Bell Ringer In December 2000, the US Census Bureau listed the levels of educational attainment for Americans over 65. 9,945,000 had no high school diploma, 11,701,000 graduated high school but had no college experience, 4,481,000 had some college but no degree, 1,390,000 had a 2-year degree, 3,133,000 had a 4 year degree, 1,213,000 had a master’s degree, and 757,000 had a Ph.D. a. b.
Create a relative frequency table for this data. Plot the relative frequency distribution of educational attainment.
a. Create a relative frequency table Education Level
Count (Thousands)
No High School Diploma
9945
HS Graduate, No College
11701
Education Level
Count (Thousands)
No High School Diploma
9945/32620
HS Graduate, No College 11701/32620
Education Level
Count (Thousands)
No High School Diploma
30.49%
HS Graduate, No College
35.87% 13.74%
Some College, No Degree
4481
Some College, No Degree
4481/32620
2-year Degree
1390
2-year Degree
1390/32620
Some College, No Degree
4-year Degree
3133
4-year Degree
3133/32620
2-year Degree
4.26%
Master's Degree
1213
Master's Degree
1213/32620
4-year Degree
9.60%
Ph.D or Professional Degree
Master's Degree
3.72%
757/32620
Ph.D or Professional Degree
2.32%
Ph.D or Professional Degree
757
Total = 32620
b. Plot the relative frequency distribution
Everything You Never Wanted to Know About Contingency Tables
What is a contingency table? (Two Way Frequency Table) Pretending to be smart
Definitely not thinking about contingency tables
Pretentiously scratching goatee
But really, what is a contingency table? ● ●
Sometimes we need to need to keep track of data in a regular frequency table. For example, the number of students with each eye color. Eye Color
Count
Blue
10
Brown
36
Green/Hazel/Other
18
... ● ● ●
But sometimes we need to see how data is spread out among more than one variable. For example, we might want to know how eye color is distributed by gender instead of how eye color is distributed among students. This is a perfect time for a contingency table! Eye Color
Blue Sex
Brown
Green/Hazel/Other
Males
6
20
6
Females
4
16
12
Types of Distributions 1. 2. 3.
Marginal Joint Conditional
Marginal Distribution ●
Distribution along the “margins.” Eye Color
Blue
Sex
Brown
Green/Hazel/Other Total
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
Eye Color
Marginal Distribution Analysis
Green/Haz Blue Brown el/Other Total Sex
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
● ● ● ●
Marginal frequency of Brown eyes is 36. Relative marginal frequency of Brown eyes is 36/64 = 56.25% Marginal frequency of males is 32. Relative marginal frequency of males is 32/64 = 50%.
Marginal frequency refers to just one variable (ie. just blue eyes, or just females) Marginal distribution refers to how the variables are distributed along the margins. Ie. the marginal distribution of eye color is 10 blue, 36 brown, and 18 other. The marginal distribution of sex is 32 males, 32 females.
Eye Color
Plot of the Marginal Distribution(s)
Green/Haz Blue Brown el/Other Total Sex
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
Male Blue
Brown
Other
Female
Joint Distributions ●
Joint Distributions refer to the inner entries of the contingency table because they join the variables from one side of the table to the other. What does each cell “mean” in words?
Eye Color
Blue Sex
Green/Hazel/Othe r Total
Brown
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
Conditional Distributions ●
There is some condition to be satisfied. Ie. “Among males, what is the distribution of eye color?” or “Out of all the blue eyed people, how many are female?”
“Among males, what is the distribution of eye color?” Eye Color Blue Sex
Brown
Green/Hazel/Other Total
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
You do it! Calculate the relative distribution of eye color among males.
6 Blue, 20 Brown, 6 Green/Hazel/Other
“What is the relative distribution of sex among brown eyed people?” Eye Color ●
Solve it. Blue Sex
Steps to solving: 1. 2. 3.
Green/Hazel/ Other Total
Brown
Males
6
20
6
32
Females
4
16
12
32
10
36
18
64
Total
What is the condition? Isolate the row or column. Calculate.
Independence ● ● ●
We say that two variables are independent if one does not depend on the other. Example - we want to know if ice cream preference is independent of class. Then we would expect that approximately the same percent of students would prefer chocolate ice cream vs. vanilla ice cream in each class. Freshman Vanilla
Chocolate
Freshman
20
10
Sophomore
24
12
Junior
18
9
Senior
22
11
Vanilla
Chocolate
0.667
0.333
Sophomore Junior Senior
Vanilla
Chocolate
0.667
0.333
Vanilla
Chocolate
0.667
0.333
Vanilla
Chocolate
0.667
0.333
