Represent and interpret data

Here you will learn how to represent and interpret data, including how to show categorical and discrete data with tally charts, pictographs and bar graphs.

Students will first learn how to represent and interpret data as part of measurement and data in 1 st grade and continue to grow in their skills through elementary school.

What does it mean to represent and interpret data?

To represent and interpret data, first collect data and then show it visually – in a table or on a graph. This is representing data. Interpreting data is using data analysis to answer questions.

One easy way to collect and represent data is with a tally chart. To do this, sort the data into categories and use tally marks to show the frequencies.

For example,

A benefit of the tally chart is that it is very clear which category occurs the most or least frequently, and can be helpful when analyzing data.

For example,

• Which color is most popular? Green ← This can be clearly seen by the amount of tally marks

*Note: Tally charts can be useful both during data collection and data analysis.

Step-by-step guide: Tally Chart

A pictograph or a picture graph uses pictures to show data.

To draw a pictograph, you use a symbol to represent the frequency. The key of the pictograph shows the value of the symbol.

A benefit of the pictograph is that it is very clear which category occurs the most or least frequently, and can be helpful when analyzing data.

For example,

What is the most popular sport among the students? Tennis ← This can be clearly seen by the amount of smiley faces

Pictographs are different from tally marks, in that pictographs do not always show the exact number.

For example,

• How many students play golf? 4 ← Even though there are only 2 smiley faces, each smiley face is 2 students, so the total is 4

Step-by-step guide: Pictograph

A bar graph shows a data set by using vertical or horizontal bars. The longer the bar, the higher the value for the individual category.

To draw a bar graph:

• Draw a pair of axes. Usually the horizontal axis is labeled with the categories of the data set and the vertical axis is the frequency.
• The frequencies need to be labeled on the vertical axis in equal intervals.
• The bars need to have equal gaps between them.
• The bars need to be of equal width.
• The chart needs a title.

Step-by-step guide: Bar Graph

What is represent and interpret data

Common Core State Standards

How does this relate to elementary school math?

• Grade 1: Measurement and Data (1.MD.C.4)
  Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

• Grade 2: Operations and algebraic thinking (2.OA.C.3)
  Determine whether a group of objects ( up to 20) has an odd or even number of members, example, by pairing objects or counting them by 2 s; write an equation to express an even number as a sum of two equal addends.

• Grade 3 – Measurement and Data (3.MD.B.3)
  Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

• Grade 4 – Number and Operations – Fractions (4.NF.B.3.b)
  Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, example, by using a visual fraction model.

• Grade 5 – Number and Operations – Base Ten (5.NBT.A.2)
  Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.

How to represent and interpret data

There are a lot of ways to represent and interpret data. For more specific step-by-step guides, check out the pages linked in the “What does it mean to represent and interpret data?” section above or read through the examples below.

Represent and interpret data examples

Example 1: favorite school subject

Below is a list of the favorite school subjects for students in class 2 a.

math, science, english, english, social studies, science, social studies, math, science, social studies, social studies, english, science, english, social studies, math, science, science, english, social studies, social studies, english, science, math, social studies, math, science, science, english.

Draw a tally chart to display this information.

1. Draw a data table with \bf{3} columns.

Here you are looking at modes of transport, so the title of the first column is ‘Subject’, then you have our ‘Tally’ column, and then the ‘Frequency’ column:

2Write the category names into each row of the table.

The data can be divided into four categories: math, science, social studies, english.

These are the labels for each row.

3Record the data into the table using five-bar gate tally marks.

Tally each value in the data one at a time.

The final tally chart should look like:

4Work out the frequency for each category by counting the tally marks.

Remember that a five-bar gate represents 5 tally marks.

Example 2: tally chart word problems

The tally chart below shows the favorite pet of first grade students in Ms. Ortega’s class.

(a) How many students were asked about their favorite pet?

(b) How many more students wanted a cat as a pet than a bird?

Example 3: read/interpret pictograph

The pictograph below shows the number of cars sold over the course of 7 days. How many cars were sold on Day 4?

Interpret the data to answer the question.

On the pictograph, Day 4 has 5 cars.

You can skip count starting with 3.

OR

You can add,

3+3+3+3+3=15

There were 15 cars sold on Day 4.

Example 4: construct a pictograph with graphic symbol equal to 1 unit

Use the information in the following table to construct a pictogram of the number of servings of fruit eaten each day over 5 days.

Make sure the table has the correct number of rows.

There are 5 days on the tally chart, so there should be 5 rows on the pictograph.

Label the table.

Make a key.

Place the graphic symbols into the pictograph.

Example 5: constructing a standard bar graph

Draw a bar graph to represent the favorite dessert of students.

Draw the axes with a ruler and label them.

Use a ruler to draw each bar with the correct height.

There are 5 in the first category, so draw a bar with the height of 5 units on the vertical axis. Repeating this for each category, you get the bar graph:

Give the chart a title.

Example 6: solving problems from a bar graph

Mr. Li’s class voted on which activity was their favorite to play at recess. The bar graph below represents the results.

How many more students voted for tag than soccer?

Locate the necessary bar(s).

The two bars that you need to locate are tag and soccer.

Read the frequency from the vertical axis.

The frequency is read from the top of the bar.

7 students voted for tag as their favorite activity.

4 students voted for soccer as their favorite activity.

Teaching tips to represent and interpret data

• Use back to school math activities to introduce students to each other and data in a fun way that involves problem solving. Ask each student to collect data about their new classmates – what is their favorite food, how many siblings do they have, how many books did they read over the summer, etc.
  
  Then have students represent their collected data with a tally chart, pictograph or bar graph. Lastly students can create questions that prompt their classmates to interpret the data.

• Instead of just using math printable worksheets to teach students how to collect, represent and interpret data, utilize interactive online resources and apps in your lesson plans to make learning fun and engaging.

Easy mistakes to make

• Confusing the total frequency and the number of categories
  The number of categories is not the same as the total frequency. Be careful when asked for the total number. This means the total frequency NOT the number of categories.
  
  For example,
  In example 1, there are 4 different school subjects – this is not the total frequency. The total frequency was the total number of students represented by the data.

• Forgetting to group tally marks in fives
  A fifth tally line should always be drawn through the previous four tally marks, creating a five bar gate.
  
  For example,
  To represent 8\text{:}
  ➜ Correct: |||| |||
  ➜ Incorrect: ||||||||

• Thinking part of a pictograph symbol cannot be a whole number
  The use of part of a symbol, does not mean that the actual value is fractional.
  For example,
  
  

  

  In the pictograph above, the smiley face represents 4 people, so half of a smiley face would represent 2 people, which is NOT fractional, but a whole number.

• Not leaving space between bars in a bar graph
  It’s important to leave space between the bars so that it is clear that each bar represents a different group. This also makes the data easier to read and ensures that the bar graph does not look like a histogram.

• Not labeling an axis
  The horizontal and vertical axes need to have labels to name what they represent. This allows you to correctly use and interpret the data given.

The next lessons are

• Ratio
• Percent
• Converting fractions, decimals, and percentages

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