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Pre-Algebra View details

9. Categorical Data

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eCourses /

Pre-Algebra /

Chapter 12

Categorical Data

This lesson focuses on categorical data analysis, exploring methods to organize and interpret it effectively. It covers relative frequencies, two-way frequency tables, and the roles of joint and marginal relative frequencies. By learning these concepts, users can better understand the relationships between variables and draw meaningful insights from data in diverse contexts.

Lesson Settings & Tools

	12 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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Categorical Data

Slide of 12

Data comes in a variety of ways such as in surveys where observations yield different types of data. If the data can be described using numbers, it is called numerical data. If it is better to describe the variables by their characteristics, this is called categorical data. This lesson will focus on the later, and how to distinguish the two.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Numerical Data
Categorical Data

Challenge

A Cozy Beauty Store

It is Friday afternoon, and Tiffaniqua just finished finals week at school. It is time to relax! As she is leaving school, she thinks of a nice bath with a bath bomb. Just thinking of how the water changes as the bath bomb dissolves already places Tiffaniqua in a state of relaxation!

Her aunt happens to own a beauty store, so she goes there to look for the bath bomb. As she is there, Tiffaniqua is amazed by how each individual product comes in different scents. Her aunt knows that Tiffaniqua loves statistics, so she shows her a table with information on how many of their different products were sold in the last month.

According to the table, which scent of bath bomb is the most popular?

In general, which is the most popular scent when taking into account all the products sold in the store?

Discussion

Categorical Data

Categorical data, also called qualitative data, is data that can be split into groups. Categorical data belongs to one or more categories that have a fixed number of possible outcomes or values. Human blood groups are one example of categorical data.

This classification is based on whether certain antigens are present or absent on the surface of red blood cells. A person's blood type can be categorized into one of these groups.

Example

Is it Categorical or Numerical?

The beauty store has plenty to offer, and Tiffaniqua wonders how her aunt manages such a wide variety of products. Auntie has a database where she stores information about each individual product for sale. For example, consider one particular bottle of rose hand lotion.

Auntie stores in her database the information about the type of product (hand lotion), the scent (rose), the content volume (

10.1

fl. oz.), and the price (

$ 5

). Which of these variables are categorical?

Hint

What variables are not described using numbers?

Solution

The store sells different types of different scented products. Consider the given bottle of hand lotion.

There are four variables present in the given picture. These are the type of product, the scent, the content volume, and the price. Notice that the type of product is hand lotion, and the scent is rose. These variables are described using words, rather than numbers. This means that these variables correspond to categorical data.

On the other hand, the content volume is given in fluid ounces, and the price in dollars. Since these variables are described using numbers, they correspond to numerical data.

Discussion

Relative Frequency

Exploring the proportion of occurrences of a group, value, or set of values in a data set provides valuable insights. This is called the relative frequency, which is given by the ratio of an observed category or value's frequency to the total number of observations in a data set.

Relative frequency $= \frac{Frequency}{Number of observations}$

For example, suppose categorical data is explored in a survey made on a classroom about the favorite color of the students.

Color	Frequency
Blue	$9$
Red	$7$
Green	$5$
Yellow	$4$
Purple	$3$
Other	$2$

Knowing that there are $30$ students in the classroom, the relative frequency of each category can be found by dividing the frequency of each category by $30 .$

Color	Frequency	Relative Frequency
Blue	$9$	$\frac{9}{3 0} = 0.3$
Red	$7$	$\frac{7}{3 0} = 0.23$
Green	$5$	$\frac{5}{3 0} = 0.17$
Yellow	$4$	$\frac{4}{3 0} = 0.13$
Purple	$3$	$\frac{3}{3 0} = 0.1$
Other	$2$	$\frac{2}{3 0} = 0.07$

Relative frequencies are typically written as percentages.

Color	Frequency	Relative Frequency
Blue	$9$	$30 %$
Red	$7$	$23 %$
Green	$5$	$17 %$
Yellow	$4$	$13 %$
Purple	$3$	$10 %$
Other	$2$	$7 %$

From the table it can be seen that blue is the most popular color, as it has the highest relative frequency. Notice that the relative frequencies add up to

100 % .

Example

Relaxing Artisan Candles

Outstandingly, Tiffaniqua's auntie hand makes the candles herself! She buys a huge block of soy wax and uses it to make different scented candles.

To give each candle their characteristic scent, she needs to add essential oils. It is vital that she knows in advance how much of each oil to buy. Otherwise, she ends up with too much. Consider her previous month's sales sheet.

Scent	Number of Candles Sold
Lavender	$15$
Citrus	$5$
Vanilla	$30$
Rose	$20$
Jasmine	$10$

Match each scent with its corresponding relative frequency.

Hint

Begin by finding how many candles were sold. Divide each frequency by the total number of candles sold. Write each relative frequency as a percentage.

Solution

In order to find the relative frequency of the sales of each different scent of candle, begin by finding how many candles were sold last month. To do so, add the frequencies of each scent.

15 + 5 + 30 + 20 + 10 = 80

This means that

80

candles were sold last month. Next, find the relative frequency by dividing the respective frequency by this total. Since

15

lavender candles were sold, dividing

15

80

yields the relative frequency of lavender candles.

\frac{1 5}{8 0} = 0.1875

Next, this number will be written as a percentage in order to compare it to the others.

0.1875 = 18.75 %

Do the same for the rest of the scents.

Scent	Number of Candles Sold	Divide	Relative Frequency
Lavender	$15$	$\frac{1 5}{8 0}$	$0.1875 = 18.75 %$
Citrus	$5$	$\frac{5}{8 0}$	$0.0625 = 6.25 %$
Vanilla	$30$	$\frac{3 0}{8 0}$	$0.375 = 37.5 %$
Rose	$20$	$\frac{2 0}{8 0}$	$0.25 = 25 %$
Jasmine	$10$	$\frac{1 0}{8 0}$	$0.125 = 12.5 %$

Knowing this, each scent can be paired with its relative frequency.

Scent	Relative Frequency
Lavender	$18.75 %$
Citrus	$6.25 %$
Vanilla	$37.5 %$
Rose	$25 %$
Jasmine	$12.5 %$

Tiffaniqua's auntie can now use these relative frequency percentages to prioritize how much of each oil to purchase!

Discussion

Two-Way Frequency Table

A two-way frequency table, also known as a two-way table, displays categorical data that can be grouped into two categories. One of the categories is represented in the rows of the table, the other in the columns. For example, the table below shows the results of a survey where $100$ participants were asked if they have a driver's license and if they own a car.

Here, the two categories are car and driver's license. Both have possible responses of yes and no. The numbers in the table are called joint frequencies. Also, two-way frequency tables often include the total of the rows and columns — these are called marginal frequencies. Select any frequency in the table below to display more information.

The sum of the Total row and the Total column, which in this case is

100,

equals the sum of all joint frequencies. This is called the grand total. A joint frequency of

43

shows that

43

people have a driver's license and own a car. A marginal frequency of

53

shows that

53

people do not have a car. The rest of the numbers from the table can also be interpreted.

Discussion

Drawing a Two-Way Frequency Table

Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. To draw a two-way frequency table, three steps must be followed.

Determine the categories.
Fill the table with the given data.
Determine if there are any missing frequencies. If so, find those.

Suppose that $53$ people took part in an online survey, where they were asked whether they prefer top hats or berets. Out of the $18$ males that participated, $12$ prefer berets. Also, $15$ of the females chose top hats as their preference. The steps listed above will now be used to analyze and present the data.

Determine the Categories

First, the two categories of the table must be determined, after which the table can be drawn without frequencies. Here, the participants gave their hat preference and their gender, which are the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male.

The total row and total column are included to write the marginal frequencies.

Fill the Table With Given Data

The given joint and marginal frequencies can now be added to the table.

Find Any Missing Frequencies

Using the given frequencies, more information can potentially be found by reasoning. For instance, because

12

out of the

18

males prefer berets, the number of males who prefer top hats is equal to the difference between these two values.

18 - 12 = 6

Therefore, there are

6

males who prefer top hats. Since there are

15

females who prefer top hats, the number of participants who prefer this type of hat is the sum of these two values.

6 + 15 = 21

It has been found that

21

participants prefer top hats. Continuing with this reasoning, the entire table can be completed.

Example

Filling the Hand Lotion Bottles

The hand lotion sold at auntie's store comes in two sizes: large and small. She actually buys the lotion in bulk. Then, she fills it in their respective bottles, one by one. Auntie needs to restock on vanilla and rose hand lotions, and Tiffaniqua will help!

Out of a total of $25$ vanilla bottles, they filled $18$ small ones. They also filled $12$ large rose bottles. In total, $60$ bottles where filled.

a How many rose bottles in total were filled?

b How many small rose bottles were filled?

c How many large bottles in total were filled?

Hint

a Make a two-way frequency table with the given information. Include the known marginal frequency and the grand total. Use the information in the table to find the marginal frequency corresponding to rose bottles.

b Make a two-way frequency table with the given information. Include the known marginal frequency and the grand total. Use the answer from Part A to find the joint frequency corresponding to small rose bottles.

c Make a two-way frequency table with the given information. Include the known marginal frequencies and the grand total. Use the information in the table to find the marginal frequency corresponding to large bottles.

Solution

a Begin by making a two way table. There are two variables in this case: the scent, and the bottle size.

Next, add a row and a column to include the marginal frequencies, which correspond to the totals of each individual category.

It is given that out of $25$ vanilla bottles, Tiffaniqua filled $18$ small ones. This means that the marginal frequency of vanilla bottles is $25$ and that the joint frequency of small vanilla bottles is $18 .$ Auntie also filled $12$ large rose bottles, which is also a joint frequency. The grand total corresponds to $60$ bottles. Add all this information to the table.

There is now enough information to fill the table. To find how many rose bottles there are note that out of the

60

bottles,

25

are vanilla, so subtract

25

from

60 .

60 - 25 = 35

There are

35

rose bottles.

b It was found in Part B that there are

35

rose bottles,

12

of which are large. Find the number of small rose bottles by subtracting

12

from

35 .

35 - 12 = 23

This means they filled

23

small rose bottles.

c Finally, to find how many large bottles were filled begin by noting that there are

18 + 23 = 41

small bottles. This means that the number of large bottles can be found by subtracting

41

from

60 .

60 - 41 = 19

Finally, while not necessary, find how many large vanilla bottles there are by subtracting

12

from

19 .

19 - 12 = 7

With this information, the table can now be completed.

Discussion

Joint and Marginal Relative Frequencies

In a two-way frequency table, a joint relative frequency is the ratio of a joint frequency to the grand total. Similarly, a marginal relative frequency is the ratio of a marginal frequency to the grand total. Consider the following example of a two-way table.

Here, the grand total is $100 .$ The joint and marginal frequencies can now be divided by $100$ to obtain the $joint$ and $marginal$ relative frequencies. Clicking in each cell will display its interpretation.

Relative frequencies can also be made either by columns or by rows. In the case it is made by rows, each joint frequency is divided by the marginal frequency of its corresponding row.

On the other hand, if the table is to be made by columns, each joint frequency needs to be divided by the marginal frequency of its respective column.

Note that when a relative frequencies table is made by rows, the values in the columns do not add up to

100 % .

Similarly, the values in the rows of a relative frequencies table made by columns do not add up to

100 % .

Example

Advertising with Free Samples

After filling up the bottles, auntie was left with some leftover lotion of vanilla and rose. She recalled that there was some leftover wax as well. She decides to give free samples to Tiffaniqua to share with friends. The following two-way table summarizes the items auntie gave to Tiffaniqua.

a What percentage of the samples are rose candles?

b What percentage of the samples are vanilla scented?

c What percentage of the samples are hand lotion bottles?

Hint

a Add all the frequencies to find the grand total. Find the relative frequency of rose candles by dividing the number of rose candles by the grand total.

b Add the frequencies of the vanilla scented products to find the marginal frequency of the vanilla scent. Divide it by the grand total to find its corresponding marginal relative frequency.

c Add the frequencies of the hand lotions to find the marginal frequency of the hand lotion product. Divide it by the grand total to find its corresponding relative marginal frequency.

Solution

a Begin by adding all the frequencies in the table to find the grand total. This corresponds to the number of items auntie gave to Tiffaniqua.

6 + 3 + 4 + 7 = 20

Auntie gave Tiffaniqua a total of

20

items. To find what percentage of the samples are rose candles divide its corresponding frequency by the grand total.

\frac{7}{2 0} = 0.35

This corresponds to the joint relative frequency of rose candles. This can now be written as a percentage.

0.35 = 35 %

b The percentage of vanilla scented samples corresponds to the marginal relative frequency of the vanilla items. Begin by adding the marginal frequencies to the given two-way table.

A total of

10

vanilla samples were given to Tiffaniqua. Find the relative marginal frequency by dividing this number by the grand total.

\frac{1 0}{2 0} = 0.5

Do not forget to write it as a percentage.

0.5 = 50 %

Tiffaniqua found that

50 %

of the samples have a vanilla scent.

c Consider the two-way table with marginal frequencies obtained in Part B.

The marginal frequency corresponding to hand lotion bottles is

9 .

Find the marginal relative frequency by dividing this number by the grand total.

\frac{9}{2 0} = 0.45

The marginal relative frequency is

0.45,

45 % .

This means that

45 %

of the samples are hand lotion bottles.

Closure

The Most Relaxing Scent

Consider the two-way table given at the start of the lesson.

To determine which scent of bath bomb is the most popular focus on the bath bomb row. Look for which bath bomb scent has the greatest joint frequency.

In order to determine which scent is the most popular in general, the marginal frequencies need to be added to the table.

From here, it can be seen that the vanilla scent has the greatest marginal frequency.

This means that the most popular bath bomb is lavender, but the most popular scent in general is vanilla. Vanilla might be an all-time favorite, but it seems that a lavender bath is the best bet when it comes to relaxation!

Level 1

Level 2

Level 3

Tiffaniqua wants to see a movie in an online streaming service. The platform summarizes the following characteristics of each individual movie: title, movie genre, duration (in minutes), filming studio, and number of awards received. Which of the variables correspond to categorical data?

Let's begin by recalling what is categorical data.

Categorical Data |- Categorical data, also called qualitative data, is data that can be split into groups. Categorical data belongs to one or more categories that have a fixed number of possible outcomes or values.

In other words, categorical data is data that can be described using words rather than numbers. The duration and the number of awards are described using numbers, so these do not correspond to categorical data. & Title & Movie Genre *& Duration & Filming Studio *& Number of Awards On the other hand, the title, the movie genre, and the filming studio are all names and words that describe the movie. ✓& Title ✓& Movie Genre & Duration ✓& Filming Studio & Number of Awards

A jewelry store sells different types of necklaces depending on which metal is used for the chain and what kind of stone is used to decorate it. The following two-way table contains information about last month sales.

How many gold ruby necklaces were sold?

How many silver necklaces were sold?

Let's begin by taking a look at the given two-way frequency table.

We are asked for how many gold ruby necklaces were sold. This corresponds to the joint frequency of gold and ruby, let's use x to represent it. We will now focus on the ruby column. Note that the marginal frequency of ruby is 5.

This means that adding 3 to our missing number x equals 5. x+3=5 Let's find x by subtracting 3 from both sides of the above equation!

We found that x=2. This means that 2 ruby gold necklaces were sold at the jewelry shop.

This time we are asked to find how many silver necklaces were sold. This corresponds to the marginal frequency of silver necklaces. Let's focus on the silver row of the given table.

At first glance, it looks like we do not have enough information to find the marginal frequency. However, note that we can find how many gold necklaces were sold. 2+4+5 = 11 Let's add this to our table.

Out of the 25 necklaces sold at the store, 11 were made of gold. This means that 25-11=14 were made of silver.

Tiffaniqua made a bunch of sandwiches to give out as charity for the needed. She used two different type of breads, white and whole grain. She also used two different types of deli meat, ham and turkey. The following table summarizes what types of sandwiches Tiffaniqua made.

How many sandwiches did Tiffaniqua make?

How many slices of white bread did Tiffaniqua use?

Let's take a look at the given two-way frequency table.

To find how many sandwiches Tiffaniqua made, we need to add all the joint frequencies. 13+7+6+4 = 30 This means that Tiffaniqua made a total of 30 sandwiches.

We are now asked to find how many slices of white bread Tiffaniqua used. Let's begin by adding the marginal frequencies to the table.

Tiffaniqua made 20 white bread sandwiches. Keep in mind that two slices of bread are used for every sandwich, which means that Tiffaniqua actually used 40 white bread slices to make the sandwiches.

A coffee shop sells two types of cappuccinos, vanilla and Irish cream. The coffee used for these beverages can be either regular or decaf. The following table summarizes last week's sales.

What percentage of the sales correspond to regular vanilla cappuccinos?

What percentage of the sales correspond to decaf cappuccinos?

Unfortunately, for budget reasons, the shop needs to stick to one flavor. Which one is the best bet?

Consider the given two-way frequency table.

We are asked to find what percentage of the sales correspond to regular vanilla cappuccinos. This means that we need to find its respective joint relative frequency. Begin by finding how many cappuccinos were sold. 88+67+25+20 = 200 Find the relative joint frequency of regular vanilla cappuccinos by dividing its respective joint frequency by the grand total. 88/200 = 0.44 Do not forget to write this as a percentage. 0.44 = 44 %

Since we are asked to find what percentage of the sales correspond to decaf cappuccinos we will begin by adding the marginal frequencies to the table.

A total of 87 decaf cappuccinos were sold. Find its corresponding marginal relative frequency by dividing it by 200. We will write it as a percentage as well. 87/200 &= 0.435 &= 43.5 %

Let's take a look at the two-way table including marginal frequencies.

Since one flavor needs to be removed, it is better to stick to the one that has the highest amount of sales. The marginal frequency of vanilla flavored cappuccinos is 155, while the Irish cream is only 45. This means that the store should keep the vanilla flavor.

The following two-way relative frequency table summarizes the results of a survey about pets.

What percentage of the population owns a pet?

Knowing that

500

people participated in the survey, how many people own both a cat and a dog?

Let's begin by taking a look at the given two-way table. Keep in mind that this is a relative frequency table.

Owning a pet in this context means owning either a cat, a dog, or both.

This means that we will add all the frequencies except those who do not own neither a dog nor a cat. 5 % + 40 % + 35 % = 80 % We found that 80 % of the surveyed people own a pet.

We are told that 500 people participated in the survey. From the table, we can see that 5 % of these persons own both a cat and a dog. To find how many people fall under this category we will first write the percentage as a decimal. 5 % = 0.05 Next, we multiply this number by the number of people that participated in the survey. 0.05 * 500 = 25 This means that 25 people own both a cat and a dog.

Categorical Data

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