1. Chapter 2: Descriptive Statistics
Lesson 2.1: Frequency Distributions and their Graphs (Part 1)

2. Frequency Distributions
A frequency distribution is a table that shows the classes or intervals of data entries with the number of entries in each class.
Example: 20, 21, 23, 25, 25,
27, 27, 27, 31, 33, 35, 35,
35, 38, 39, 39, 39, 40, 42, 42
All the classes MUST have the
same class width. Here the
class width is 5.
Class
Frequency
20-24 3
25-29 5
30-34 2
35-39 7
40-44

3. Relative Frequency
Relative frequency: frequency of the class divided by the total frequency (or sample size)
Class
Frequency
Relative frequency
Cumulative frequency
Class Boundaries
Class Midpoint
20-24 3
3/20 = 0.15
25-29 5
5/20 = 0.25
30-34 2
2/20 = 0.1
35-39 7
7/20 = 0.35
40-44

4. Cumulative Frequency
Cumulative frequency: the sum of the frequencies of that class and all previous classes
Class
Frequency
Relative frequency
Cumulative frequency
Class Boundaries
Class Midpoint
20-24 3
3/20 = 0.15
25-29 5
5/20 = 0.25
3+5 = 8
30-34 2
2/20 = 0.1
8+2 = 10
35-39 7
7/20 = 0.35
10+7 = 17
40-44
17+3 = 20

5. Class Boundaries
Class boundaries: The numbers that separate classes without forming gaps between them. If the class values are integers subtract 0.5 from the lower class value and add 0.5 to the upper class value.
Class
Frequency
Relative frequency
Cumulative frequency
Class Boundaries
Class Midpoint
20-24 3
3/20 = 0.15
19.5 – 24.5
25-29 5
5/20 = 0.25
3+5 = 8
24.5 – 29.5
30-34 2
2/20 = 0.1
8+2 = 10
29.5 – 34.5
35-39 7
7/20 = 0.35
10+7 = 17
34.5 – 39.5
40-44
17+3 = 20
39.5 – 44.5

6. Class Midpoint
Class midpoint: The middle of the class. This can be found by averaging the lower and upper class values
Class
Frequency
Relative frequency
Cumulative frequency
Class Boundaries
Class Midpoint
20-24 3
3/20 = 0.15
19.5 – 24.5 22
25-29 5
5/20 = 0.25
3+5 = 8
24.5 – 29.5 27
30-34 2
2/20 = 0.1
8+2 = 10
29.5 – 34.5 32
35-39 7
7/20 = 0.35
10+7 = 17
34.5 – 39.5 37
40-44
17+3 = 20
39.5 – 44.5 42

7. Chapter 2: Descriptive Statistics
Lesson 2.1: Frequency Distributions and their Graphs (Part 2)

8. Frequency Histograms Class ƒ Boundaries 67-78 3 66.5-78.5 79-90 5
91-102 8 9
Class ƒ
Midpoint
67-78 3
72.5
79-90 5
84.5
91-102 8
96.5 9
108.5
120.5

9. Relative Frequency Histograms
Time on Phone
Relative frequency
minutes
Class
Frequency
Relative Frequency
Cumulative Frequency
67-78 3
0.10
79-90 5
0.17 8
91-102
0.27 16 9
0.30 25 30

10. Frequency Polygons How to construct a frequency Polygon:
Time on Phone 9
Class ƒ
Midpoint
67-78 3
72.5
79-90 5
84.5
91-102 8
96.5 9
108.5
120.5 8 7 6 5 4 3 2 1
60.5
72.5
84.5
96.5
108.5
120.5
132.5
minutes
How to construct a frequency Polygon:
Plot the midpoint and frequency.
Connect consecutive midpoints.
Extend the graph to the axis by one class.

11. Ogives (Cumulative Frequency Graphs)
Class
Frequency
Cumulative Frequency
Cumulative Rel. Frequency
67-78 3
0.1
79-90 5 8
0.267
91-102 16
0.534 9 25
0.834 30
1.001

12. Classwork Using the data on page 37, fill in the following table:
On a sheet of graph paper construct the following:
Histogram
Frequency Polygon
Ogive
Class
Frequency
Relative F.
Cumulative F.
Midpoints
Boundaries
30 – 39
40 – 49
50 – 59
60 – 69
70 – 79
80 – 89

13. Chapter 2: Descriptive Statistics
Lesson 2.2: More Graphs and Displays

14. Constructing Stem & Leaf Plots
Example: 102, 124, 108, 86, 103, 82, 71, 104, 112, 118, 87, 95, 103, 116, 85, 122, 87, 100, 105, 97, 107, 67, 78, 125, 109, 99, 105, 99, 101, 92
Stem Leaf
6 7
7 1
7 8
8 2
9 2
11 2
12 5
STEM LEAF
6 7
Key: 6 | 7 means 67

15. Interpreting Stem & Leaf Plots
Suppose the following stem-and-leaf plot set represents the scores on a statistics quiz (out of 50).
a) How many students took the quiz?
b) How many students scored a perfect score of 50?
c) What is the lowest score on the quiz?
d) What score occurs the most frequently (the “mode”)
e) What advantage(s) does a stem-and-leaf plot have compared to a histogram?

16. Side-by-side Stem & Leaf Plots
What is the highest grade a man earned on the quiz?
How many women were in the class?
What percent of the class is male?
If a passing grade is 35 or higher, what percent of the females passed?
Overall do you think women or men did better on this quiz?

17. Dotplots 66 76 86 96
106
116
126
minutes
Duplicates:
87, 99, 103, 105

18. Interpreting Dotplots
The following is a dotplot for a different group of students taking the 50-point statistics quiz
a) How many students took the quiz?
b) How many students scored a perfect score of 50?
c) What is the lowest score on the quiz?
d) What score represents the mode?

19. Bar Graphs and Pareto Charts
A Bar chart is constructed by labeling each category of data on either the horizontal or vertical axis and the frequency (or relative frequency) on the other axis.
A Pareto chart is a bar graph whose bars are drawn in decreasing order .

20. Pie Charts
A pie chart is a circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category.
What percentage of the
M&M’s are green?
What color was the most
common M&M color?
Suppose 17% of the M&Ms
are blue. If you have a bag
with 400 M&M’s. How many
of them would be blue?

21. Constructing Pie Charts
Category Budget Degrees
Human Space Flight
Technology
Mission Support
Total:
Pie charts help visualize the relative proportion of each category. Find the relative frequency for each category and multiply it by 360 degrees to find the central angle.
Central Angle for each segment =
# in category
X 360 degrees
total number

22. Chapter 2: Descriptive Statistics
Lesson 2.3: Measures of Central Tendency

23. Practice Mean, Median, & Mode
Find the mean, median and mode:
Mean:
Median:
Mode:

24. Shapes of distributions
Bell-shaped
Symmetric
Uniform
Symmetric
When describing the center of a data set use the mean for symmetrical distributions and the median for skewed distributions
Mean = Median
Mean = Median
Skewed Left
Skewed Right
Mean < Median
Mean > Median

25. Weighted Mean 1
Example: You are taking AP Biology and your grade is determined by five sources: 50% test, 15% midterm, 20% final exam, 10% lab work, and 5% homework. You scores are 86 (test), 96 (midterm), 82 (final), 98 (lab work), and 100 (homework). What is your weighted mean? .
Source X W XW
Tests
Midterm
Final
Labs
Homework

26. Weighted Mean 2
The average starting salaries (by degree attained) for 20 employees at a company are given below:
2 with a doctorate: $105,000
7 with a masters degree: $63,000
11 with a bachelors degree: $41,000
What is the mean starting salary for these employees?
Source X W XW
Σw =
Σxw =

27. Mean of a frequency distribution
Class x f xf
7 – 18
12.5 6
19 – 30
24.5 10
31 – 42
36.5 13
43 – 54
48.5 8
55 – 66
60.5 5
67 – 78
72.5
79 – 90
84.5 2
n = 50
*If you are given a class, use the midpoint for x.

28. Practice
Example: Use the frequency distribution to approximate the mean age of the residents of Bow, Wyoming.
Age x f xf
0 – 9 57
10 – 19 68
20 – 29 36
30 – 39 55
40 – 49 71
50 – 59 44
60 – 69
70 – 79 14
80 – 89 8
n =
Σ xf =

29. Chapter 2: Descriptive Statistics
Lesson 2.4: Measures of Variation (Part 1)

30. Compare Team 1 & Team 2 Measures of Central Tendency:
Mean: 75
Median: 76
Mode: 76
Measures of Variation:
Range
Variance
Standard Deviation
Team 1
Team 2 72 67 73 76 78 84

31. Range Simplest measure of variation Range = maximum – minimum
Team 1 Range: 6 inches
Team 2 Range: 17 inches
Disadvantages of the Range
Ignores the ____________________________
Only uses _____________________________
Sensitive to __________________

32. Deviations
To learn to calculate measures of variation that use every value in the data set, you first want to know about deviations.
The deviation for each value x is the difference between the value of x and the mean of the data set.
In a population, the deviation for each value x is:
In a sample, the deviation
for each value x is:

33. Variance Population Variance: Sample Variance:
If team 1 were a population what is the variance?
Team 1
(x - µ)^2 72
(72 – 75)^2 = 73
(73 – 75)^2 = 76
(76 – 75)^2 = 78
(78 – 75)^2 =
Variance:

34. Standard Deviation Population Standard deviation:
Sample Standard deviation:
Therefore team 1 has a standard deviation of:
SQRT(24/5) = 2.19 inches
What is the standard deviation of team 2?
Team 2
(x - µ)^2 67 72 76 84

35. Using the Calculator Enter Data into a list Calculate values
“STAT” -> Edit menu
67, 72, 76, 76, 84
Calculate values
“STAT” ->Calc menu -> 1:1-Var Stats
OUTPUT
𝑥 = mean
Sx = sample standard deviation
𝜎 𝑥 = population standard deviation
n = sample size
minX = minimum value in the data set
Q1 = quartile 1
Med = median = quartile 2
Q3 = quartile 3
MaxX = maximum value in the data set

36. Empirical Rule
About 68% of the data lies within __ standard deviation of the mean About 95% of the data lies within __ standard deviations of the mean About 99.7% of the data lies within __ standard deviations of the mean

37. Chapter 2: Descriptive Statistics
Lesson 2.4: Measures of Variation (Part 2)

38. Measures of Variation for Grouped Data
Population Variance:
Sample Variance:
Take the square root of variance to get standard deviation
𝜎 2 = 𝑥−𝜇 2 𝑓 𝑁

39. Sample Standard Deviationx f xf
𝒙− 𝒙 𝟐 𝒇 10 1 13 2 8 3 5 4
First find the ______. Do this by _____________________ _______________________.
Then create a column for the ______________________________________________.
Find variance by dividing the value in step 2 by _______.
Lastly get standard deviation by taking the __________ of variance
Mean = ∑xf/∑f =
Variance =
Standard Deviation =

40. Population Standard Deviation
Class x f xf
𝒙−𝝁 𝟐 𝒇
0-9
4.5 3
13.5
675
10-19
14.5 8
116
200
20-29
24.5 6
147
150
30-39
34.5 2
450
40-49
44.5 1 89
625
First find the ______. Do this by _____________________ _______________________.
Then create a column for the ______________________________________________.
Find variance by dividing the value in step 2 by _______.
Lastly get standard deviation by taking the __________ of variance
Mean = ∑xf/∑f =
Variance =
Standard Deviation =

41. Time between eruptions (min)
Classwork
In studying the behavior of Old Faithful geyser in Yellowstone National Park, geologists collect data for the time (in minutes) between eruptions. The table below summarizes actual data that were obtained.
(a) What is the mean time between eruptions?
(b) What is the standard deviation for the time between
eruptions?
Time between eruptions (min)
Frequency
40-49 8
50-59 44
60-69 23
70-79 6
80-89
107
90-99 11 1

42. Chapter 2: Descriptive Statistics
Lesson 2.5: Measures of Position

43. Fractiles
Fractiles are numbers that divide an ordered data set into equal parts. Common fractiles are:
Quartiles: Divide a data set into 4 equal parts
Q1, Q2, Q3
Deciles: Divide a data set into 10 equal parts
D1, D2, D3, …, D9
Percentiles: Divide a data set into 100 equal parts
P1, P2, P3, …, P99
D3 means ___% of the data falls _____________.

44. Finding Quartiles
You are managing a store. The average sale for each of 27 randomly selected days in the last year is given. Find Q1, Q2, and Q3.
The data in ranked order are:
The median = Q2 = ___
Q1 is ___ and Q3 is ___
The Interquartile range is Q3 – Q1 = _____________
Upper half
Lower half

45. Box and Whisker Plots
A box and whisker plot uses 5 key values to describe a set of data. Q1, Q2 and Q3, the minimum value and the maximum value.
5 Number Summary
Minimum value
First Quartile Q1
The median Q2
Third Quartile Q3
Maximum value

46. Box and Whisker Plot Practice
Construct a box and whisker plot using the following data:
5 Number Summary
Minimum value:
First Quartile Q1:
The median Q2:
Third Quartile Q3:
Maximum value: 90 80 70 60 50
100
Interquartile Range =____________

47. The Standard Score (z-score)
The z-score represents the number of standard deviations a value x falls from the mean
z-scores are calculated using the formula:
z-scores can be positive, negative, or 0
A ________ z-score means the data value falls above the mean
A ________ z-scores means the data value falls below the mean
A z-score of _____ means the data value equals the mean
z-scores that fall between ________ are considered usual

48. z-score practice
The average height for males is 70 inches with a standard deviation of 1.5 inches
The average height for females is 64 inches with a standard deviation of 2 inches.
Would it be unusual to find a man
who is 72 inches tall?
Mrs. Hallbach is 69 inches tall.
Is this unusual?
Respective to gender, who is
“taller” a man who is 72 inches
tall or a woman who is 69 inches
tall?