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Tally charts DecimalsCollecting data

Inequalities Mean, median and mode RangeHere you will learn about grouped frequency tables, including what grouped frequency tables are and how to make them.

Students will first learn about grouped frequency tables as part of statistics and probability in 6 th grade.

A **grouped frequency table** (grouped frequency distribution) is a way of organizing a large set of data into more manageable groups.

For example,

Here are the number of correct answers, out of 20, scored by 20 students on a test.

They can be organized into a grouped frequency table like this:

The organized groups of numerical data are called **class intervals**. They can have the same or different class widths and must not overlap. The lower limit is the smallest value in the data class, and the upper limit is the greatest (highest) data value.

To make a grouped frequency table, you have the following columns:

**Intervals**: The first column is for the groups where you write the class intervals.**Tally marks**: This column is for the tally marks.**Frequency**: The frequency column is where you can add up the tally marks.

Grouped frequency tables help to organize the data in order to create graphs such as dot plots, pie charts, and histograms. You can add two more columns (cumulative frequency column and cumulative relative frequency column) to the grouped frequency table to further organize the data.

**Cumulative Frequency**: The cumulative frequency column is used to find the**total frequency**of the data set.**Cumulative relative frequency**: The cumulative relative frequency is a percentage that is found by dividing the frequency by the total number of data values.

Organizing the grouped data will help when drawing and analyzing graphs.

You can use grouped frequency tables to create dot plots and histograms. When creating a dot plot or histogram, adding the cumulative frequency column can help.

For example,

You can use the grouped frequency table here and add the cumulative frequency to the table.

To create the following dot plot,

and histogram,

You can use grouped frequency tables to create a pie chart. When creating a pie chart, adding the cumulative relative frequency column can help.

You find the cumulative relative frequency by dividing the frequency by the total number of data points.

You can use the grouped frequency table here,

To create the following pie chart,

How does this relate to 6 th grade math?

**Grade 6: Statistics and Probability (6.SP.B.4)**Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Use this quiz to check your 2nd and 6th grade students’ understanding of frequency tables. 10+ questions with answers covering a range of 2nd and 6th grade frequency table topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your 2nd and 6th grade students’ understanding of frequency tables. 10+ questions with answers covering a range of 2nd and 6th grade frequency table topics to identify areas of strength and support!

DOWNLOAD FREEIn order to make a grouped frequency table, you need to:

**Using the set of data, account for each item with a tally mark in the table.****Fill in the frequency column.****Find the cumulative frequency and the cumulative relative frequency, if needed.**

Sort these 20 items into the grouped frequency table.

**Using the set of data, account for each item with a tally mark in the table.**

2**Fill in the frequency column.**

Add up the tally marks and fill in the frequency column.

3**Find the cumulative frequency and the cumulative relative frequency, if needed.**

The question doesn’t ask for the cumulative frequency and you are not using the data to create a graph, so you do not need to add the cumulative frequency or cumulative relative frequency column.

Sort these 25 items into the grouped frequency table and find the cumulative frequency.

**Using the set of data, account for each item with a tally mark in the table.**

**Fill in the frequency column.**

Add up the tally marks and fill in the frequency column.

**Find the cumulative frequency and the cumulative relative frequency, if needed.**

To find the cumulative frequency table, add the frequency from the previous group to the current group.

For group 0-4, there is no previous group, so the cumulative frequency will be 3.

For the next group, 5 - 9, you will add the previous group frequency, 3, to the current frequency 3.

3+3=6

For group 10 - 14, add the previous group frequency, 6, to the current frequency, 5.

6+5=11

Continue for the remaining groups.

The cumulative frequency is 25.

Sort these 25 items into the grouped frequency table and find the cumulative relative frequency.

**Using the set of data, account for each item with a tally mark in the table.**

Be careful to check the inequality signs to see which numbers are included in the interval.

For example, the item ‘10’ would be in the first group as the class interval includes 10.

**Fill in the frequency column.**

Add up the tally marks and fill in the frequency column.

**Find the cumulative frequency and the cumulative relative frequency, if needed.**

To compute the cumulative relative frequency, divide the frequency of each by the total number of data points and multiply by 100 to find the percentage.

For example,

For group 0 < x \leq 10, you will first divide 4 by 25.

4 \div 25=0.16

Then to find the percentage, you will multiply 0.16 by 100.

0.16 \times 100=16

The cumulative relative frequency for group 0 < x \leq10 is 16 \%.

Sort these 15 items into the grouped frequency table.

**Using the set of data, account for each item with a tally mark in the table.**

Be careful to check the inequality signs to see which numbers are included in the interval.

**Fill in the frequency column.**

Add up the tally marks and fill in the frequency column.

**Find the cumulative frequency and the cumulative relative frequency, if needed.**

The question doesn’t ask for the cumulative frequency and you are not using the data to create a graph, so you do not need to add the cumulative frequency or cumulative relative frequency column.

In order to create a histogram from a grouped frequency table, you need to:

**Find the cumulative frequency and the cumulative relative frequency on a given grouped frequency table.****Decide what bin size to use and how many bins are needed.****Draw and label the \textbf{x} and \textbf{y} axes with units.****Draw in the bins of the histogram.**

25 randomly chosen workers in Illinois were asked how many minutes they traveled to work.

The data below are the responses.

\begin{aligned} &\bf{10, 20, 45, 30, 25, 20, 15, 10, 30, 45, 50, 5, 40,} \\ &\bf{35, 45, 50, 5, 65, 80, 20, 30, 15, 10, 10, 65} \end{aligned}The data has been organized in a grouped frequency table.

**Decide what bin size to use and how many bins are needed.**

Since the groups on the grouped frequency table are in tens, the width of each bin will be 10. The height of each bin will correspond to the frequency.

**Draw and label the \textbf{x} and \textbf{y} axes with units.**

**Draw in the bins of the histogram.**

In order to create a dot plot from a grouped frequency table, you need to:

**Find the cumulative frequency of the given grouped frequency table.****Decide what bin size to use and how many bins are needed.****Draw and label the \textbf{x} axis with units.****Plot the data on the dot plot and create a key.**

Mr. James asked his 2 nd period class how much time they spent completing homework on an average night. The data below shows the responses.

The data has been organized into a frequency table. Create a dot plot to represent the amount of time students spend on homework on an average night.

**Find the cumulative frequency of the given grouped frequency table.**

To find the cumulative frequency, add the frequency of the current group with the frequency of the group before it.

Because the group 0 < x \leq 10 does not have a group before it, the cumulative frequency will be the same as its frequency.

For all groups following, add the frequency to the frequency of the group before it.

The cumulative frequency is 25, which means that 25 students were asked about the amount of time they spend doing homework each night.

**Decide what bin size to use and how many bins are needed.**

**Draw and label the \textbf{x} axis with units.**

**Plot the data on the dot plot and create a key.**

- Provide project based activities for students where they have opportunity to collect and sort their own data based on their own interests.

- Infuse Google sheets or other digital tools where students can create digital graphs and tables instead of doing them by hand.

- For practice focused on creating graphs, providing students with worksheets that include frequency tables can allow students to focus on creating graphs.

- Step-by-step instructions of how to use grouped frequency tables should be displayed in the classroom for students to refer to when needed. This information can also be included in student math journals for personal reference.

**Not grouping tally marks in the table**

Each tally mark represents one item.

You can group the tally marks into 5 s.

**Writing the classes for grouped frequency tables incorrectly**

The class intervals used in grouped frequency tables can be written in different ways. Take care with inequalities that the item goes in the correct group.

E.g.

0 to 5

0 - 5

0 \leq x < 5

0<x \leq 5

1. Which is the correct grouped frequency table for the following data set?

Checking the first group, there are 3 numbers in the 0 to 9 class. Checking the second group, there are 5 numbers in the 10 to 19 class. And finally there are 7 numbers in the 20 to 29 class.

2. Which is the correct grouped frequency table for the following data set?

Checking the first group, there are 3 numbers in the 0 \leq x < 10 class. Checking the second group, there are 6 numbers in the 10 \leq x < 20 class. And finally there are 6 numbers in the 20 \leq x < 30 class.

3. Which is the correct grouped frequency table for the following set of data?

Checking the first group, there are 5 numbers in the 0 \leq x < 10 class. Checking the second group, there are 7 numbers in the 10 \leq x < 20 class. Finally there are 3 numbers in the 20 \leq x < 30 class.

4. Which is the correct grouped frequency table for the following set of data including the cumulative frequency and the cumulative relative frequency?

Checking the first group, there are 3 numbers in the 0 \leq x < 20 class. Checking the second group, there are 8 numbers in the 20 \leq x < 40 class. And finally there are 4 numbers in the 40 \leq x < 60 class.

To find the cumulative frequency, add the previous group’s frequency to the frequency of the current group, creating a running count. The highest frequency equals the cumulative frequency.

To find the cumulative relative frequency, divide the number of data points in the group by the cumulative frequency. Then multiply by 100 to find the percentage.

5. Which of the following grouped frequency tables matches the given dot plot?

First, check that the intervals given in the grouped frequency tables match the dot plot.

Next, check that the tally and the frequency match.

\begin{aligned} & 20 < x \leq 30-5 \\\ & 30 < x \leq 40-3 \\\ & 40 < x \leq 50-5 \\\ & 50 < x \leq 60-7 \\\ & 60 < x \leq 70-5 \end{aligned}

Last, check that the cumulative frequency is correct. You are left with one grouped frequency tables,

6. Which of the following grouped frequency tables matches the given histogram?

First, check that the intervals given in the grouped frequency tables match the histogram.

Next, check that the frequency and the cumulative relative frequency match the histogram.

51-60:

Frequency: 3

Cumulative frequency: 15 \%

61-70:

Frequency: 5

Cumulative frequency: 25 \%

71-80:

Frequency: 4

Cumulative frequency: 20 \%

81-90:

Frequency: 3

Cumulative frequency: 15 \%

91-100:

Frequency: 5

Cumulative frequency: 25 \%

You are left with one grouped frequency table,

Grouped data is data that is given in the form of class intervals, such as 1-10, 11-20, etc. Ungrouped data is when the data is given as individual data points, values or numbers.

Yes, there are many different real world contexts that would include data with fractions and decimals.

The standard deviation is a value that summarizes the variation around the mean. It is another measure that can be calculated from the data in frequency tables.

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