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## Exploratory Data Analysis: One Variable

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Plan of attack

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

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

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?

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

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
- examples:
- 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

Example

- Age: quantitative
- Gender: categorical
- Race: categorical
- Salary: quantitative
- Job type: categorical

Variable types in JMP

- Qualitative/categorical
- JMP uses Nominal
- Quantitative
- Discrete
- JMP uses Ordinal
- Continuous
- JMP uses Continuous

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

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)

Summary table

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

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

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

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

Quantitative variables

- Graphical summary
- Histogram
- Stemplots
- Time plots
- more
- Numerical sumary
- Mean
- Median
- Quartiles
- Range
- Standard deviation
- more

Histograms

The bins are:

3.0 ≤ rate < 4.0

4.0 ≤ rate < 5.0

5.0 ≤ rate < 6.0

6.0 ≤ rate < 7.0

7.0 ≤ rate < 8.0

8.0 ≤ rate < 9.0

9.0 ≤ rate < 10.0

10.0 ≤ rate < 11.0

11.0 ≤ rate < 12.0

12.0 ≤ rate < 13.0

13.0 ≤ rate < 14.0

14.0 ≤ rate < 15.0

Histograms

The bins are:

3.0 ≤ rate < 4.0

4.0 ≤ rate < 5.0

5.0 ≤ rate < 6.0

6.0 ≤ rate < 7.0

7.0 ≤ rate < 8.0

8.0 ≤ rate < 9.0

9.0 ≤ rate < 10.0

10.0 ≤ rate < 11.0

11.0 ≤ rate < 12.0

12.0 ≤ rate < 13.0

13.0 ≤ rate < 14.0

14.0 ≤ rate < 15.0

Histograms

The bins are:

2.0 ≤ rate < 4.0

4.0 ≤ rate < 6.0

6.0 ≤ rate < 8.0

8.0 ≤ rate < 10.0

10.0 ≤ rate < 12.0

12.0 ≤ rate < 14.0

14.0 ≤ rate < 16.0

16.0 ≤ rate < 18.0

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

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

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

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

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

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

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

Mean

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

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?

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

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.
- Does the median age change?
- If so, what is the new median age?

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?

Mean vs Median

- Symmetric distribution

Mean vs Median

- Right skewed distribution

Mean vs Median

- Left skewed distribution

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

$25,000 $27,000 $29,000

$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

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?

Measures of spread

- 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?

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

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?

Problem from text (p. 74, #2)

- Which of the following sets of numbers has the smaller SD’

a) 50, 40, 60, 30, 70, 25, 75b) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50

- Repeat for these two sets

c) 50, 40, 60, 30, 70, 25, 75d) 50, 40, 60, 30, 70, 25, 75, 99, 1

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.

Properties of SD

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

Mean and SD

- What happens to the mean if you add 5 to every number in a list?
- What happens to the SD?

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?

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)

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

Properties of normal curves

- For a N(0,1) the following holds

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

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

More IQ

- Reggie Jackson, one of the greatest baseball players ever, has an IQ of 140. What percentage of people have bigger IQs than Reggie?
- Marilyn vos Savant, self-proclaimed smartest person in the world, has a reported IQ of 205. 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?

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

Measurement error

- Measurement error model
- 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

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