MATH 1040 - Introduction to Statistics

Lesson 9.2 Relative Frequency (Proportions)

Reading

Reading sections are from the Introductory Statistics Textbook

3.1.1 Probability (pages 88-90)

Lesson

A Probability is the ratio of the number of successful outcomes to the number possible outcomes. However, if you do an experiment in real life, the randomness causes numbers to not always follow probabilities. For example, if I roll a die 100 times and count the 5’s, I could get 20 out of the 100 rolls as 5 even though the probability of rolling a 5 says I should have 16.67 rolls of my 100 be 5’s.

In that example, we saw what is called the Relative Frequency, or the fraction of events that match our successful outcome. To calculate the relative frequency, take the number of successes in your event and divide it by the total number of events.

Example 1: You have a bucket of 6-sided dice. You just dump them out then count the occurrences of each number. This table displays the counts:

Roll 1 2 3 4 5 6

Count 83 48 27 12 76 33

Adding these up, there is a total of 279 rolls. So, the relative frequencies are:

Roll Count Relative Frequency

1 83 83/279 = 0.297 = 29.7%

2 48 48/279 = 0.172 = 17.2%

3 27 27/279 = 0.097 = 9.7%

4 12 12/279 = 0.043 = 4.3%

5 76 76/279 = 0.272 = 27.2%

6 33 33/279 = 0.118 = 11.8%

Roll	1	2	3	4	5	6
Count	83	48	27	12	76	33

Roll	Count	Relative Frequency
1	83	83/279 = 0.297 = 29.7%
2	48	48/279 = 0.172 = 17.2%
3	27	27/279 = 0.097 = 9.7%
4	12	12/279 = 0.043 = 4.3%
5	76	76/279 = 0.272 = 27.2%
6	33	33/279 = 0.118 = 11.8%

This table is what we call a Probability Distribution. We will see them more in Lesson 12. But for now, note the following 2 rules of probability distributions:

All relative frequencies (or probabilities) must be between 0 and 1 (that is, 0% and 100%) - no negatives and nothing larger than 1.
All relative frequencies (or probabilities) must add up to 1 (100%).

Example 2: There are 15 students in a class. Below is a table of their grade levels:

Grade Freshman Sophomore Junior Senior

Count 4 8 2 1

The relative frequencies are just the number in each class divided by the total of 15:

Grade Count Relative Frequency

Freshman 4 4/15 = 0.267 = 26.7%

Sophomore 8 8/15 = 0.533 = 53.3%

Junior 2 2/15 = 0.133 = 13.3%

Senior 1 1/15 = 0.067 = 6.7%

Grade	Freshman	Sophomore	Junior	Senior
Count	4	8	2	1

Grade	Count	Relative Frequency
Freshman	4	4/15 = 0.267 = 26.7%
Sophomore	8	8/15 = 0.533 = 53.3%
Junior	2	2/15 = 0.133 = 13.3%
Senior	1	1/15 = 0.067 = 6.7%

Practice

Practice Question 9.2.1

A survey was conducted to find out people’s preferred type of pet. The results are as follows:

Pet Type	People who prefer pet
Dogs	40 people
Cats	25 people
Birds	10 people
Fish	15 people
Reptiles	10 people

After solving on your own, check the solution.

You have a large bag of colored marbles. You randomly grab a handful of marbles and count the colors. Calculate the relative frequency of each color:

Color	Red	Blue	Green	Yellow	Black	White
Count	22	13	10	9	7	18

After solving on your own, check the solution.

A teacher recorded the number of books read by students in five different genres over the course of a semester. The data is shown below:

Genre	Fiction	Non-Fiction	Mystery	Science Fiction	Biography
Count	24	18	12	6	10

After solving on your own, check the solution.