# 11.02 Frequency tables

Lesson

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.

Frequency table

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

To find the median, we can find the cumulative frequency for each score. The cumulative frequency is the sum of the frequencies of the score and each of the scores below it. The cumulative frequency of the first row will be the frequency of that row. For each subsequent row, add the frequency to the cumulative frequency of the row before it.

Score ($x$x) Frequency ($f$f) Cumulative frequency
$1$1 $6$6 $6$6
$2$2 $9$9 $15$15
$3$3 $1$1 $16$16
$4$4 $6$6 $22$22
$5$5 $8$8 $30$30
$6$6 $6$6 $36$36
$7$7 $6$6 $42$42
$8$8 $2$2 $44$44
$9$9 $8$8 $52$52

The final row has a cumulative frequency of $52$52, so there are $52$52 scores in total. This means that the median will be the mean of the $26$26th and $27$27th scores in order.

Looking at the cumulative frequency table, there are $22$22 scores less than or equal to $4$4 and $30$30 scores less than or equal to $5$5. This means that the $26$26th and $27$27th scores are both $5$5, so the median is $5$5.

Finally, we can find the range just by looking at the score column. The highest score is $9$9 and the lowest is $1$1, so the range will be $9-1=8$91=8.

### Grouped frequency tables

When the data are more spread out, sometimes it doesn't make sense to record the frequency for each separate result and instead we group results together to get a grouped frequency table.

Grouped frequency table

A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.

#### Exploration

A teacher wants to express the heights (in cm) of her students in a table using the following data points:

$189,154,146,162,165,156,192,175,167,174$189,154,146,162,165,156,192,175,167,174

$161,153,184,177,155,192,169,166,148,170$161,153,184,177,155,192,169,166,148,170

$168,151,186,152,195,169,143,164,170,177$168,151,186,152,195,169,143,164,170,177

She realises that if each result has its own frequency then the table would have too many rows, so instead she grouped the results into sets of $10$10 cm. As a result, her grouped frequency table looked like this:

Height (cm) Frequency
$140-149$140149
$150-159$150159
$160-169$160169
$170-179$170179
$180-189$180189
$190-199$190199

To fill in the frequency for each group, the teacher counted the number of results that fell into the range of each group.

For example, the group $150-159$150159 would include the results:

$154,156,153,155,151,152$154,156,153,155,151,152

Since there are $6$6 results that fall into the range of this group, this group has a frequency of $6$6.

Using this method, the teacher filled in the grouped frequency table to get:

Height (cm) Frequency
$140-149$140149 $3$3
$150-159$150159 $6$6
$160-169$160169 $9$9
$170-179$170179 $6$6
$180-189$180189 $3$3
$190-199$190199 $3$3

Looking at the table, she can see that the modal class is the group $160-169$160169, since it has the highest frequency.

By adding the frequencies in the bottom two rows she could also see that $6$6 students were at least $180$180 cm tall. There are $30$30 students in the class in total, so she now knows that $\frac{6}{30}$630 of her students, or one fifth of the class, are taller than $180$180 cm.

Modal class

The modal class in a grouped frequency table is the group that has the highest frequency.

If there are multiple groups that share the highest frequency then there will be more than one modal class.

As we can see, grouped frequency tables are useful when the data are more spread out. While the teacher could have obtained the same information from a normal frequency table, the grouping of the results condensed the data into an easier to interpret form.

However, the drawback of a grouped frequency table is that the data becomes less precise, since we have grouped multiple data points together rather than looking at them individually.

### Summarising data from a grouped frequency table

When finding the mean and median of grouped data we want to first find the class centre of each group. The class centre is the mean of the highest and lowest possible scores in the group.

#### Exploration

Estimate the mean and median of the following data.

Group Frequency ($f$f)
$1-5$15 $7$7
$6-10$610 $2$2
$11-15$1115 $4$4
$16-20$1620 $7$7

First we find the class centre for each group. This is just the average of the endpoints of the group. For example, the first group is $1$1 to $5$5, so the class centre is $\frac{1+5}{2}=3$1+52=3.

Group Class Centre ($x$x) Frequency ($f$f)
$1-5$15 $3$3 $7$7
$6-10$610 $8$8 $2$2
$11-15$1115 $13$13 $4$4
$16-20$1620 $18$18 $7$7

Notice that we've given the class centre the pronumeral $x$x this time. This is because we will use the class centre in the same way that we used the score for ungrouped data.

To find the mean, we want to make an $xf$xf column again. In this case, $x$x is the class centre.

Group Class Centre ($x$x) Frequency ($f$f) $xf$xf
$1-5$15 $3$3 $7$7 $21$21
$6-10$610 $8$8 $2$2 $16$16
$11-15$1115 $13$13 $4$4 $52$52
$16-20$1620 $18$18 $7$7 $126$126

Dividing the sum of the $xf$xf column by the sum of the $f$f column gives us $\frac{21+16+52+126}{7+2+4+7}=\frac{215}{20}=10.75$21+16+52+1267+2+4+7=21520=10.75.

Similarly for the median we want to make a cumulative frequency table.

Group Class Centre ($x$x) Frequency ($f$f) Cumulative frequency
$1-5$15 $3$3 $7$7 $7$7
$6-10$610 $8$8 $2$2 $9$9
$11-15$1115 $13$13 $4$4 $13$13
$16-20$1620 $18$18 $7$7 $20$20

Since there are $20$20 scores, we look for the $10$10th and $11$11th scores, which are both in the group $11-15$1115. While we don't know the exact score of the median, we can use the class centre $13$13 as our estimate for the median.

#### Practice questions

##### Question 1

Fill in the frequency table using the data set below.

$77,54,53,56,73,55,94,95,76,52,72,46,85,61,48,90,64,70,40,52,57,88,59,95,61$77,54,53,56,73,55,94,95,76,52,72,46,85,61,48,90,64,70,40,52,57,88,59,95,61

1. Class Frequency Cumulative frequency
$40$40$-$$4949 \editable{} \editable{} 5050-$$59$59 $\editable{}$ $\editable{}$
$60$60$-$$6969 \editable{} \editable{} 7070-$$79$79 $\editable{}$ $\editable{}$
$80$80$-$$8989 \editable{} \editable{} 9090-$$99$99 $\editable{}$ $\editable{}$
##### Question 2

Find the median for this data set.

Score Frequency Cumulative frequency
$2$2 $3$3 $3$3
$3$3 $5$5 $8$8
$4$4 $3$3 $11$11
$5$5 $4$4 $15$15
$6$6 $8$8 $23$23
$7$7 $2$2 $25$25
##### Question 3

Find the mean for this data set.

Score ($x$x) Frequency ($f$f) $xf$xf
$2$2 $7$7 $14$14
$3$3 $2$2 $6$6
$4$4 $8$8 $32$32
$5$5 $5$5 $25$25
$6$6 $4$4 $24$24
$7$7 $7$7 $49$49

### 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