1. Discrete or Continuous
Types of data
Quantitative
(numerical)
Categorical
Discrete
Continuous
Discrete

2. Two Types of Variables
A Numerical Variable describes quantities of the objects of interest. Data values are numbers.
Weight of an infant
Number of sexual partners
Time to run the mile
A Categorical Variable describes qualities of the objects of interest. Data values are usually words.
Skin color
Birth city
Last Name

3. Example: Numerical or Categorical?
Age
Gender
Major
Units
Housing
GPA 18
Male
Psychology 16
Dorm
3.6 21
Nursing 15
Parents
3.1 20
Female
Business
Apartment
2.8
Numerical
Age
Units
GPA
Categorical
Gender
Major
Housing

4. Numerical or Categorical?
Why are you in college? Answer:
Person Growth Career Opportunities
3. Parental Pressure 4. Personal Networking
Results from 12 participants:
1, 4, 3, 2, 2, 1, 2, 3, 3, 1, 4, 2
Coding Categorical Data with Numbers: Although the above data values are numbers, the variable is still categorical.
Reason for Coding: Easier to input into a computer.

5. Scales of Measurement Nominal Ordinal Scale Characteristics Examples
Label and categorize
No quantitative distinctions
Gender
Diagnosis
Experimental or Control
Ordinal
Categorizes observations
Categories organized by size or magnitude
Rank in class
Clothing sizes (S,M,L,XL)
Olympic medals
Interval
Ordered categories
Interval between categories of equal size
Arbitrary or absent zero point
Temperature
This material is arguably in the “Top Ten Most Important” concepts the students will encounter in the study of statistics and may merit identifying it as such.
Ratio
Ordered categories
Equal interval between categories
Absolute zero point
Number of correct answers
Time to complete task
Gain in height since last year

6. What kinds of data are typically collected?
Ratio
Nominal
Ordinal
Interval
Continuous
Categorical
Nominal Data
no ordering, e.g. it makes no sense to state that F > M
arbitrary labels, e.g., m/f, 0/1, etc
Ordinal Data
ordered but differences between values are not important
e.g., Likert scales, rank on a scale of 1..5 your degree of satisfaction
Interval Data
ordered, constant scale, but no natural zero
differences make sense, but ratios do not (e.g., 30°-20°=20°-10°, but 20°/10° is not twice as hot!
Ratio Data
ordered, constant scale, natural zero
e.g., height, weight, age, length

7. Example 2.3 Frequency, Proportion and Percent
p = f/N
percent = p(100) 5 1
1/10 = .10
10% 4 2
2/10 = .20
20% 3
3/10 = .30
30%

8. Displaying distributions Qualitative variables
Pie Charts
Bar Graphs

9. PIE CHART FOR THE TASTE TEST
Coca-Cola
Pepsi
Others
Seven up
Dr Pepper

10. Graphs for Nominal or Ordinal Data
For non-numerical scores (nominal and ordinal data), use a bar graph
without a particular order (nominal)
non-measurable width (ordinal)

11. Bar graph
FIGURE 2.6 A bar graph showing the distribution of personality types in a sample of college students. Because personality type is a discrete variable measured on a nominal scale, the graph is drawn with space between the bars.

12. BAR CHART FOR THE AIDS DATA
1 ATLANTA
2 AUSTIN
3 DALLAS
4 HOUSTON
5 NY, NY.
6 SAN. FRAN.
7 WASH D.C.
8 W. P. BEACH

13. Figure 2.7 Bar Graph of Relative Frequencies
FIGURE 2.7 A frequency distribution showing the relative frequency for two types of fish. Notice that the exact number of fish is not reported; the graph simply says that there are twice as many bluegill as there are bass.

14. A Misleading Bar Graph
Problem
The bar graph that follows presents the total sales figures for three realtors. When the bars are replaced with pictures, often related to the topic of the graph, the graph is called a pictogram.
Realtor 3
(a) The height for Realtor 1 is just slightly over twice that of Realtor 3. The heights are at the correct total sales levels.
(b) To avoid distortion of the pictures, the area of the home for Realtor 1 is more than four times the area of the home for Realtor 3.
What We’ve Learned: When you see a pictogram, be careful to interpret the results appropriately, and do not allow the area of the pictures to mislead you.
Realtor 2
Realtor 1
(a) How does the height of the home for Realtor 1 compare to that for Realtor 3?
(b) How does the area of the home for Realtor 1 compare to that for Realtor 3?

15. Displaying Distributions Quantitative Variables
Histograms
Polygons
Frequency plots
Stem and Leaf Plots
Time plots
Scatterplots

16. Histogram Histogram of Age
31,36,36,40, 41,41,41,44,44,44,44,45, 45, 45,46,47,48,49, 51,51
CLASS TALLY # OBSERVATIONS PERCENTAGE
[30,35) / /20 = %
[35,40) // /20 = %
[40,45) //////// /20 = %
[45,50) /////// /20 = %
[50,55) // /20 = %

18. Figure 2.3 Frequency Distribution Block Histogram
FIGURE 2.3 A frequency distribution in which each individual is represented by a block placed directly above the individual’s score. For example, three people had scores of X = 2.

19. Histogram versus Bar Graph
GRAPH I
GRAPH II
GRAPH III
GRAPH IV

20. Misleading Histograms
FIGURE 2.9 Two graphs showing the number of homicides in a city over a 4-year period. Both graphs show exactly the same data. However, the first graph gives the appearance that the homicide rate is high and rising rapidly. The second graph gives the impression that the homicide rate is low and has not changed over the 4-year period.

21. Figure 2.4 Frequency Distribution Polygon
FIGURE 2.4 An example of a frequency distribution polygon. The same set of data is presented in a frequency distribution table and in a polygon.

22. Figure 2.5 Grouped Data Frequency Distribution Polygon
FIGURE 2.5 An example of a frequency distribution polygon for grouped data. The same set of data is presented in a grouped frequency distribution table and in a polygon.

23. Describe The Distribution

24. What eyes see
Describe
1) with words
and
2) with numbers

25. Describing with WORDS

26. Three Aspects of a Distribution
Shape
Symmetry
How a many bumps or modes?
Other distinguishing features
Center
What is a typical value?
The bulk of the data
Spread
Is the data all close together or spread out?
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27. Distribution Shapes
FIGURE Examples of different shapes for distributions.

28. SHAPE ~ Symmetric Distributions
A distribution is symmetric if the left hand side is roughly the mirror image of the right hand side.
Symmetric Distributions
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29. Symmetric Is the histogram symmetric?
If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

30. SHAPE ~ Normal Distributions
A Normal distribution has the following properties
Symmetric
Unimodal
Mound or Bell Shaped
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31. SHAPE ~ Skewness
A distribution is Skewed Right if most of the data values are small and there is a “tail” of larger values to the right.
A distribution is Skewed Left if most of the data values are large and there is a “tail” of smaller values to the left.
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32. Skewed
The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.
In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

33. SHAPE ~ How Many Mounds A Unimodal distribution has one mound.
A Bimodal distribution has two mounds.
A Multimodal distribution has more than two mounds.
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34. Peaks: Modes
Does the histogram have a single, central peak or several separated peaks?
Peaks in a histogram are called modes.
A histogram with one main peak is called unimodal; histograms with two peaks are bimodal;
histograms with three or more peaks are
called multimodal.

35. Center
For now, we look at the most common value in each distribution. We will develop more precise ways to describe the center of a distribution in the next section.
What is the center of this distribution?

36. Center What is a typical value
Center not a typical value for bimodal or skewed.
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37. Center
For now, we look at the most common value in each distribution. We will develop more precise ways to describe the center of a distribution in the next section.
What is the center of this distribution?

38. SPREAD~ Range
The range of the data is the difference between the maximum and minimum values

39. Spread: Range
Always report a measure of spread along with a measure of center when describing a distribution numerically.
The range of the data is the difference between the maximum and minimum values:
Range = max – min
A disadvantage of the range is that a single extreme value can make it very large and, thus, not representative of the data overall.
For example, if my test scores were 10, 87, 94, 88, 85, 82, 85, 92 my range would be 94-10=84. This is a large spread, but most of my scores are in the 80. We will soon discuss different measures of spread.

40. Quiz Scores Please see replacement activity
Which class (A or B) has more variability?
2014 Summer Training Institute
College of the Canyons

41. Hypothetical Quiz Scores
Please see replacement activity
Which class has the least? Which the most?
2014 Summer Training Institute
College of the Canyons

42. Outliers
An Outlier is a data value that is either much smaller or much larger than the rest of the data.
Some reasons for outliers
Error in data collection
No error. For example, the owner’s salary could be an outlier if the rest of the employees are all low wage workers
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43. Anything Unusual? (cont.)
The following histogram has possible outliers to the left.

44. Describing a Distribution with words Using Stats language in Context.
What is the shape?
Is it Symmetric, Skewed, or Neither?
Unimodal, Bimodal, or Multimodal?
Normal?
Are there outliers?
Where is the center? Is the center a typical value?
Is there low or high variability?
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45. Describe The Distributions
It is always more interesting to compare groups. Below are daily wind speeds at a National Park.

46. Describe The Distribution
The dotplots below show drive times for 3 different routes.
Describe these dotplots.
What route would you take and why?

48. Shape center and Spread activity

49. Describe The Distributions
It is always more interesting to compare groups. Below are daily wind speeds at a National Park.

50. Describe The Distribution
The dotplots below show drive times for 3 different routes.
Describe these dotplots.
What route would you take and why?