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11.03 Frequency histograms and polygons

Lesson

Histograms

The histogram is a graph that represents the value of a data type using a column. The label underneath the column tells us what data type the column refers to while the height of the column tells us the value.

In addition to this, a histogram also has labels for both axes that provide information about the axes and tell us what type of values we have.

Exploration

Consider the histogram below:

We can quickly see that, since the column labelled $1$1 is the tallest, the mode of the data is $1$1. We can also see that the column labelled $0$0 has a value of three, and since column $4$4 is at the same height, both $0$0 and $4$4 have a value of three.

The vertical axis label tells us that the values represent the "number of families" while the horizontal axis label tells us that each column represents a specific "number of children in the family".

Putting this information together, we can see that in the survey there were an equal number of families that had $0$0 and $4$4 children; three families in each case.

 

 

Frequency polygons

Frequency polygons are like histograms in that they display the scores on the horizontal axis and the frequencies on the vertical axis. We show the frequency of each score by marking the point above the score to the right of the frequency and connecting the points with line segments.

Histogram

Frequency polygon with the same data

Notice that the vertices of the frequency polygon are in the same place as the tops of the bars of the histogram. We can use frequency polygons in the same way that we use histograms.

Worked example

Create a frequency table from the frequency polygon above.

Think: Since the vertices show the frequencies for each score, we can work backwards and read off the frequencies from the vertices.

Do: Looking at the horizontal axis, the first score is $0$0. We can see that there is a vertex with a height of $2$2. That means that the score $0$0 occurred twice. We can put a $2$2 as the frequency for the score $0$0 in our frequency table. Following the same process for every score gives us this table.

Score Frequency
$0$0 $2$2
$1$1 $10$10
$2$2 $8$8
$3$3 $7$7
$4$4 $3$3
$5$5 $3$3
$6$6 $2$2
$7$7 $9$9
$8$8 $8$8
$9$9 $8$8

Reflect: We fill the frequency table the same way when we're reading a frequency polygon as a histogram. Now that we have a frequency table, we can use it to find the mean or median.

 

Practice questions

Question 1

Fill in the frequency table using the histogram below.

  1. Score Frequency
    $2$2 $\editable{}$
    $3$3 $\editable{}$
    $4$4 $\editable{}$
    $5$5 $\editable{}$
    $6$6 $\editable{}$
    $7$7 $\editable{}$
Question 2

What is the mode of this data set?

Question 3

Find the median for this data set.

Outcomes

MA4-19SP

collects, represents and interprets single sets of data, using appropriate statistical displays

MA4-20SP

analyses single sets of data using measures of location, and range

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