What are data representations?

• 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?

• 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.

  Example

  Language	Number of Students
  Spanish	$200$?
  French	$80$?
  Mandarin	$280$?
  Latin	$40$?

  The left column contains the languages and the right column contains the number of students who want to study each particular language.

  Using the table, we can conclude that the number of students want to study Mandarin ($280$?) and the number of students who want to study Spanish or French ($200+80)$? are the same.
• 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.

  Example

  The heights of the bars show the number of students who want to study each language.

  Using the bar chart, we can conclude that the greatest number of students want to study Mandarin and the least number of students want to study Latin.
• 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.

  Example

  Each

  represents $40$? students.

  The number of pictures shows the number of students who want to study each language.

  Using the pictograph, we can conclude that twice as many students want to study French as want to study Latin.
• 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.

  Example

  The area of each section represents the fraction of students who want to study that language.

  Using the circle graph, we can conclude that just under ${\displaystyle \frac{1}{2}}$? the students want to study Mandarin and about${\displaystyle \frac{1}{3}}$? want to study Spanish.

How are quantitative data displayed?

• 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.

  Example

  Each dot represents the typical travel time to school for one student.

  Using the dotplot, we can conclude that the most common travel time is $10$? minutes. We can also see that the values for travel time range from $5$? to $35$? minutes.
• 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.

  Example

  The height of each bar represents the number of students having a typical travel time within the corresponding interval.

  Using the histogram, we can conclude that the most common travel time is between $10$? and $15$? minutes and that all typical travel times are between $5$? and $40$? minutes.

How are trends over time displayed?

How are relationships between variables displayed?

Your turn!

Things to remember

• 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.