We have seen many types of data displays. Each display has different benefits and drawbacks. Let's look at all the representations we have seen so far.

Favorite color	Frequency
\text{red}	5
\text{blue}	3
\text{green}	10
\text{yellow}	20

\,\\\\ \, \\\\For categorical data, it can be helpful to first organize the data in a table.

A pictograph showing pizza choices of students. One whole pizza is equal to 4 students. Cheese has 2 and 1/4 pizzas, pepperoni has 3 and 3/4 pizzas, vegetarian has 1 whole pizza, shawarma has 1 and 1/4 pizzas and hawaiian has one whole pizza.

\,\\\\ \, \\\\A pictograph show us how many items are in each category using pictures and a key that tells us what the pictures mean.

A bar graph is titled What type of pet do you have? There is no title on the vertical axis but divided into scale from 0 to 8 in an interval of 2. The horizontal axis has no label but contains the categories with boxes of different heights to represent the frequency. A vertical bar is shown for each category and height: Fish, 2; Dog, 6; Cat, 4; Lizard, midway between 0 and 2.

\,\\\\ \, \\\\In a bar graph, the height of each bar represents the count or frequency of that category.

\text{Age}

\text{Hops on one foot}

A line graph shows how a value changes over time. We can look for patterns, peaks, and valleys.

A line (dot) plot showing the number of minutes to eat breakfast from 0-12, each having the following number of dots in order, starting from 0: 5,1,4,5,2,3,0,0,1,3,7,2,3

Line plots how many data points are at each value. This display is best used for discrete values with a small range.

The number line must count by equal amounts with no gaps, even if no data exists for a value
Dots or X's represent each data point and should be equal size.

Age when first new cell phone purchased
Stem	Leaf
1	0\ 1\ 4\ 5\ 8\ 8\ 8\ 9\ 9
2	0\ 2\ 5
3	1\ 4\ 9
4
5
6	1
Key 1\vert 2 = 12 years

\,\\\\ \, \\\\Stem-and-leaf plots show all of the raw data in an organized and sorted way.

A circle graph about favorite colors. The colors and percentages are: red 13%, yellow 12%, pink 12%, purple 6%, orange 8%, blue 30%, and green 19%.

In a circle graph, each piece represents the percentage or proportion of a category and all pieces add to 100\%.

Exploration

This table tells us how many students have birthdays in the different seasons.

Seasons	Number of students
\text{Winter}	5
\text{Spring}	8
\text{Summer}	2
\text{Fall}	5

The data was also organized into these two displays.

A circle graph and a bar graph that shows how many students have birthdays in different seasons. Both displays show the following number of students per season: Winter 5, fall 5, summer 2, and spring 8.

Which season had the most birthdays? Which display did you use, the table, circle graph, or bar graph?
Which season had the fewest birthdays? Which display did you use?
How do winter and fall compare? Which display did you use?

Let's look at when different displays are useful.

A circle graph is useful for showing proportions and parts of a whole.

Advantages:

We can easily compare the proportions of different categories visually. They are commonly used.

Disadvantages

It can be difficult to compare similar groups if they are not labeled. We may lose the original totals if we just show the percentages.

A bar graph is useful for showing the count or frequency of different categories.

Advantages:

Easily display large data sets. The highest and lowest categories are easy to see.

Disadvantages

For large scales, it can be difficult to read off exact values.

A line plot (dot plot) is useful for showing frequency of different categories

Advantages:

The highest and lowest categories are easy to see.

Disadvantages

Do not work well for large data sets.

A pictograph is useful for representing data in an easy to interpret and visual way.

Advantages:

The highest and lowest categories are easy to see. Can be more visual interesting.

Disadvantages

Do not work well for large data sets or with many categories.

Stem-and-leaf plots show all of the raw data in an organized and sorted way.

Age when first new cell phone purchased
Stem	Leaf
1	0\ 1\ 4\ 5\ 8\ 8\ 8\ 9\ 9
2	0\ 2\ 5
3	1\ 4\ 9
4
5
6	1
Key 1\vert 2 = 12 years

Advantages:

Can see individual data values. The highest and lowest values are easy to see.

Disadvantages

Do not work well for large data sets.

A line graph can show how values vary over time.

\text{Age}

\text{Hops on one foot}

Advantages:

Can easily see how a value changes over time.

Disadvantages

Cannot be used for categorical data.

The same type of display with a different scale, coloring, or order can also make it easier or more difficult to answer questions from it.

Examples

Example 1

These bar and circle graphs show the number of purchases for each tire brand at an auto supply store.

Bar graph

Circle graph

In which display is it easier to see the tire brand(s) that had around 35 sales?

Worked Solution

Create a strategy

The bar graphs tell us how many sales were made for each brand. The circle graph tells us what portion of sales were each brand.

Apply the idea

We can easily only see this information from the bar graph as the circle graph tells us about the portion not the amount.

The correct answer is A.

Reflect and check

We could use the circle graph to check our answer if we knew the total number of sales.

In which display is it easier to see the tire brand that was sold the most?

Worked Solution

Create a strategy

In the bar graph, we need to look for the bar with the greatest height, and for the circle graph, we need to look for the biggest section.

Apply the idea

We can see this information easily from both graphs. So either A or B is a correct answer.

In which display is it easier to see the tire brands that made up half of the sales?

Worked Solution

Create a strategy

In the bar graph, we can see the totals, and for the circle graph, we can see the portions of a whole.

Apply the idea

The circle graph makes it easier to see fractions or portions, so the correct answer is option B.

Reflect and check

Brands C and E make up one half of the sales and Brands A, B, and D make up the other half.

Example 2

Five teams are completing a walking challenge. A circle graph and dot plot are used to show the distances each team walked.

A dot plot and a circle graph showing the distance walked by 5 different teams. Ask your teacher for more information.

Which team walked the furthest?

Worked Solution

Create a strategy

We need to identify the team with the greatest distance on the graphs. Select the graph where we can easily identify the highest number.

Apply the idea

The Wombats team walked the furthest as clearly shown by the dot plot.

Reflect and check

We can also identify the team that walked the furthest using the circle graph, but using the dot plot is ideal as it better displays the data we need to answer the question.

The part of the circle graph with the biggest section or highest percentage is the team that walked the furthest.

Which display would you use to compare the Eagles and Lions? Explain.

Worked Solution

Create a strategy

The dot plot shows us the number, but we need to count the dots. The circle graph shows the proportions.

Apply the idea

In the dot plot, we can see that there are two more points for Eagles compared to the Lions, but it is a bit difficult because they are not beside each other. The circle graph shows the proportion for each team, and the Eagles and Lions are beside one another which is nice.

It may depend on the question we are looking to answer, but in general using the circle graph to compare the Eagles and Lions, would be a good choice.

Complete this table that would show the same information as the two displays.

Team	Distance (miles)	Percentage of miles
Eagles
Wombats
Hyenas
Roosters
Lions

Worked Solution

Create a strategy

We can get the distance from the dot plot and the percentage from the circle graph.

Apply the idea

Fill in the distance column using the data from the dot plot and the percentage column using the data from the circle graph.

Team	Distance (miles)	Percentage of miles
\text{Eagles}	11	22\%
\text{Wombats}	15	30\%
\text{Hyenas}	8	16\%
\text{Roosters}	7	14\%
\text{Lions}	9	18\%

Reflect and check

We should always double check that the percentages add up to 100\%.

From this we can also see that the total distance walked by all teams was 50 miles.

Idea summary

Different displays are useful to represent different types of data or to answer different questions.

	Strengths	Drawbacks
Circle graph	We can compare the proportions of different categories visually. They are commonly used.	It can be difficult to compare similar groups if they are not labeled. We may lose the original totals if we just show the percentages.
Bar graph	Easily display large data sets. The highest and lowest categories are easy to see.	For large scales, it can be difficult to read off exact values.
Dot plot (line plot)	Useful for individual data values. The highest and lowest categories are easy to see.	Do not work well for large data sets. Can be slower to make by hand.
Pictograph	Can see which categories are the most of least popular. Can be more visually interesting.	Does not work for very large data sets, with too many categories, or for numerical data.
Stem-and-leaf plot	Useful for individual data values. The highest and lowest values are easy to see.	Do not work well for large data sets.
Line graph	Shows how a value changes over time	Cannot be used for categorical data

Outcomes

6.PS.1

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs.

6.PS.1a

Formulate questions that require the collection or acquisition of data with a focus on circle graphs.

6.PS.1b

Determine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).

6.PS.1c

Determine the factors that will ensure that the data collected is a sample that is representative of a larger population.

6.PS.1d

Organize and represent data using circle graphs, with and without the use of technology tools. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20, 7/20, or 35%).

6.PS.1e

Analyze data represented in a circle graph by making observations and drawing conclusions.

6.PS.1f

Compare data represented in a circle graph with the same data represented in other graphs, including but not limited to bar graphs, pictographs, and line plots (dot plots), and justify which graphical representation best represents the data.

9.03 Compare representations of data

Ideas

Compare representations of data

Exploration

Examples

Example 1

Example 2

Outcomes

6.PS.1

6.PS.1a

6.PS.1b

6.PS.1c

6.PS.1d

6.PS.1e

6.PS.1f

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