1. Methods for Describing Sets of Data
Chapter 2
Methods for Describing Sets of Data
Slides for Optional Sections
Section 2.8 Methods for Detecting Outliers Slides 31-34
Section 2.9 Graphing Bivariate Relationships Slide 35
Section The Time Series Plot Slide 36

2. Objectives
Describe Data using Graphs
Describe Data using Charts

3. Describing Qualitative Data
Qualitative data are nonnumeric in nature
Best described by using Classes
2 descriptive measures
class frequency – number of data points in a class
class relative = class frequency
frequency total number of data points in data set
class percentage – class relative freq. x 100
Add discussion of class percentage

4. Describing Qualitative Data – Displaying Descriptive Measures
Summary Table
Class
Frequency
Class percentage – class relative frequency x 100

5. Describing Qualitative Data – Qualitative Data Displays
Bar Graph

6. Describing Qualitative Data – Qualitative Data Displays
Pie chart

7. Describing Qualitative Data – Qualitative Data Displays
Pareto Diagram

8. Graphical Methods for Describing Quantitative Data
The Data

9. Graphical Methods for Describing Quantitative Data
For describing, summarizing, and detecting patterns in such data, we can use three graphical methods:
dot plots
stem-and-leaf displays
histograms

10. Graphical Methods for Describing Quantitative Data
Dot Plot

11. Graphical Methods for Describing Quantitative Data
Stem-and-Leaf Display

12. Graphical Methods for Describing Quantitative Data
Histogram

13. Graphical Methods for Describing Quantitative Data
More on Histograms
Number of Observations in Data Set
Number of Classes
Less than 25
5-6
25-50
7-14
More than 50
15-20

14. Summation Notation Used to simplify summation instructions
Each observation in a data set is identified by a subscript
x1, x2, x3, x4, x5, …. xn
Notation used to sum the above numbers together is

15. Summation Notation
Data set of 1, 2, 3, 4
Are these the same? and

16. Numerical Measures of Central Tendency
Central Tendency – tendency of data to center about certain numerical values
3 commonly used measures of Central Tendency:
Mean
Median
Mode

17. Numerical Measures of Central Tendency
The Mean
Arithmetic average of the elements of the data set
Sample mean denoted by
Population mean denoted by
Calculated as and

18. Numerical Measures of Central Tendency
The Median
Middle number when observations are arranged in order
Median denoted by m
Identified as the observation if n is odd, and the mean of the and observations if n is even

19. Numerical Measures of Central Tendency
The Mode
The most frequently occurring value in the data set
Data set can be multi-modal – have more than one mode
Data displayed in a histogram will have a modal class – the class with the largest frequency

20. Numerical Measures of Central Tendency
The Data set
Mean
Median is the or 5th observation, 8
Mode is 8

21. Numerical Measures of Variability
Variability – the spread of the data across possible values
3 commonly used measures of Variability:
Range
Variance
Standard Deviation

22. Numerical Measures of Variability
The Range
Largest measurement minus the smallest measurement
Loses sensitivity when data sets are large
These 2 distributions have the same range.
How much does the range tell you about the data variability?

23. Numerical Measures of Variability
The Sample Variance (s2)
The sum of the squared deviations from the mean divided by (n-1). Expressed as units squared
Why square the deviations? The sum of the deviations from the mean is zero

24. Numerical Measures of Variability
The Sample Standard Deviation (s)
The positive square root of the sample variance
Expressed in the original units of measurement

25. Numerical Measures of Variability
Samples and Populations - Notation
Sample
Population
Variance s2
Standard Deviation s

26. Numerical Measures of Relative Standing
Descriptive measures of relationship of a measurement to the rest of the data
Common measures:
percentile ranking
z-score

27. Numerical Measures of Relative Standing
Percentile rankings make use of the pth percentile
The median is an example of percentiles.
Median is the 50th percentile – 50 % of observations lie above it, and 50% lie below it
For any p, the pth percentile has p% of the measures lying below it, and (100-p)% above it

28. Numerical Measures of Relative Standing
z-score – the distance between a measurement x and the mean, expressed in standard units
Use of standard units allows comparison across data sets

29. Numerical Measures of Relative Standing
More on z-scores
Z-scores follow the empirical rule for mounded distributions

30. Methods for Detecting Outliers
Outlier – an observation that is unusually large or small relative to the data values being described
Causes:
Invalid measurement
Misclassified measurement
A rare (chance) event
2 detection methods:
Box Plots
z-scores

31. Methods for Detecting Outliers
Box Plots
based on quartiles, values that divide the dataset into 4 groups
Lower Quartile QL – 25th percentile
Middle Quartile - median
Upper Quartile QU – 75th percentile
Interquartile Range (IQR) = QU - QL

32. Methods for Detecting Outliers
Box Plots
Not on plot – inner and outer fences, which determine potential outliers
QU (hinge)
QL (hinge)
Median
Potential Outlier
Whiskers

33. Methods for Detecting Outliers
Rules of thumb
Box Plots
measurements between inner and outer fences are suspect
measurements beyond outer fences are highly suspect
Z-scores
Scores of 3 in mounded distributions (2 in highly skewed distributions) are considered outliers

34. Graphing Bivariate Relationships
Bivariate relationship – the relationship between two quantitative variables
Graphically represented with the scattergram

35. The Time Series Plot
Time Series Data – data produced and monitored over time
Graphically represented with the time series plot
Time on x axis
Order on x axis

36. Summary Graphical methods for Qualitative Data
Pie chart
Bar graph
Pareto diagram
Graphical methods for Quantitative Data
Dot plot
Stem-and-leaf display
Histogram

37. Summary Numerical measures of central tendency
Mean
Median
Mode
Numerical measures of variation
Range
Variance
Standard Deviation

38. Summary Measures of relative standing Methods for detecting Outliers
Percentile ranking
z-scores
Methods for detecting Outliers
Box plots
Method for graphing the relationship between two quantitative variables
Scatterplot