7. The Range

11. Chapter Data Analysis
Learning Goal: To be able to describe the general shape of a distribution in terms of its number of modes, skewness, and variation.

12. Number of Modes
One way to describe the shape of a distribution is by its number of peaks, or modes.
Uniform distribution—has no mode because all data values have the same frequency.

13. Any peak is considered a mode, even if all peaks do not have the same height.
A distribution with a single peak is called a single-peaked, or unimodal, distribution.
A distribution with two peaks, even though not the same size, is a bimodal distribution.
What is the following distribution?

14. Symmetry or Skewness
A distribution is symmetric if its left half is a mirror image of its right half.
A symmetric distribution with a single peak and a bell shape is known as a normal distribution.

16. Symmetry or Skewness A distribution is left-skewed
(or negatively skewed) if the values
are more spread out on the left,
meaning that some low values are
likely to be outliers.
A distribution is right skewed
or positively skewed if the
values are more spread out
on the right. It has a tail
pulled toward the right.

24. Data can be "skewed", meaning it tends to have a long tail on one side or the other:

25. Example

26. Stem and Leaf Plot

29. 5a

30. 6 a,b

31. Box and Whisker Plot
A box-and-whisker plot shows the distribution of data. The middle half of the data is represented by a “box” with a vertical line at the median.
The box extends to the upper and lower quartiles.
The upper quartile is the median of the upper half of the data. The lower quartile is the median of the lower half of the data.
*It helps you to interpret and represent data.
*It gives a visual representation of data.

32. The box extends to the upper and lower quartiles.
A box-and-whisker plot shows the distribution of data. The middle half of the data is represented by a “box” with a vertical line at the median.
The box extends to the upper and lower quartiles.
The upper quartile is the median of the upper half of the data. The lower quartile is the median of the lower half of the data.
Data set: 85,92,78,88,90,88,89
Lower quartile
Median
Upper quartile
What is a box-and-whisker plot?
What are the upper and lower quartiles?

34. Example Back to Back S & L Plot

35. The first step to creating any Stem and Leaf Plot is to write our values out from lowest to highest, and group them into Tens; eg. ones, tens, twenties, thirties, etc

36. Stem and Leaf Plot for the “Brand A” Phone.

38. Stem and Leaf Plot for the “Brand B” phone

39. Two Stem and Leaf Plots and the final steps we need to take to combine them into a single plot.

44. Back-to back Stem and Leaf Plots

45. Variation
Variation describes how widely data are spread out about the center of a data set.
How would you expect the variation to differ between times in a 5K city run and a 5K run in a state meet?

46. What is the relationship between mean, median and mode of a left-skewed distribution?
Find the mean, median and mode of:
0, 5, 10, 20, 40, 45, 45, 50, 50, 50, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 70
The mean is 51.5.
The median is 60.
The mode is 70.

47. What is the relationship between mean, median and mode of a right-skewed distribution?
Find the mean, median, and mode of:
20, 20, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 45, 45, 45, 50, 50, 60, 70, 90
The mean is 36.1.
The median is 30.
The mode is 20.

48. House prices in the United States.
For each of the following situations, state whether you expect the distributions to be symmetric, left-skewed or right-skewed.
House prices in the United States.
Weight in a sample of 30 year old men.
The heights of all players in the NBA.

49. Copyright © 2009 Pearson Education, Inc.
Which is a better measure of “average” (or of the center of the distribution) for a skewed distribution:
the median or the mean? Why?
Page 160
Copyright © 2009 Pearson Education, Inc.

50. The body temperature of 2000 randomly selected college students
How many modes would you expect for each of the following distributions? Why? Make a rough sketch with clearly labeled axes?
The body temperature of 2000 randomly selected college students
The attendance at Disney World during a year
The last digit of your phone number

51. To summarize--
The general shape of a distribution can be discussed using:
The number of modes
Symmetry or skewness
Variation.

52. What is the relationship between mean, median and mode for a normal distribution?
Find the mean median and mode of:
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7
Mean is 4.
Median is 4.
Mode is 4.