If you were evaluating a set of data describing the numbers of “A’s”, “B’s”, “C’s”, and “D’s” that students earned on a particular test, and needed to display the data on a ** relative frequency table**, how would you go about it?

### Relative Frequencies

Frequency tables are closely related to histograms and stem-and-leaf plots. A relative frequency table is specifically designed to display the ratio of each individual frequency to the total frequency of the data. To begin building a relative frequency table, start by grouping values into categories, classes, or intervals, depending on the type of data. You should try to limit the number of intervals or classes to less than a dozen in most cases, and you can use the square root of the number of actual data points as a guide if you wish.

Once you have all of your data separated into separate classes or categories, (often known as “**binning**”, since the data is divided up into multiple “bins,” one for each specified class, category, or interval), tally the number of values in each category and the total number of values all together.

To calculate the relative frequency of each category, divide the category, class, or interval frequency by the overall frequency. The decimal you get will represent the part of the entire sample that is represented by that category. Once you have calculated all of the relative frequencies for every category, add them up to make sure they total 1.0.

**Note!** If you are graphing the relative frequencies of a **continuous variable**, you will need to specify how to handle any values that fall right on the boundary of a category (also commonly called a class). Here are a couple of ways to do this:

- You can specify on your table that values equal to lower class limits are included in a class, but values equal to upper class limits are not (this is the conventional method). This means that a value of 5 would be considered part of a 5-10 class, but not part of a 1-5 class.
- You can also define each category so that there are no overlapping values:

1-4.99 5-9.99 10-14.99 15-20

**Determining Relative Frequencies **

1. You are given a bag of marbles in multiple colors, if there are 25 red, 22 yellow, 17 green, and 28 blue marbles, what are the relative frequencies of each color?

Start by totaling the number of marbles: \begin{align*}25 + 22 + 17 + 28 = 92\end{align*}

Divide the number of each color by the total number of marbles:

\begin{align*}\frac{25 \ red \ marbles}{92 \ total \ marbles} = .272\end{align*}

\begin{align*}\frac{22 \ yellow \ marbles}{92 \ total marbles} = .239\end{align*}

\begin{align*}\frac{17 \ green \ marbles}{92 \ total \ marbles} = .185\end{align*}

\begin{align*}\frac{28 \ blue \ marbles}{92 \ total \ marbles} = .304\end{align*}

Add your totals together to verify that they equal 1:

\begin{align*}.272+.239+.185+.304=1.0\end{align*}

Note that each of the relative frequencies can also be understood as percentages:

.272 = 27.2% red marbles

.239 = 23.9% yellow marbles

.185 = 18.5% green marbles

.304 = 30.4% blue marbles

\begin{align*}27.2\% + 23.9\% + 18.5\% + 30.4\% = 100\%\end{align*}

2. A police officer is reviewing accident statistics for her city. She notes that there were a total of 23 incidents involving teen drivers between ages sixteen and twenty-one, 19 incidents involving drivers aged twenty-two through twenty-six, 19 involving twenty-seven to forty-year-olds, and 18 for ages above forty-one.

What are the relative frequencies for each age range?

The total number of accidents is:

\begin{align*}23+19+19+18=79 \ \text{total accidents}\end{align*}

The relative frequencies are:

\begin{align*}\frac{23 \ in \ age \ range \ 16-21}{79 \ total} = .291\end{align*}

\begin{align*}\frac{19 \ in \ age \ range \ 22-26}{79 \ total} = .241\end{align*}

\begin{align*}\frac{19 \ in \ age \ range \ 27-40}{79 \ total} = .241\end{align*}

\begin{align*}\frac{18 \ in \ age \ range \ 41+}{79 \ total} = .228\end{align*}

Verify that the relative frequencies total 1.0:

\begin{align*}.291+.241+.241+.228=1.001 (due \ to \ rounding)\end{align*}

**Calculating Probability **

A local high school has 150 students who drive to school. Examining the parking lot, you note that there are 25 white cars, 35 red cars, 13 green cars, 19 blue cars, and 58 others.

What are the chances, expressed as percentages that randomly chosen students have each of the different colored cars?

We are given the total number of cars in the question: 150

Divide each of the individual colors by the total and convert the decimal answers to percentages:

\begin{align*}\frac{25 \ white \ cars}{150 \ total \ cars} = .167 = 16.7\%\end{align*}

\begin{align*}\frac{35 \ red \ cars}{150 \ total \ cars} = .233 = 23.3\%\end{align*}

\begin{align*}\frac{13 \ green \ cars}{150 \ total \ cars} = .087 = 8.7\%\end{align*}

\begin{align*}\frac{19 \ blue \ cars}{150 \ total \ cars}= .127 =12.7\%\end{align*}

\begin{align*}\frac{58 \ others}{150 \ total \ cars} .387 = 38.7\%\end{align*}

**Earlier Problem Revisited**

If you were evaluating a set of data describing the numbers of “A’s”, “B’s”, “C’s”, and “D’s” that students earned on a particular test, and needed to display the data on a relative frequency table, how would you go about it?

Add up the number of entries in each category, A, B, C, and D, to get the total number of data points. Divide the number of values in each category by the total to get the relative frequencies. Convert the decimal values to percentages if necessary.

### Examples

#### Example 1

The Sackmore and Headbut village football teams have played each other 50 times. Sackmore has won 10 times, Headbut has won 35 times, and the teams have drawn 5 times. Based on past performance, what is the probability that Sackmore will win the next match?

So far, Sackmore has won 35 out of the 50 matches. We can write this as a fraction, whic (reduced) is: \begin{align*}\frac{7}{10}\end{align*}. This fraction isn't really the probability of Sackmore winning, but it is an estimate of that probability. We say that the relative frequency of Sackmore winning \begin{align*}\frac{7}{10}\end{align*}.

#### Example 2

Tony estimates that the probability that there will be an empty space in the car park when he arrives at work is \begin{align*}\frac{4}{5}\end{align*}. His estimate is based on 50 observations. On how many of these 50 days was he *unable* to find an empty space in the car park?

If Tony has figured that he is able to find a space 4 of every 5 times he arrives, then he is not able to find a space 1 in every 5 times. if we set the ratio: \begin{align*}\frac{1}{5}=\frac{x}{50}\end{align*}, we can solve for \begin{align*}x\end{align*} to find that he did not have a space 10 times.

#### Example 3

A pair of dice (one red, one green) is cast 30 times, and on 4 of these occasions, the sum of the numbers facing up is 7. What is the relative frequency that the sum is 7?Out of thirty throws, four of them were 7's. The relative frequency is \begin{align*}\frac{4}{30}\end{align*} or \begin{align*}\frac{2}{15}\end{align*}.

#### Example 4

The students in a class were asked what kind of music they liked. 18 liked rock, 11 liked pop, 5 liked hip hop, and 8 liked country. Create a frequency and relative frequency table using this information.

To create the frequency table, we just need one column for each category.

Rock | Pop | Hip Hop | Country |

18 | 11 | 5 |
8 |

To convert to a relative frequency table, just divide each frequency by the total:

Rock | Pop | Hip Hop | Country |

\begin{align*}\frac{18}{42}=.43\end{align*} | \begin{align*}\frac{11}{42}=.26\end{align*} | \begin{align*}\frac{5}{42}=.12\end{align*} | \begin{align*}\frac{8}{42}=.19\end{align*} |

#### Example 5

In 1990, there were approximately 10,000 fast food outlets in the US that specialized in Mexican food. Of these, the largest were Taco Bell with 4809 outlets, Taco John's with 430 outlets and Del Taco with 275 outlets. The relative frequency that a fast food outlet that specializes in Mexican food is none of the above is:

The likelihood that a restaurant is *not* one of the top three would equal the number of Mexican fast food restaurants that are not one of the three: **\begin{align*}10,000 - 4809 - 430 - 275 = 4486\end{align*}**, divided by the total number of Mexican fast food restaurants, **10,000**:

\begin{align*}\frac{4,486}{10,000}=.4486 \ or \ 44.86\%\end{align*}

### Review

30 Students in a class surveyed each other to find out their favorite movie series, and recorded the results in a table like the one shown below.

Movie Series |
Number of Likes |

Twilight | 7 |

Lord of the Rings | 5 |

Pirates of the Caribbean | 9 |

Harry Potter | 6 |

Narnia | 2 |

High School Musical | 1 |

1. What was the relative frequency for Narnia?

2. What was the relative frequency for Pirates of the Caribbean?

3. 100 people were asked whether they were left-handed. 8 people answered yes. What is the relative frequency of left-handed people in the survey?

4. The relative frequency of getting a white candy from a particular bag is 0.3. If the bag contains 100 candies, estimate the number of whites.

5. Kyle observed 80 cars as they drove by his bedroom window. 24 of them were red. What is the relative frequency of red cars?

6. The relative frequency of rain in April is .6. There are 30 days in April. Estimate the number of days of rain expected in April.

Use the table below listing the heights of 100 male semiprofessional soccer players.

HEIGHTS (INCHES) |
FREQUENCY OF STUDENTS |
RELATIVE FREQUENCY |

59.95-61.95 | 5 | |

61.95-63.95 | 3 | 3100 = 0.03 |

63.95-65.95 | 15100 = 0.15 | |

65.95-67.95 | 40 | 40100 = 0.40 |

67.95-69.95 | 17 | |

69.95-71.95 | 12 | 12100 = 0.12 |

71.95-73.95 | 7100 = 0.07 | |

73.95-75.95 | 1 | 1100 = 0.01 |

Total = 100 | Total = |

7. Fill in the blanks and check your answers.

8. The percentage of heights that are from 67.95 to 71.95 inches is:

9. The percentage of heights that are from 67.95 to 73.95 inches is:

10. The percentage of heights that are more than 65.95 inches is:

11. The number of players in the sample who are between 61.95 and 71.95 inches tall is:

12. What kind of data does this chart highlight, qualitative or quantitative?

13. What is the height interval for the players who fall under the frequency of .03?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.3.