8, 8,

3 0, 1, 1, 2, 3, 5, 6, 7, 8, 9, 9,

4 2, 2, 5, 7, 8, 9, 9,

22 31 21 49 26 42

42 30 28 31 39 39

20 37 32 36 35 33

45 47 49 38 28 48

Ages of Personnel

Example 1b: Back-to-Back Stem

2 0, 8

3 2, 3, 5, 6, 7, 8,

4 5, 7, 8, 9,

22 31 21 49 26 42

42 30 28 31 39 39

20 37 32 36 35 33

45 47 49 38 28 48

Office B: Ages of PersonnelOffice A: Ages of Personnel

1, 2, 6, 8

0, 1, 1, 9, 9

2, 2, 9

Example 2

Below are times obtained from a mail-order company's shipping records concerning time from receipt of order to delivery (in days) for items from their catalogue?

a) Construct a stem plot of the delivery times

b) Construct a split stem plot of the delivery times

c) Construct a histogram of the delivery times

3 7 10 5 14 12

6 2 9 22 25 11

5 7 12 10 22 23

14 8 5 4 7 13

27 31 13 21 6 8

3 10 19 12 11 8

Example 2: Stem and Leaf Part

0 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9

1 0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 9

2 1, 2, 2, 3, 5, 7

3 1

Days to Deliver

3 7 10 5 14 12

6 2 9 22 25 11

5 7 12 10 22 23

14 8 5 4 7 13

27 31 13 21 6 8

3 10 19 12 11 8

Example 2b: Split Stem and Leaf

0 2, 3, 3, 40 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 91 0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4 1 92 1, 2, 2, 32 5, 73 1

Days to Deliver

3 7 10 5 14 12

6 2 9 22 25 11

5 7 12 10 22 23

14 8 5 4 7 13

27 31 13 21 6 8

3 10 19 12 11 8

Day 1 Summary and Homework

• Summary– Categorical data

• Data where adding/subtracting makes no sense• Pie charts and bar graphs

– Quantitative data • Data where arithmetic operations make sense• Stem plots and histograms

– Some graphs can work for both types of data• Frequency and dot plots• Ogive and Pareto

• Homework– pg 46 – 48 problems 1-5