1. EXPLORING DATA WITH GRAPHS AND NUMERICAL SUMMARIES
Math Chapter 2
EXPLORING DATA WITH GRAPHS AND NUMERICAL SUMMARIES
Chapter 2

3. 2.1 What Are the Types of Data?

4. Variable
A variable is any characteristic that is recorded for the subjects in a study
Examples: Marital status, Height, Weight, IQ
A variable can be classified as either
Categorical or
Quantitative
Discrete or
Continuous

5. Categorical Variable
A variable is categorical if each observation belongs to one of a set of categories.
Examples:
Gender (Male or Female)
Religion (Catholic, Jewish, …)
Type of residence (Apt, Condo, …)
Belief in life after death (Yes or No)

6. Quantitative Variable
A variable is called quantitative if observations take numerical values for different magnitudes of the variable.
Examples:
Age
Number of siblings
Annual Income

7. Quantitative vs. Categorical
Math Chapter 2
Quantitative vs. Categorical
For Quantitative variables, key features are the center (a representative value) and spread (variability).
For Categorical variables, a key feature is the percentage of observations in each of the categories .

8. Discrete Quantitative Variable
A quantitative variable is discrete if its possible values form a set of separate numbers: 0,1,2,3,….
Examples:
Number of pets in a household
Number of children in a family
Number of foreign languages spoken by an individual
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9. Continuous Quantitative Variable
A quantitative variable is continuous if its possible values form an interval
Measurements
Examples:
Height/Weight
Age
Blood pressure

10. Proportion & Percentage (Rel. Freq.)
Proportions and percentages are also called relative frequencies.

11. Math Chapter 2
Frequency Table
A frequency table is a listing of possible values for a variable, together with the number of observations or relative frequencies for each value.

12. 2.2 Describe Data Using Graphical Summaries

13. Graphs for Categorical Variables
Use pie charts and bar graphs to summarize categorical variables
Pie Chart: A circle having a “slice of pie” for each category
Bar Graph: A graph that displays a vertical bar for each category
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14. Pie Charts Summarize categorical variable
Drawn as circle where each category is a slice
The size of each slice is proportional to the percentage in that category

15. Bar Graphs Summarizes categorical variable
Vertical bars for each category
Height of each bar represents either counts or percentages
Easier to compare categories with bar graph than with pie chart
Called Pareto Charts when ordered from tallest to shortest

16. Graphs for Quantitative Data
Math Chapter 2
Graphs for Quantitative Data
Dot Plot: shows a dot for each observation placed above its value on a number line
Stem-and-Leaf Plot: portrays the individual observations
Histogram: uses bars to portray the data

17. Which Graph? Dot-plot and stem-and- leaf plot: Histogram
Math Chapter 2
Which Graph?
Dot-plot and stem-and- leaf plot:
More useful for small data sets
Data values are retained
Histogram
More useful for large data sets
Most compact display
More flexibility in defining intervals
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18. Dot Plots To construct a dot plot Draw and label horizontal line
Mark regular values
Place a dot above each value on the number line
Sodium in Cereals

19. Stem-and-leaf plots Summarizes quantitative variables
Separate each observation into a stem (first part of #) and a leaf (last digit)
Write each leaf to the right of its stem; order leaves if desired
Sodium in Cereals

20. Histograms
Graph that uses bars to portray frequencies or relative frequencies of possible outcomes for a quantitative variable

21. Constructing a Histogram
Sodium in Cereals
Divide into intervals of equal width
Count # of observations in each interval

22. Constructing a Histogram
Math Chapter 2
Constructing a Histogram
Label endpoints of intervals on horizontal axis
Draw a bar over each value or interval with height equal to its frequency (or percentage)
Label and title
Sodium in Cereals

23. Interpreting Histograms
Assess where a distribution is centered by finding the median
Assess the spread of a distribution
Shape of a distribution: roughly symmetric, skewed to the right, or skewed to the left
Left and right sides are mirror images

24. Examples of Skewness

25. Math Chapter 2
Shape and Skewness
Consider a data set containing IQ scores for the general public. What shape?
Symmetric
Skewed to the left
Skewed to the right
Bimodal
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26. Shape and Skewness
Consider a data set of the scores of students on an easy exam in which most score very well but a few score poorly. What shape?
Symmetric
Skewed to the left
Skewed to the right
Bimodal

27. Shape: Type of Mound

28. Outlier
An outlier falls far from the rest of the data

29. Time Plots Display a time series, data collected over time
Plots observation on the vertical against time on the horizontal
Points are usually connected
Common patterns should be noted
Time Plot from 1995 – of the # worldwide who use the Internet

30. 2.3 Describe the Center of Quantitative Data

31. Mean
The mean is the sum of the observations divided by the number of observations
It is the center of mass

32. Median
Midpoint of the observations when ordered from least to greatest
Order observations
If the number of observations is:
Odd, the median is the middle observation
Even, the median is the average of the two middle observations

33. Comparing the Mean and Median
Mean and median of a symmetric distribution are close
Mean is often preferred because it uses all
In a skewed distribution, the mean is farther out in the skewed tail than is the median
Median is preferred because it is better representative of a typical observation

34. Resistant Measures
A measure is resistant if extreme observations (outliers) have little, if any, influence on its value
Median is resistant to outliers
Mean is not resistant to outliers

35. Mode Value that occurs most often Highest bar in the histogram
Mode is most often used with categorical data

36. 2.4 Describe the Spread of Quantitative Data

37. Range Range = max - min The range is strongly affected by outliers.
Math Chapter 2
Range
Range = max - min
The range is strongly affected by outliers.

38. Math Chapter 2
Standard Deviation
Each data value has an associated deviation from the mean,
A deviation is positive if it falls above the mean and negative if it falls below the mean
The sum of the deviations is always zero

39. Math Chapter 2
Standard Deviation
Standard deviation gives a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations:
Find mean
Find each deviation
Square deviations
Sum squared deviations
Divide sum by n-1
Take square root

40. Standard Deviation Metabolic rates of 7 men (calories/24 hours)
Math Chapter 2
Standard Deviation
Metabolic rates of 7 men (calories/24 hours)

41. Properties of Sample Standard Deviation
Measures spread of data
Only zero when all observations are same; otherwise, s > 0
As the spread increases, s gets larger
Same units as observations
Not resistant
Strong skewness or outliers greatly increase s

42. Empirical Rule: Magnitude of s
Math Chapter 2
Empirical Rule: Magnitude of s

43. 2.5 How Measures of Position Describe Spread

44. Percentile
The pth percentile is a value such that p percent of the observations fall below or at that value

45. Finding Quartiles Splits the data into four parts
Math Chapter 2
Finding Quartiles
Splits the data into four parts
Arrange data in order
The median is the second quartile, Q2
Q1 is the median of the lower half of the observations
Q3 is the median of the upper half of the observations

46. Measure of Spread: Quartiles
Math Chapter 2
Measure of Spread: Quartiles
Quartiles divide a ranked data set into four equal parts:
25% of the data at or below Q1 and 75% above
50% of the obs are above the median and 50% are below
75% of the data at or below Q3 and 25% above
Q1= first quartile = 2.2
M = median = 3.4
We are going to start out with a very general way to describe the spread that doesn’t matter whether it is symmetric or not - quartiles.
Just as the word suggests - quartiles is like quarters or quartets, it involves dividing up the distribution into 4 parts.
Now, to get the median, we divided it up into two parts. To get the quartiles we do the exact same thing to the two halves.
Use same rules as for median if you have even or odd number of observations.
Now, what an we do with these that helps us understand the biology of these diseases?
Q3= third quartile = 4.35

47. Calculating Interquartile Range
The interquartile range is the distance between the thirdand first quartile, giving spread of middle 50% of the data: IQR = Q3 - Q1

48. Criteria for Identifying an Outlier
An observation is a potential outlier if it falls more than 1.5 x IQR below the first or more than 1.5 x IQR above the third quartile.

49. 5 Number Summary The five-number summary of a dataset consists of:
Minimum value
First Quartile
Median
Third Quartile
Maximum value

50. Boxplot Box goes from the Q1 to Q3
Line is drawn inside the box at the median
Line goes from lower end of box to smallest observation not a potential outlier and from upper end of box to largest observation not a potential outlier
Potential outliers are shown separately, often with * or +

51. Comparing Distributions
Boxplots do not display the shape of the distribution as clearly as histograms, but are useful for making graphical comparisons of two or more distributions

52. Math Chapter 2
Z-Score
An observation from a bell-shaped distribution is a potential outlier if its z-score < -3 or > +3

53. 2.6 How Can Graphical Summaries Be Misused?

54. Misleading Data Displays

55. Guidelines for Constructing Effective Graphs
Label axes and give proper headings
Vertical axis should start at zero
Use bars, lines, or points
Consider using separate graphs or ratios when variable values differ