1. Unit 4 Statistical Analysis Data Representations

2. 3 RULES of EXPLORATORY DATA ANALYSIS
MAKE A PICTURE – find patterns difficult to see in a chart
MAKE A PICTURE – show important features in graph
MAKE A PICTURE- communicates your data to others

3. Concepts to know! Bar graph Histogram Dot plot Stem leaf plot Boxplots
Scatter Plots

4. Categorical Data
The objects being studied are grouped into categories based on some qualitative trait.
The resulting data are merely
labels or categories.

5. Categorical Data (Single Variable)
Eye Color
BLUE
BROWN
GREEN
Frequency
(COUNTS) 20 50 5
Relative Frequency
20/75 =
.27
50/75=
.66
5/75=
.07

6. Pie Chart (Data is Counts or Percentages)

7. Bar Graph Summarizes categorical data.
Horizontal axis represents categories, while vertical axis represents either counts (“frequencies”) or percentages (“relative frequencies”).
Used to illustrate the differences in percentages (or counts) between categories.

8. Bar Graph (Shows distribution of data)
Bin Width

9. Contingency Table (How data is distributed across multiple variables)
Class
Survival
First
Second
Third
Crew
Total
ALIVE
203
118
178
212
711
DEAD
122
167
528
673
1490
325
285
706
885
2201

10. What can go wrong when working with categorical data?
Pay attention to the variables and what the percentages represent
(9.4% of passengers who were in first class survived is different from 67% of survivors were first class passengers!!!)
Make sure you have a reasonably large data set (67% of the rats tested died and 1 lived)
Marginal distribution

11. Bar chart is to categorical data as histogram is to ...
Analogy
Bar chart is to categorical data as histogram is to ...
quantitative data.

12. Histogram

13. Histogram
Divide measurement up into equal-sized categories (BIN WIDTH)
Determine number (or percentage) of measurements falling into each category.
Draw a bar for each category so bars’ heights represent number (or percent) falling into the categories.
Label and title appropriately.

14. Histogram Use common sense in determining number of categories to use.
Between 6 & 15 intervals is preferable
(Trial-and-error works fine, too.)

15. Too few categories

16. Too many categories

17. Dot Plot Summarizes quantitative data.
Horizontal axis represents measurement scale.
Plot one dot for each data point.

18. Dot Plot

19. Stem-and-Leaf Plot Summarizes quantitative data.
Each data point is broken down into a “stem” and a “leaf.”
First, “stems” are aligned in a column.
Then, “leaves” are attached to the stems.

20. High temperatures for the last week: 72, 78, 87, 90, 88, 86, 87, 89
Stem 7 8 9
Leaf
2 8
7 2 = 72 degrees

21. Box Plot Summarizes quantitative data.
Vertical (or horizontal) axis represents measurement scale.
Lines in box represent the 25th percentile (“first quartile”), the 50th percentile (“median”), and the 75th percentile (“third quartile”), respectively.

22. Box Plot

23. 5 Number Summary Minimum Q1 (25th percentile) Median (50th percentile)
Maximum

24. An aside... Roughly speaking:
The “25th percentile” is the number such that 25% of the data points fall below the number.
The “median” or “50th percentile” is the number such that half of the data points fall below the number.
The “75th percentile” is the number such that 75% of the data points fall below the number.

25. Using Box Plots to Compare
Outliers

26. Strengths and Weaknesses of Graphs for Quantitative Data
Histograms
Uses intervals
Good to judge the “shape” of a data
Not good for small data sets
Stem-Leaf Plots
Good for sorting data (find the median)
Not good for large data sets

27. Strengths and Weaknesses of Graphs for Quantitative Data
Dotplots
Uses individual data points
Good to show general descriptions of center and variation
Not good for judging shape for large data sets
Boxplots
Good for showing exact look at center, spread and outliers
Not good for judging shape or overall data analyses

28. Contingency table is to categorical data with two variables as
Analogy
Contingency table is to categorical data with two variables as
scatterplot is to ..
quantitative data with two variables.

29. Scatter Plots

30. Scatter Plots
Summarizes the relationship between two quantitative variables.
Horizontal axis represents one variable and vertical axis represents second variable.
Plot one point for each pair of measurements.

31. No relationship

32. Summary Many possible types of graphs.
Use common sense in reading graphs.
When creating graphs, don’t summarize your data too much or too little.
When creating graphs, label everything for others.
Remember you are trying to communicate something to others!

33. How to Compare Distributions
When you’re visualizing data, you have lots of options as to how we display it. If we are comparing data on the same type of graph, it is important we focus on the relevant qualities. In order to do this, we need to CUSS!

34. Center: The area where about half of the observations (data) are on either side.
Unusual features: Gaps (where there is no data) and outliers.
Spread: The variability of the data. If the data has a wide range, it has a larger spread. If the data has a narrow range, it has a smaller spread.
Shape: Described by symmetry, skewness, number of peaks, etc.

35. Center
The center is probably at about 4. We’d need to do some calculations to be more precise.
The center is at 5.

36. Unusual features
Gap – a space in the data, somewhat balanced on the sides
Outlier – a big gap with one, maybe two, pieces of data on the far side

37. Less Spread More Spread Spread
This measure mostly exists as a comparative, not an absolute.

38. Shape
Symmetrical
Uniform

39. Negative Skew left – it tails off to the left
Positive Skew right – it tails off to the right

40. Nonsymmetric – Skewed Negative -- Skewed Left

41. Nonsymmetric – Skewed Positive -Skewed Right

42. Symmetrical Bi-Modal – two peaks, but balanced on both peaks
Nonsymmetrical Bi-Modal – two peaks, but unbalanced on both peaks