1. STA 291 Spring 2008
Lecture 3
Dustin Lueker

2. Stratified Sampling
Suppose the population can be divided into separate, non-overlapping groups (“strata”) according to some criterion
Select a simple random sample independently from each group
Usefulness
We may want to draw inference about population parameters for each subgroup
Sometimes, (“proportional stratified sample”) estimators from stratified random samples are more precise than those from simple random samples
STA 291 Spring 2008 Lecture 3 2

3. Proportional Stratification
The proportions of the different strata are the same in the sample as in the population
Mathematically
Population size N
Subpopulation Ni
Sample size n
Subpopulation ni
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4. Descriptive Statistics
Summarize data
Condense the information from the dataset
Graphs
Table
Numbers
Interval data
Histogram
Nominal/Ordinal data
Bar chart
Pie chart
STA 291 Spring 2008 Lecture 3

5. Data Table: Murder Rates
Alabama
Alaska
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
D C
Florida
Georgia
Hawaii …
Difficult to see the “big picture” from these numbers
We want to try to condense the data
STA 291 Spring 2008 Lecture 3

6. Frequency Distribution
A listing of intervals of possible values for a variable
Together with a tabulation of the number of observations in each interval.
STA 291 Spring 2008 Lecture 3

7. Frequency Distribution
Murder Rate
Frequency
0-2.9 5
3-5.9 16
6-8.9 12
9-11.9 4 1
>21
Total 51
STA 291 Spring 2008 Lecture 3

8. Frequency Distribution
Conditions for intervals
Equal length
Mutually exclusive
Any observation can only fall into one interval
Collectively exhaustive
All observations fall into an interval
Rule of thumb:
If you have n observations then the number of intervals should approximately
STA 291 Spring 2008 Lecture 3

9. Relative Frequencies Relative frequency for an interval
Proportion of sample observations that fall in that interval
Sometimes percentages are preferred to relative frequencies
STA 291 Spring 2008 Lecture 3

10. Frequency, Relative Frequency, and Percentage Distribution
Murder Rate
Frequency
Relative Frequency
Percentage
0-2.9 5
.10 10
3-5.9 16
.31 31
6-8.9 12
.24 24
9-11.9 4
.08 8 1
.02 2
>21
Total 51
100
STA 291 Spring 2008 Lecture 3

11. Frequency Distributions
Notice that we had to group the observations into intervals because the variable is measured on a continuous scale
For discrete data, grouping may not be necessary
Except when there are many categories
Intervals are sometimes called classes
Class Cumulative Frequency
Number of observations that fall in the class and in smaller classes
Class Relative Cumulative Frequency
Proportion of observations that fall in the class and in smaller classes
STA 291 Spring 2008 Lecture 3

12. Frequency and Cumulative Frequency
Murder Rate
Frequency
Relative Frequency
Cumulative
Relative
Cumulative Frequency
0-2.9 5
.10
3-5.9 16
.31 21
.41
6-8.9 12
.24 33
.65
9-11.9 45
.89 4
.08 49
.97 1
.02 50
.99
>21 51
Total
STA 291 Spring 2008 Lecture 3

13. Histogram (Interval Data)
Use the numbers from the frequency distribution to create a graph
Draw a bar over each interval, the height of the bar represents the relative frequency for that interval
Bars should be touching
Equally extend the width of the bar at the upper and lower limits so that the bars are touching.
STA 291 Spring 2008 Lecture 3

14. Histogram
STA 291 Spring 2008 Lecture 3

15. Histogram w/o DC
STA 291 Spring 2008 Lecture 3

16. Bar Graph (Nominal/Ordinal Data)
Histogram: for interval (quantitative) data
Bar graph is almost the same, but for qualitative data
Difference:
The bars are usually separated to emphasize that the variable is categorical rather than quantitative
For nominal variables (no natural ordering), order the bars by frequency, except possibly for a category “other” that is always last
STA 291 Spring 2008 Lecture 3

17. Pie Chart (Nominal/Ordinal Data)
First Step
Create a frequency distribution
Highest Degree Obtained
Frequency
(Number of Employees)
Grade School 15
High School
200
Bachelor’s
185
Master’s 55
Doctorate 70
Other 25
Total
550
STA 291 Spring 2008 Lecture 3

18. We could display this data in a bar chart…
Bar graph
If the data is ordinal, classes are presented in the natural ordering
STA 291 Spring 2008 Lecture 3

19. Pie Chart Pie is divided into slices
Area of each slice is proportional to the frequency of each class
STA 291 Spring 2008 Lecture 3

20. Pie Chart for Highest Degree Achieved
STA 291 Spring 2008 Lecture 3

21. Sample/Population Distribution
Frequency distributions and histograms exist for the population as well as for the sample
Population distribution vs. sample distribution
As the sample size increases, the sample distribution looks more and more like the population distribution
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22. Population Distribution
The population distribution for a continuous variable is usually represented by a smooth curve
Like a histogram that gets finer and finer
Similar to the idea of using smaller and smaller rectangles to calculate the area under a curve when learning how to integrate
Symmetric distributions
Bell-shaped
U-shaped
Uniform
Not symmetric distributions:
Left-skewed
Right-skewed
Skewed
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23. Skewness Symmetric Right-skewed Left-skewed
STA 291 Spring 2008 Lecture 3

24. Summary of Graphical and Tabular Techniques
Discrete data
Frequency distribution
Continuous data
Grouped frequency distribution
Small data sets
Stem and leaf plot
Interval data
Histogram
Categorical data
Bar chart
Pie chart
Grouping intervals should be of same length, but may be dictated more by subject-matter considerations
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25. Good Graphics Present large data sets concisely and coherently
Can replace a thousand words and still be clearly understood and comprehended
Encourage the viewer to compare two or more variables
Do not replace substance by form
Do not distort what the data reveal
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26. Bad Graphics Don’t have a scale on the axis Have a misleading caption
Distort by using absolute values where relative/proportional values are more appropriate
Distort by stretching/shrinking the vertical or horizontal axis
Use bar charts with bars of unequal width
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