1. Dotplots Horizontal axis with either quanitative scale or categories
Each dot represent one data point
If more than one data point has the same value, “stack” the dots
CASA, AP Stats 05-06, Section 1.1.1

2. Stemplots
Each data point is represented by a single number (leaf) associated with a more significant number (stem)
Rounding can help the shape of the distribution become more noticeable.
CASA, AP Stats 05-06, Section 1.1.1

3. Histograms
Each row of a stemplot represents a number of data points found in a range. What kind of ranges might a stemplot have?
Histograms have the same general idea, but histograms are more flexible in terms of the ranges.
Ranges have equal width
CASA, AP Stats 05-06, Section 1.1.1

4. Histograms
Histograms may be less useful than stemplots because you can’t see the actual data.
CASA, AP Stats 05-06, Section 1.1.1

5. Histograms
Ranges for histograms generally include the lower end of the range, but does not include the upper end of the range
CASA, AP Stats 05-06, Section 1.1.1

6. Using the TI-83 to make histograms
The TI-83 can be used to make histograms, and will allow you to change the location and widths of the ranges.
CASA, AP Stats 05-06, Section 1.1.1

7. Using the TI-83 to make histograms
Start by entering data into a list
Example: Enter the presidential data from page 19 into any list; in this case, we will use L1
CASA, AP Stats 05-06, Section 1.1.1

8. Using the TI-83 to make histograms
Choose 2nd:Stat Plot to choose a histogram plot
Caution: Watch out for other plots that might be “turned on” or equations that might be graphed
CASA, AP Stats 05-06, Section 1.1.1

9. Using the TI-83 to make histograms
Turn the plot “on”,
Choose the histogram plot.
X-list should point to the location of the data.
CASA, AP Stats 05-06, Section 1.1.1

10. Using the TI-83 to make histograms
Under the “Zoom” menu, choose option 9: ZoomStat
CASA, AP Stats 05-06, Section 1.1.1

11. Using the TI-83 to make histograms
The result is a histogram where the calculator has decided the width and location of the ranges
You can use the Trace key to get information about the ranges and the frequencies
CASA, AP Stats 05-06, Section 1.1.1

12. Using the TI-83 to make histograms
You can change the size and location of the ranges by using the Window button
Press the Graph button to see the results
CASA, AP Stats 05-06, Section 1.1.1

13. Using the TI-83 to make histograms
Voila!
Of course, you can still change the ranges if you don’t like the results.
CASA, AP Stats 05-06, Section 1.1.1

14. Using Percents
It is sometimes difficult to compare the “straight” numbers. For example:
Ty Cobb in his 24 year baseball career had 4,189 hits. Pete Rose also played for 24 years but he collected 4,256 hits. Was Pete Rose a better batter than Ty Cobb?
CASA, AP Stats 05-06, Section 1.1.1

15. Using Percents
Ty Cobb had a career batting average of (He got a hit 36.6% of the time)
Pete Rose had a career batting average of .303.
Comparing the percentage, we get a clearer picture.
CASA, AP Stats 05-06, Section 1.1.1

16. Relative Frequency
When constructing a histogram we can use the “relative frequency” (given in percent) instead of “count” or “frequency”
Using relative frequency allows us to do better comparisons.
Histograms using relative frequency have the same shape as those using count.
CASA, AP Stats 05-06, Section 1.1.1

17. Finding Relative Frequency
For each count in a class, divide by the total number of data points in the data set.
Convert to a percentage.
CASA, AP Stats 05-06, Section 1.1.1

18. Finding Relative Frequency
Class
Frequency
40-44 2
45-49 6
50-54 13
55-59 12
60-64 7
65-69 3
Total 43
CASA, AP Stats 05-06, Section 1.1.1

19. Finding Relative Frequency
Class
Frequency
Relative Frequency
40-44 2
2/43=4.7%
45-49 6
6/43=14.0%
50-54 13
13/43=30.2%
55-59 12
12/43=27.9%
60-64 7
7/43=16.3%
65-69 3
3/43=7.0%
Total 43
CASA, AP Stats 05-06, Section 1.1.1

20. Histograms
CASA, AP Stats 05-06, Section 1.1.1

21. Finding Cumulative Frequency
Class
Frequency
Relative Frequency
Cumulative Frequency
40-44 2
2/43=4.7%
45-49 6
6/43=14.0% 8
50-54 13
13/43=30.2% 21
55-59 12
12/43=27.9% 33
60-64 7
7/43=16.3% 40
65-69 3
3/43=7.0% 43
Total
CASA, AP Stats 05-06, Section 1.1.1

22. Finding Relative Cumulative Frequency
Class
Frequency
Relative Frequency
Cumulative Frequency
Relative Cumulative Frequency
40-44 2
2/43=4.7%
45-49 6
6/43=14.0% 8
8/43=18.6%
50-54 13
13/43=30.2% 21
21/43=48.8%
55-59 12
12/43=27.9% 33
33/43=76.7%
60-64 7
7/43=16.3% 40
40/43=93.0%
65-69 3
3/43=7.0% 43
43/43=100%
Total
CASA, AP Stats 05-06, Section 1.1.1

23. Percentiles
“The p-th percentile of a distribution is the value such that p percent of the observations fall at or below it.”
If you scored in the 80th percentile on the SAT, then 80% of all test takers are at or below your score.
CASA, AP Stats 05-06, Section 1.1.1

24. Relative Cumulative Frequency
Percentiles
Class
Relative Cumulative Frequency
40-44
2/43=4.7%
45-49
8/43=18.6%
50-54
21/43=48.8%
55-59
33/43=76.7%
60-64
40/43=93.0%
65-69
43/43=100%
Total
It is easy to see the percentiles at the breaks.
“A 64 year old would be at the 93rd percentile.”
What do you do for a 57 year old?
CASA, AP Stats 05-06, Section 1.1.1

25. Ogives or “Relative Cumulative Frequency Graph”
Class
Relative Cumulative Frequency
40-44
2/43=4.7%
45-49
8/43=18.6%
50-54
21/43=48.8%
55-59
33/43=76.7%
60-64
40/43=93.0%
65-69
43/43=100%
Total
CASA, AP Stats 05-06, Section 1.1.1

26. AP Statistics September 3, 2014 Mr. Calise
Section 1.2 Part 1
AP Statistics
September 3, 2014
Mr. Calise

27. Describing Distributions with Numbers: Center/Mean
The Mean
The Average
The Arithmetical Mean
The mean is not a resistance measure of center
AP Statistics, Section 1.2, Part 1

28. Describing Distributions with Numbers: Center/Median
When numbers are ordered from low to high, the median is the middle number (if n is odd) or the average of the two middle numbers (if n is even)
Resistant measure of center
AP Statistics, Section 1.2, Part 1

29. Describing Distributions with Numbers: Spread/IQR
The first quartile (Q1) is the median of the first half of the distribution
The third quartile (Q3) is the median of the second half of the distribution
The interquartile range (IQR) is the distance between the first and third quartiles. (IQR = Q3 – Q1)
AP Statistics, Section 1.2, Part 1

30. AP Statistics, Section 1.2, Part 1
Finding M, Q1, Q3, IQR
Barry Bonds’ home run counts per season (average=35.4): 16, 19, 24, 25, 25, 33, 33, 34, 34, 37, 37, 40, 42, 46, 49, 73
Hank Aaron’s home run counts per season (average=34.9): 13, 20, 24, 26, 27, 29, 30, 32, 34, 34, 38, 39, 39, 40, 40, 44, 44, 44, 44, 45, 47
AP Statistics, Section 1.2, Part 1

31. AP Statistics, Section 1.2, Part 1
Finding M, Q1, Q3, IQR
Barry Bonds: Q1=25, M=34, Q3=41, IQR=16
Hank Aaron: Q1=28, M=38, Q3=44, IQR=16
What does this say about comparing Bonds and Aaron?
AP Statistics, Section 1.2, Part 1

32. Describing Distributions with Numbers: Outliers
If a data point falls more than 1.5 * IQR above the Q3 or falls less than 1.5 * IQR below the Q1.
Barry Bonds: M=34, Q1=25, Q3=41, IQR=16
* 16 = 65 (anything above 65)
* 16 = 1 (anything below 1)
AP Statistics, Section 1.2, Part 1

33. AP Statistics, Section 1.2, Part 1
Side by Side Boxplots
AP Statistics, Section 1.2, Part 1

34. AP Statistics, Section 1.2, Part 1
Modified Boxplot
AP Statistics, Section 1.2, Part 1

35. Describing with Numbers: Spread/Standard Deviation
s2 is called the variance, and its in units2
s is called the standard deviation, and its in units.
AP Statistics, Section 1.2, Part 1

36. Describing with Numbers: Spread/Standard Deviation
s measures spread about the mean and should be used only when the mean is chosen as the measure of center
s equals 0 only when there is no spread. This happens only when all observations have the same value.
s is not resistant. Outliers effect it adversely.
AP Statistics, Section 1.2, Part 1

37. AP Statistics, Section 1.2, Part 1
Calculating s by hand
Observation
(x-bar)
Deviation
(x-sub-i – x-bar)
Squared Deviation
1792
1666
1362
1614
1460
1867
1439
Sum = 11200
Mean = 1600
AP Statistics, Section 1.2, Part 1

38. AP Statistics, Section 1.2, Part 1
Calculating s by hand
Observation
(x-bar)
Deviation
(x-sub-i – x-bar)
Squared Deviation
1792
= 192
1666
=66
1362
=-238
1614
=14
1460
=-140
1867
=267
1439
=-161
Sum = 11200
Mean = 1600
Sum = 0
AP Statistics, Section 1.2, Part 1

39. AP Statistics, Section 1.2, Part 1
Calculating s by hand
Observation
(x-bar)
Deviation
(x-sub-i – x-bar)
Squared Deviation
1792
= 192
192 * 192 = 36864
1666
=66
4356
1362
=-238
56644
1614
=14
196
1460
=-140
36864
1867
=267
71289
1439
=-161
25921
Sum = 11200
Mean = 1600
Sum = 0
Sum =
AP Statistics, Section 1.2, Part 1

40. AP Statistics, Section 1.2, Part 1
Calculating s by hand
AP Statistics, Section 1.2, Part 1

41. Calculating s using TI-83
AP Statistics, Section 1.2, Part 1

42. Calculating s using TI-83
AP Statistics, Section 1.2, Part 1

43. Calculating s using TI-83
AP Statistics, Section 1.2, Part 1

44. Linear Transformations
Adding “a” does not change the shape or spread
Adding “a” does change the center by “a”
AP Statistics, Section 1.2, Part 1

45. Linear Transformations
Multiplying by “b” does change the shape or spread by “b”
Multiplying by “b” does change the center by “b”
AP Statistics, Section 1.2, Part 1

46. Comparing Distributions
We have already seen the usefulness of comparing two distribution in the answering difficult questions like:
“Who was a better home run hitter, Barry Bonds or Hank Aaron?”
We used side-by-side boxplots to answer that question.
AP Statistics, Section 1.2, Part 1

47. Back-to-Back Stemplots
We can use stemplots as a way of doing comparisons between two distributions.
For example, which baseball league is more competitive, the American League or the National League?
AP Statistics, Section 1.2, Part 1

48. Back-To-Back Stemplots
AP Statistics, Section 1.2, Part 1

49. Back-To-Back Stemplots
AP Statistics, Section 1.2, Part 1

50. AP Statistics, Section 1.2, Part 1
Reminders
If there are outliers, don’t use mean or standard deviation to compare distributions.
You can’t compare a standard deviation of one distribution to the IQR of a different distribution.
You can’t compare a mean of one distribution to the median of a different distribution.
AP Statistics, Section 1.2, Part 1

51. AP Statistics, Section 1.2, Part 1
Homework Assignment
Exercises 1.31 (a) & (b)
1.33, 1.34, 1.35
1.36 (a)
AP Statistics, Section 1.2, Part 1