1. Exploratory Data Analysis: One Variable
FPP 3-6

2. Plan of attack Distinguish different types of variables
Summarize data numerically
Summarize data graphically
Use theoretical distributions to potentially learn more about a variable.

3. The five steps of statistical analyses
Form the question
Collect data
Model the observed data
We start with exploratory techniques.
Check the model for reasonableness
Make and present conclusions

4. Just to make sure we are on the same page
More (or repeated) vocabulary
Individuals are the objects described by a set of data
examples: employees, lab mice, states…
A variable is any characteristic of an individual that is of interest to the researcher. Takes on different values for different individuals
examples: age, salary, weight, location…
How is this different from a mathematical variable?

5. Just to make sure we are on the same page #2
Measurement The value of a variable obtained and recorded on an individual
Example: 145 recorded as a person’s weight, 65 recorded as the height of a tree, etc.
Data is a set of measurements made on a group of individuals
The distribution of a variable tells us what values it takes and how often it takes these values

6. Two Types of Variables
a categorical/qualitative variable places an individual into one of several groups or categories
examples:
Gender, Race, Job Type, Geographic location…
JMP calls these variables nominal
a quantitative variable takes numerical values for which arithmetic operations such as adding and averaging make sense
Height, Age, Salary, Price, Cost…
Can be further divided to ordinal and continuous
Why two types?
Both require their own summaries (graphically and numerically) and analysis.
I can’t emphasis enough the importance of identifying the type of variable being considered before proceeding with any type of statistical analysis

7. Example Age: quantitative Gender: categorical Race: categorical
Salary: quantitative
Job type: categorical

8. Variable types in JMP Qualitative/categorical Quantitative
JMP uses Nominal
Quantitative
Discrete
JMP uses Ordinal
Continuous
JMP uses Continuous

9. Exploratory data analysis
Statistical tools that help examine data in order to describe their main features
Basic strategy
Examine variables one by one, then look at the relationships among the different variables
Start with graphs, then add numerical summaries of specific aspects of the data

10. Exploratory data analysis: One variable
Graphical displays
Qualitative/categorical data: bar chart, pie chart, etc.
Quantitative data: histogram, stem-leaf, boxplot, timeplot etc.
Summary statistics
Qualitative/categorical: contingency tables
Quantitative: mean, median, standard deviation, range etc.
Probability models
Qualitative: Binomial distribution(others we won’t cover in this class)
Quantitative: Normal curve (others we won’t cover in this class)

11. Example categorical/qualitative data

12. Summary table
we summarize categorical data using a table. Note that percentages are often called Relative Frequencies.

13. Bar graph
The bar graph quickly compares the degrees of the four groups
The heights of the four bars show the counts for the four degree categories

14. Pie chart A pie chart helps us see what part of the whole group forms
To make a pie chart, you must include all the categories that make up a whole

15. Summary of categorical variables
Graphically
Bar graphs, pie charts
Bar graph nearly always preferable to a pie chart. It is easier to compare bar heights compared to slices of a pie
Numerically: tables with total counts or percents

16. Quantitative variables
Graphical summary
Histogram
Stemplots
Time plots
more
Numerical sumary
Mean
Median
Quartiles
Range
Standard deviation

17. Histograms The bins are: 3.0 ≤ rate < 4.0 4.0 ≤ rate < 5.0

18. Histograms The bins are: 3.0 ≤ rate < 4.0 4.0 ≤ rate < 5.0

19. Histograms The bins are: 2.0 ≤ rate < 4.0 4.0 ≤ rate < 6.0

20. Histograms Where did the bins come from?
They were chosen rather arbitrarily
Does choosing other bins change the picture?
Yes!! And sometimes dramatically
What do we do about this?
Some pretty smart people have come up with some “optimal” bin widths and we will rely on there suggestions

21. Histogram The purpose of a graph is to help us understand the data
After you make a graph, always ask, “What do I see?”
Once you have displayed a distribution you can see the important features

22. Histograms
We will describe the features of the distribution that the histogram is displaying with three characteristics
Shape
Symmetric, skewed right, skewed left, uni-modal, multi-modal, bell shaped
Center
Mean, median
Spread (outliers or not)
Standard deviation, Inter-quartile range

23. Body temperatures of 30 people

24. Incomes from 500 households in 2000 current population survey

25. Histogram vs. Bar graph
Spaces mean something in histograms but not in bar graphs
Shape means nothing with bar graphs
The biggest difference is that they are displaying fundamentally different types of variables

26. Time Plots Many variables are measured at intervals over time
Examples
Closing stock prices
Number of hurricanes
Unemployment rates
If interest is a variable is to see change over time use a time plot

27. Time Plots Patterns to look for
Patterns that repeat themselves at known regular intervals of time are called seasonal variation
A trend is a persistant, long-term rise or fall

28. Time plots
number of hurricanes each year from

29. Numerical summaries of quantitative variables
Want a numerical summary for center and spread
Center
Mean
Median
Mode
Spread
Range
Inter-quartile range
Standard deviation
5 number summary is a popular collection of the following
min, 1st quartile, median, 3rd quartile, max

30. Mean
To find the mean of a set of observations, add their values and divide by the number of observations
equation 1:
equation 2:

31. Mean example
The average age of 20 people in a room is 25. A 28 year old leaves while a 30 year old enters the room.
Does the average age change?
If so, what is the new average age?

32. Median The median is the midpoint of a distribution
The number such that half the observations are smaller and the other half are larger
Also called the 50th percentile or 2nd quartile
To compute a median
Order observations
If number of observations is odd the median is the center observation
If number of observations is even the median is the average of the two center observations

33. Median example
The median age of 20 people in a room is 25. A 28 year old leaves while a 30 year old enters the room.
Does the median age change?
If so, what is the new median age?
The median age of 21 people in a room is 25. A 28 year old leaves while a 30 year old enters the room.

34. Mean vs Median When histogram is symmetric mean and median are similar
Mean and median are different when histogram is skewed
Skewed to the right mean is larger than median
Skewed to the left mean is smaller than median
The business magazine Forbes estimates that the “average” household wealth of its readers is either about $800,000 or about $2.2 million, depending on which “average” it reports. Which of these numbers is the mean wealth and which is the median wealth? Why?

35. Mean vs Median
Symmetric distribution

36. Mean vs Median
Right skewed distribution

37. Mean vs Median
Left skewed distribution

38. Extreme example Income in small town of 6 people
$25,000 $27,000 $29,000
$35,000 $37,000 $38,000
Mean is $31,830 and median is $32,000
Bill Gates moves to town
$35,000 $37,000 $38,000 $40,000,000
Mean is $5,741,571 median is $35,000
Mean is pulled by the outlier while the median is not. The median is a better of measure of center for these data

39. Is a central measure enough?
A warm, stable climate greatly affects some individual’s health. Atlanta and San Diego have about equal average temperatures (62o vs. 64o). If a person’s health requires a stable climate, in which city would you recommend they live?

40. Measures of spread Range: Inter-quartile range:
subtract the largest value form the smallest
Inter-quartile range:
subtract the 3rd quartile from the 1st quartile
Standard Deviation (SD):
“average” distance from the mean
Which one should we use?

41. Standard Deviation
The standard deviation looks at how far observations are from their mean
It is the square root of the average squared deviations from the mean
Compute distance of each value from mean
Square each of these distances
Take the average of these squares and square root
Often we will use SD to denote standard deviation

42. Example

43. Standard deviation
Order these histograms by the SD of the numbers they portray. Go from smallest largest
What is a reasonable guess of the SD for each?

44. Histograms on same scale

45. Problem from text (p. 74, #2)
Which of the following sets of numbers has the smaller SD’
a) 50, 40, 60, 30, 70, 25, 75 b) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50
Repeat for these two sets
c) 50, 40, 60, 30, 70, 25, 75 d) 50, 40, 60, 30, 70, 25, 75, 99, 1

46. More intuition behind the SD
This is a variance contest. You must give a list of six numbers chosen from the whole numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with repeats allowed.
Give a list of six numbers with the largest standard deviation such a list described above can possibly have.
Give a list of six numbers with the smallest standard deviation such a list can possibly have.

47. Properties of SD SD ≥ 0. (When is SD = 0)?
Has the same unit of measurement as the original observations
Inflated by outliers

48. Mean and SD
What happens to the mean if you add 5 to every number in a list?
What happens to the SD?

49. Standard deviation SDs are like measurement units on a ruler
Any quantitative variable can be converted into “standardized” units
These are often called z-scores and are denoted by the letter z
Important formula
Example
ACT versus SAT scores
Which is more impressive
A 1340 on the SAT, or a 32 on the ACT?

50. The normal curve
When histogram looks like a bell-shaped curve, z-scores are associated with percentages
The percentage of the data in between two different z-score values equals the area under the normal curve in between the two z-score values
A bit of notation here.
N(, ) is short hand for writing normal curve with mean  and standard deviation  (get used to this notation as it will be used fairly regularly through out the course)

51. Normal curves

52. Normal curves

53. Properties of normal curve
In the Normal distribution with mean  and standard deviation :
68% of the observations fall within 1  of 
95% of the observations fall within 2 s of 
99.7% of the observations fall within 3 s of 
By remembering these numbers, you can think about Normal curves without constantly making detailed calculations

54. Properties of normal curves
For a N(0,1) the following holds

55. IQ A person is considered to have mental retardation when
IQ is below 70
Significant limitations exist in two or more adaptive skill areas
Condition is present from childhood
What percentage of people have IQ that meet the first criterion of mental retardation

56. IQ A histogram of all people’s IQ scores has a μ=100 and a σ=16
How to get % of people with IQ < 70

57. More IQ
Reggie Jackson, one of the greatest baseball players ever, has an IQ of What percentage of people have bigger IQs than Reggie?
Marilyn vos Savant, self-proclaimed smartest person in the world, has a reported IQ of What percentage of people have IQ scores smaller than Marilyn’s score?
Mensa is a society for “intelligent people.” To qualify for Mensa, one needs to be in at least the upper 2% of the population in IQ score. What is the score needed to qualify for Mensa?

58. Checking if data follow normal curve
Look for symmetric histogram
A different method is a normal probability plot. When normal curve is a good fit, points fall on a nearly straight line

59. Measurement error Measurement error model Outliers
Measurement = truth + chance error
Outliers
Bias effects all measurements in the same way
Measurement = truth + bias + chance error
Often we assume that the chance error follows a normal curve that is centered at 0