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# Data representations | Lesson

## What are data representations?

We collect both
and
. However, long lists of data points can be difficult to interpret.
Data representations are graphics that display and summarize data and help us to understand the data's meaning.
Data representations can help us answer the following questions:
• How much of the data falls within a specified category or range of values?
• What is a typical value of the data?
• How much spread is in the data?
• Is there a trend in the data over time?
• Is there a relationship between two variables?

## What skills are tested?

• Matching a data set to its graphical representation
• Matching a graphical representation to a description
• Using data representations to solve problems

## How are qualitative data displayed?

Data displays can relate a qualitative variable and a quantitative measure such as a count or percent. Such displays show the data for each different descriptor or category of the qualitative variable.
Example: Mordor University surveys 600 incoming students about which world language they want to study. Here, the qualitative variable is the language that the students want to study and the categories are the particular languages chosen (Spanish, French, Mandarin, and Italian).
A variety of data representations can be used to communicate qualitative (also called categorical) data.
• A table summarizes the data using rows and columns. Each column contains data for a single variable, and a basic table contains one column for the qualitative variable and one for the quantitative variable. Each row contains a category of the qualitative variable and the corresponding value of the quantitative variable.
• A vertical bar chart lists the categories of the qualitative variable along a horizontal axis and uses the heights of the bars on the vertical axis to show the values of the quantitative variable. A horizontal bar chart lists the categories along the vertical axis and uses the lengths of the bars on the horizontal axis to show the values of the quantitative variable. This display draws attention to how the categories rank according to the amount of data within each.
• A pictograph is like a horizontal bar chart but uses pictures instead of the lengths of bars to represent the values of the quantitative variable. Each picture represents a certain quantity, and each category can have multiple pictures. Pictographs are visually interesting, but require us to use the legend to convert the number of pictures to quantitative values.
• A circle graph (or pie chart) is a circle that is divided into as many sections as there are categories of the qualitative variable. The area of each section represents, for each category, the value of the quantitative data as a fraction of the sum of values. The fractions sum to 1. Sometimes the section labels include both the category and the associated value or percent value for that category.

## How are quantitative data displayed?

Data displays for quantitative data are typically oriented along two numerical axes and relate two quantitative variables. Displays of quantitative data help us understand the shape and spread of the data.
Example: Ms. Buehler asks her homeroom students how long it typically takes them to get to school (in minutes) and records their responses in the following list: 5, space, 25, space, 10, space, 20, space, 10, space, 15, space, 35, space, 10, space, 5, space, 20. Here, one quantitative variable is students' typical travel time to school.
A variety of data representations can be used to communicate quantitative data.
• Dotplots use one dot for each data point. The dots are plotted above their corresponding values on a number line. The number of dots above each specific value represents the count of that value. Dotplots show the value of each data point and are practical for small data sets.
• Histograms divide the horizontal axis into equal-sized intervals and use the heights of the bars to show the count or percent of data within each interval. By convention, each interval includes the lower boundary but not the upper one. Histograms show only totals for the intervals, not specific data points.

## How are trends over time displayed?

are data displays that show trends over time. These graphs typically present time (e.g., day, month, or year) on the horizontal axis and another quantitative variable (e.g., temperature, oil price, or income) on the vertical axis.
Each dot on a line graph represents the value of a quantitative variable at a particular time, and the dots are connected to form graph.

## How are relationships between variables displayed?

display data about two quantitative variables as a set of points in the x, y-plane.
A scatterplot is a key tool to determine if there is a relationship between the values of two variables.

TRY: MATCHING TABLE DATA TO A BAR CHART
A12
B8
C4
D2
Which of the following best represents the data above?

TRY: MATCHING A DESCRIPTION TO A GRAPH
Jitendra notices that the more months a cell phone model has been out, the lower its price gets. Which of the following graphs best represents the price of a cell phone over time ?

TRY: INTERPRETING A HISTORGRAM
The graph above shows the size (in square meters) of each house on Pine Street. How many houses on Pine Street have an area less than 150 square meters?
houses

TRY: INTERPRETING A DOTPLOT
The dotplot above shows the number of hours Kim worked each weekend since she started her new job. Each dot represents a different weekend. On what fraction of the weekends did Kim work at least 10 hours?

## Things to remember

Data representations are useful for interpreting data and identifying trends and relationships.
When working with data representations, pay close attention to both the data values and the key words in the question.
• When matching data to a representation, check that the values are graphed accurately for all categories.
• When reporting data counts or fractions, be clear whether a question asks about data within a single category or a comparison between categories.
• When finding the number or fraction of the data meeting a criteria, watch for key words such as or, and, less than, and more than.

## Want to join the conversation?

• the last one gets a little tricky.