1. Picturing Distributions with Graphs
BPS chapter 1
Chapter 1: Picturing Distributions with Graphs
© 2006 W. H. Freeman and Company

2. Objectives (BPS chapter 1)
Picturing Distributions with Graphs
Individuals and variables
Two types of data: categorical and quantitative
Levels of Data
Ways to chart quantitative data: histograms and stemplots
Interpreting histograms
Time plots
Chapter 1: Picturing Distributions with Graphs

3. Chapter 1: Picturing Distributions with Graphs
What is STATISTICS ?
Using “data” to draw a conclusion about something unknown.
Decision making in the presence of uncertainty.
Chapter 1: Picturing Distributions with Graphs

4. Chapter 1: Picturing Distributions with Graphs
Statistics- Meaning ?
Method of analysis
a collection of methods for planning experiments or observational studies, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions based on the data.
Chapter 1: Picturing Distributions with Graphs

5. Chapter 1: Picturing Distributions with Graphs
Statistics- Meaning ?
Our Book:
Statistics is the science (or ‘art’) of data.
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6. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
What Is “Data”?
Pieces of information.
Numbers.
The above are data only if the information has a meaning attached.
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7. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
The Nature of Data
Data
Data are observations that have been collected. The observation may be numerical (example: age, height, GPA) or non-numerical (example: gender, eye colour, province of residence)
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8. Individuals and variables
Individuals are the objects described by a set of data. Individuals may be people, animals, or things.
Example: Freshmen, 6-week-old babies, golden retrievers, fields of corn, cells
A variable is any characteristic of an individual. A variable can take different values for different individuals.
Example: Age, height, blood pressure, ethnicity, length, first language
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9. Chapter 1: Picturing Distributions with Graphs
Two types of variables
Chapter 1: Picturing Distributions with Graphs

10. Two types of variables Cont.
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.
categorical
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.
Chapter 1: Picturing Distributions with Graphs

11. Two types of variables Cont.
See page 5 for description of variables.
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12. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Levels of Measurements
1. Nominal: characterized by data that consist of names, labels, or categories only. The data cannot be arranged in an ordering scheme (such as low to high)
Example: Survey responses may be yes, no, or undecided. Eye colour, gender etc.
page 7 of text
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13. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
2. Ordinal: involves data that may be arranged in some order, but differences between data values either cannot be determined or are meaningless
Example: Course grades: A, B, C, D, or F
Dress size: small, medium, large, XL
Understanding the differences between the levels of data will help students later in determining what type of statistical tests to use.
Nominal and ordinal data should not be used for calculations (even when assigned ‘numbers’ for computerization) as differences and magnitudes of differences are meaningless.
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14. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
3. Interval: like the ordinal level, with the additional property that the difference between any two data values is meaningful. However, there is no natural zero starting point (where none of the quantity is present)
Example: Years 1000, 2000, 1776, and 1492
Temperature in 0C C does not mean no temperature.
Students usually have some difficulty understanding the difference between interval and ratio data. Fortunately, interval data occurs in very few instances.
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15. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
4. Ratio: the interval level modified to include the natural zero starting point (where zero indicates that none of the quantity is present). For values at this level, differences and ratios are meaningful.
Example: Prices of college textbooks.
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16. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Levels of Measurement
______________________ - categories only
__________________________- categories with some order
______________________- differences but no natural starting point
______________________- differences and a natural starting point
review of four levels of measurement
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17. Chapter 1: Picturing Distributions with Graphs
Summary
Chapter 1: Picturing Distributions with Graphs

18. Exploratory Data Analysis (EDA)
Can be a table, graph, or function
Distribution
Tells what values a variable takes and how
often it takes these values.
Can be a table, graph, or function.
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EDA Cont.
The W’s:
Who
What (in what units)
When
Where
Why
Can be a table, graph, or function
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20. “University Students are Healthier than you Think”
Math 161 Spring 2008
“University Students are Healthier than you Think”
A Spring 2002 random undergraduate classroom survey of n=810 students was conducted by the Office of Health Promotion within a University Student Health Services, Division of Student Affairs.
Statistics from this survey led to the following conclusions: - most students (67%) have 0-4 drinks when they go out - most (69%) have had 0-1 sex partners in the past year - most (76%) either don’t drink, or use designated drivers if they do
Identify the W’s.
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21. Important Characteristics of Data
The following characteristics of data are usually important:
Center: An average value that indicates where the middle of the dataset is located.
Variation/Spread: A measure of the amount of variation in the data (average variation from the center).
Distribution: The shape of the distribution of the data (symmetric, uniform or skewed).
Outliers: Sample values that lie far away from the vast majority of the other values.
Time: Trend- changing characteristics of the data over time.
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Presentation of Data
To better analyze a dataset, we first present it in a summarized form using:
Frequency Tables
Pictures or Graphs
Numerical Summaries (Center and Variation)
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Frequency Tables
A frequency table lists data values (either individually or by groups of intervals called classes ), along with the number of items that fall into each class (frequency). Example:
Test Score Frequency
This frequency table has 7 classes (0-4,5-9,10-14,15-19,20-24,25-29,30-34). The frequency represents the number of students receiving that score.
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24. Chapter 1: Picturing Distributions with Graphs
Example
The heights (in inches) of 30 students are as
follows:
Create a frequency table for the above data
using the classes 60-61, 62-63, etc.
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25. Chapter 1: Picturing Distributions with Graphs

26. RELATIVE FREQUENCY TABLES
Relative frequency = frequency / total # of items
The relative frequency gives the percent of items in each class.
A relative frequency table is a frequency table with a column for the relative frequencies. The relative frequencies might not add to 1 (100%) due to rounding.
Example: Construct a relative frequency table for our last example.
Chapter 1: Picturing Distributions with Graphs

27. Graphs for Categorical Data
A picture (a good one) is worth a thousand words.
Bar Graph
Horizontal axis represents the categories.
Vertical axis represents the frequencies.
A bar whose height is proportional to the frequency is drawn
centered at the category.
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Example
The following table gives the grade distributions of a Math 161 Test:
Grade Frequency A B C D F
Draw a bar graph for the data.
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29. Chapter 1: Picturing Distributions with Graphs
Top 10 causes of death in the U.S., 2001
Bar graph sorted by rank
 Easy to analyze
Sorted alphabetically
 Much less useful
Chapter 1: Picturing Distributions with Graphs

30. Graphs for Categorical Data
Double (Side-by-side) Bar Graphs
Used to compare two different distributions.
For each category, draw two adjacent bars (one for each
distribution).
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31. Chapter 1: Picturing Distributions with Graphs
Example
Suppose we now have the grades for two sections of Math 161:
Grade Frequency
Section 1 Section 2 A B C D F
Draw a double bar graph for the data.
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32. Chapter 1: Picturing Distributions with Graphs
Example
Are the frequency bar graphs the right way to compare the performance of the two sections? Note that the class sizes are not the same. What would be a better way to compare the two sections?
Grade Frequency
Section 1 Section 2 A B C D F
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33. Graphs for Categorical Data
Pie Chart
Shows the whole group of categories in a circle.
Shows the parts of some whole .
The area of the sector representing a category is
proportional to the frequency of the category.
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34. Chapter 1: Picturing Distributions with Graphs
Example
The following table gives the grade distributions of a Math 161 Test:
Grade Frequency A B C D F
Draw a pie chart for the data.
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35. Chapter 1: Picturing Distributions with Graphs
Example: The pie chart shows the distribution of the type of housing a
random sample of VIU students have.
If 20 students live in apartments, how many students live in shared houses? 
A) 15 B) 20 C) 45 D) 60 E) 75
What central angle does the sector for Dorm have? 
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36. Chapter 1: Picturing Distributions with Graphs
Pictographs
A picture of a set of small
figures or icons used to
represent data, and often to
represent trends.
Usually, the icons are
suggestively related to the
data being represented.
They can be misleading.
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Pictographs
Double the length, width, and height of a cube, and the volume increases by a factor of eight.
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38. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
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39. Chapter 1: Picturing Distributions with Graphs
Time (Line) Graphs
A time graph shows behavior over time.
Time is always on the horizontal axis.
Look for an overall pattern (trend).
Look for patterns that repeat at known regular intervals (seasonal variations).
Look for any striking deviations that might indicate unusual occurrences.
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40. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
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Misleading Graphs
Changing the scale of a line graph or a bar graph can make
increases or decreases appear more rapid.
Both graphs plot the same data. Which one makes the increase in cancer
deaths appear more rapid? Which graph would a cancer advocate use?
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43. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Salaries of People with Bachelor’s Degrees and with High School Diplomas
Which graph is misleading?
$40,500
$40,500
$40,000
$40,000
35,000
30,000
$24,400
30,000
20,000
$24,400
page 11 of text
Graphs whose vertical scales do not start at 0 will give a misleading representation of the differences in heights of the bars.
25,000
10,000
20,000
Bachelor High School
Degree Diploma
Bachelor High School
Degree Diploma
(a)
(b)
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44. Chapter 1: Picturing Distributions with Graphs
Important skills
Computer programs will construct plots and calculate summary statistics automatically.
The important skills for people are:
knowing what to use when.
Interpretation.
The tools used to analyze and summarize data depend upon the type of variable one is interested in.
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45. Chapter 1: Picturing Distributions with Graphs
Principles for plots
The way plots are used depends upon the purpose for which
they are being used:
Exploration
Principle: Look at the data in as many different ways as possible searching for its important features.
Communication to others (follows exploration)
Principle : Be selective. Choose the displays that best show to a reader features you have observed.
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46. Chapter 1: Picturing Distributions with Graphs
Making Good Graphs
Title your graph.
Make sure labels and legends describe variables and their measurement units. Be careful with the scales used.
Make the data stand out. Avoid distracting grids, artwork, etc.
Pay attention to what the eye sees. Avoid pictograms and tacky effects.
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47. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Key Concepts
Categorical and Quantitative Variables
Distributions
Pie Charts
Bar Graphs
Line Graphs
Techniques for Making Good Graphs
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48. Graphs for Quantitative Variables
Math 161 Spring 2008
Graphs for Quantitative Variables
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49. Stemplots (Stem-and-Leaf Plots)
For quantitative variables.
Separate each observation into a stem (first part of the number) and a leaf (the remaining part of the number).
Usually, the last digit is used as the leaf and the remaining digits form the stem.
If using the last digits as they are results in a lot of stem values, we could round the numbers to more convenient values.
Write the stems in a vertical column; draw a vertical line to the right of the stems.
Write each leaf in the row to the right of its stem; order leaves if desired.
Chapter 1: Picturing Distributions with Graphs

50. Weight Data Weights (in pounds) for a group of 40 students.
Chapter 1: Picturing Distributions with Graphs

51. Weight Data: Stemplot (Stem and Leaf Plot)
Math 161 Spring 2008 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Weight Data: Stemplot (Stem and Leaf Plot) 5 2
570
Key
20|3 means 203 pounds
Stems = 10’s Leaves = 1’s 2
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Stem-and-Leaf Plots
Double stemmed (expanded) stem-and-leaf:
If there are a lot of leaves on one stem, we could break it up into two stems one for the digits 0-4 and the other for the digits 5-9.
Back to back : Used to compare two different sets of data.
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53. Chapter 1: Picturing Distributions with Graphs
Histogram
A histogram is a bar graph in which
the horizontal axis represents the items or classes.
the vertical axis represents the frequencies.
the height of the bars are proportional to the frequencies.
There are usually no gaps between the bars (unless some classes have 0
frequencies).
To draw a histogram, we first need to construct a frequency table.
Example: draw a histogram for our weights example.
The number of classes can affect the shape of the histogram.
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54. Chapter 1: Picturing Distributions with Graphs
Same data set
Not summarized enough
Too summarized
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55. Weight Data: Frequency Table
Math 161 Spring 2008
Weight Data: Frequency Table
* Left endpoint is included in the group, right endpoint is not.
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56. Weight Data: Histogram
Math 161 Spring 2008
Weight Data: Histogram
* Left endpoint is included in the group, right endpoint is not.
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58. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Shape of the Data
Symmetric
bell-shaped
other symmetric shapes
Asymmetric
skewed to the right
skewed to the left
Unimodal, bimodal
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59. Symmetric Distributions
Math 161 Spring 2008
Symmetric Distributions
Mound-Shaped
Bell-Shaped
Uniform
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60. Asymmetric Distributions
Math 161 Spring 2008
Asymmetric Distributions
Skewed to the Left
Skewed to the Right
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61. Most common distribution shapes
Symmetric distribution
A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other.
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.
Right Skewed distribution
Complex, multimodal distribution
Not all distributions have a simple overall shape, especially when there are few observations.
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63. Number of Books Read for Pleasure
Math 161 Spring 2008
Number of Books Read for Pleasure
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64. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Outliers
Extreme values, far from the rest of the data.
May occur naturally.
May occur due to error in recording.
May occur due to error in measuring.
Observational unit may be fundamentally different.
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65. Chapter 1: Picturing Distributions with Graphs
Math 161 Spring 2008
Key Concepts
Displays (Stemplots & Histograms)
Graph Shapes
Symmetric
Skewed to the Right
Skewed to the Left
Outliers
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