12.02 Create and interpret frequency tables, histograms and polygons
When representing the frequency of different results in our data, we often choose to use a frequency table.
Exploration
Suppose that Melanie wanted to find the least common colour of car in her neighbourhood. To help her find an answer to this, she conducted a survey by observing the colours of the cars passing through her street.
By sitting in front of her house and recording the colour of the first $20$20 cars that drove past, Melanie obtained the following data:
white, black, white, black, black, blue, blue, white, red, white,
white, blue, orange, blue, white, white, orange, red, blue, red
In order to better interpret her data, Melanie converted this list of colours into a frequency table, counting the number of cars corresponding to a particular colour and writing that number in the frequency column next to that colour.
| Car colour | Frequency |
|---|---|
| Red | $3$3 |
| Black | $3$3 |
| White | $7$7 |
| Orange | $2$2 |
| Blue | $5$5 |
Looking at her table, Melanie found that orange was the least common car colour in her neighbourhood as it had the lowest frequency.
Melanie has answered her initial question, but she realises she can use the same data to answer other questions about the colours of cars in her neighbourhood.
a) What fraction of the cars were black?
We can read from the table that $3$3 cars were black. Since Melanie recorded the colour of $20$20 cars, this means that $3$3 out of $20$20 of the cars were black. We can express this as the fraction $\frac{3}{20}$320.
b) What was the most common colour of car?
Looking at the table, we can see that the result with the highest frequency is the colour white, so this was the most common colour.
A frequency table communicates the frequency of each result from a set of data. This is often represented as a column table with the far-left column describing the result and any columns to the right recording frequencies of different result types.
As seen in the exploration, frequency tables can help us find the least or most common results among categorical data. They can also allow us to calculate what fraction of the data a certain result represents.
When working with numerical data, frequency tables can also help us to answer other questions that we might have about how the data are distributed.
Summarising data from a frequency table
We can find the mode, mean, median and range from a frequency table. These will be the same as the mode, mean, median and range from a list of data but we can use the frequency table to make it quicker.
Exploration
Find the mode, mean, median and range of the following data.
| Score ($x$x) | Frequency ($f$f) |
|---|---|
| $1$1 | $6$6 |
| $2$2 | $9$9 |
| $3$3 | $1$1 |
| $4$4 | $6$6 |
| $5$5 | $8$8 |
| $6$6 | $6$6 |
| $7$7 | $6$6 |
| $8$8 | $2$2 |
| $9$9 | $8$8 |
The mode is the score with the highest frequency. Looking at the frequency table, the score $2$2 has a frequency of $9$9 and all of the other scores have a lower frequency. So the mode is $2$2.
To find the mean we add together all of the scores. Since each score occurs multiple times, we can save time by multiplying the scores by the frequencies. Notice that we've assigned the score the pronumeral $x$x and the frequency the pronumeral $f$f. We want to find the product $xf$xf for each score.
| Score ($x$x) | Frequency ($f$f) | $xf$xf |
|---|---|---|
| $1$1 | $6$6 | $6$6 |
| $2$2 | $9$9 | $18$18 |
| $3$3 | $1$1 | $3$3 |
| $4$4 | $6$6 | $24$24 |
| $5$5 | $8$8 | $40$40 |
| $6$6 | $6$6 | $36$36 |
| $7$7 | $6$6 | $42$42 |
| $8$8 | $2$2 | $16$16 |
| $9$9 | $8$8 | $72$72 |
Now if we add up the $xf$xf column, we will get the sum of all of the scores, and if we add up the frequency column we will get the total number of scores. Dividing the two sums will give us the mean.
| $\frac{\text{Sum of all scores}}{\text{Total number of scores}}$Sum of all scoresTotal number of scores | $=$= | $\frac{6+18+3+24+40+36+42+16+72}{6+9+1+6+8+6+6+2+8}$6+18+3+24+40+36+42+16+726+9+1+6+8+6+6+2+8 |
Using the definition of the mean |
| $=$= | $\frac{221}{52}$22152 |
Evaluate the sums |
|
| $\frac{\text{Sum of all scores}}{\text{Total number of scores}}$Sum of all scoresTotal number of scores | $=$= | $4.25$4.25 |
Evaluate the quotient |
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 4
Fill in the frequency table using the histogram below.
Frequency |
 |
| Score |
Score Frequency $2$2 $\editable{}$ $3$3 $\editable{}$ $4$4 $\editable{}$ $5$5 $\editable{}$ $6$6 $\editable{}$ $7$7 $\editable{}$
Question 5
What is the mode of this data set?
Question 6
Find the median for this data set.
