22.09 Grouped data (Extended)

We have already seen that data can be categorised as either numerical or categorical. While bar charts are preferred for displaying categorical data, histograms are the preferred option for data that is numerical.

 

Histograms and grouped data

Continuous numerical data, such as times, heights, weights or temperatures, are based on measurements, so any data value is possible within a large range of values. For displaying this type of data, a special chart called a histogram is used.

As an example, the following frequency distribution table and histogram represent the times taken for $72$72 runners to complete a ten kilometre race.

Class interval	Frequency
$45\le\text{time }<50$45≤time <50	$9$9
$50\le\text{time }<55$50≤time <55	$7$7
$55\le\text{time }<60$55≤time <60	$20$20
$60\le\text{time }<65$60≤time <65	$30$30
$65\le\text{time }<70$65≤time <70	$6$6

The histogram represents the distribution of the data. It allows us to see clearly where all of the recorded times fall along a continuous scale.  

 

Class intervals

What may surprise us at first is that the histogram above has only five columns, even though it represents $72$72 different data values. 

To produce the histogram, the data is first grouped into class intervals (also known as classes or bins), using the frequency distribution table.

In the table above,

• The first class interval includes the running times for $9$9 different runners. Each of their times fall within a range that is greater than or equal to $45$45 minutes, but less than $50$50 minutes. This class interval is represented by the first column in the histogram.
   
• The second class interval includes the running times for $7$7 different runners, each with times falling with a range greater than or equal to $50$50 minutes, but less than $55$55 minutes. This class interval is represented by the second column in the histogram, and so on.

 

 

There are several different ways that class intervals are defined. Here are some examples with two adjacent class intervals:

Class interval formats		Description
$45<\text{time }\le50$45<time ≤50	$50<\text{time }\le55$50<time ≤55	Upper endpoint included, lower endpoint excluded.
$45\le\text{time }<50$45≤time <50	$50\le\text{time }<55$50≤time <55	Lower endpoint included, upper endpoint excluded.
$45$45 to $<50$<50	$50$50 to $<55$<55	Lower endpoint included, upper endpoint excluded.
$45$45 - $49$49	$50$50 - $54$54	Suitable for data rounded to the nearest whole number, or discrete data.
$45$45 → $50$50	$50$50 → $55$55	Not clear which endpoints are included or excluded. Assume upper endpoint is included.

Regardless of the format used, each class interval for a given set of data should be consistent across all class intervals.

Note: In this course, class intervals for any particular set of data will be the same width. There are situations in data representation when class intervals are different widths, but this is beyond the scope of this course.

 

 

Class centre

The class centre is the average of the endpoints of each interval.

For example, if the class interval is $45\le\text{time }<50$45≤time <50, or $45$45 - $50$50, the class centre is calculated as follows:

class centre	$=$=	$\frac{45+50}{2}$45+502​
	$=$=	$47.5$47.5

 

Because the class centre is an average of the endpoints, it is often used as a single value to represent the class interval. In some histograms, it may be used for the scale on the horizontal axis, with the class centre displayed directly below the middle of each vertical column.

 

Practice question

Question 1

 

Frequency polygon

Using the example of running times, we can add a 'class centre' column to the frequency distribution table.

Class interval	Class centre	Frequency
$45\le\text{time }<50$45≤time <50	$47.5$47.5	$9$9
$50\le\text{time }<55$50≤time <55	$52.5$52.5	$7$7
$55\le\text{time }<60$55≤time <60	$57.5$57.5	$20$20
$60\le\text{time }<65$60≤time <65	$62.5$62.5	$30$30
$65\le\text{time }<70$65≤time <70	$67.5$67.5	$6$6

The class centre is used to create an alternative to the histogram, called a frequency polygon. 

A frequency polygon is a line graph that displays the frequency distribution of a set of data, and for that reason, is similar to a histogram. If we draw a frequency polygon and histogram together, the polygon begins and ends on the horizontal axis and connect the midpoints at the top of each column, as shown below. 

• Notice that the class centres have been used as the scale on the horizontal axis. Each point on the frequency polygon is a coordinate pair made up of the class centre and the frequency: $\left(\text{class centre },\text{frequency }\right)$(class centre ,frequency ).
• If we look at the way the frequency polygon 'cuts off' triangles from the columns of the histogram, we can see that the area under the frequency polygon is equal to the area of the columns.
• A frequency polygon can be drawn together with a frequency histogram or it can be displayed on its own.

 

A histogram is not a bar chart

Although a histogram looks similar to a bar chart, there are a number of important differences between them:

• Histograms show the distribution of data values, whereas a bar chart is used to compare data values.
• Histograms are used for numerical data, whereas bar charts are used for categorical data.
• A histogram has a numerical scale on both axes, while a bar chart only has a numerical scale on the vertical axis. 
• The columns in a bar chart could be re-ordered, without affecting the representation of the data. In a histogram each column corresponds with a range of values on a continuous scale, so the columns cannot be re-ordered.

Histogram		Bar chart

 

Key features of a histogram

• The horizontal axis is a continuous numerical scale (like a number line). It represents numerical data, such as time, height, mass or temperature, and may be divided into class intervals.
• The vertical axis is the frequency of each data value or class interval.
• There are no gaps between the columns, because the horizontal axis is a continuous scale. It is possible for a class interval to have a frequency of zero, but this is not the same as having gaps between each column.
• The area of each column, rather than the height, is proportional to the frequency. This is because histograms can have columns of different widths. If all the columns are equal width, then the height of each column will be proportional to the frequency.
• It is good practice, when creating a histogram, to leave a half-column-width gap between the vertical axis and the first column.

 

To better understand histograms, we will look at an example of how a histogram is created from a set of raw data.

 

Worked example

In 2016, the World Health Organisation (WHO) collected data on the average life expectancy at birth for $183$183 countries around the world.

To appreciate what the raw data looks like, here is a reduced version of the data set. 

$62.7$62.7	$76.4$76.4	$76.4$76.4	$62.6$62.6	$75.0$75.0	$76.9$76.9	$74.8$74.8
$82.9$82.9	$81.9$81.9	$73.1$73.1	$75.7$75.7	$79.1$79.1	$72.7$72.7	$75.6$75.6
	$\vdots$⋮		$\vdots$⋮		$\vdots$⋮
$62.5$62.5	$72.5$72.5	$77.2$77.2	$81.4$81.4	$63.9$63.9	$78.5$78.5	$77.1$77.1
$72.3$72.3	$72.0$72.0	$74.1$74.1	$76.3$76.3	$65.3$65.3	$62.3$62.3	$61.4$61.4

Each value represents the average life expectancy at birth (in years) for a single country. 

Before organising the data into a frequency distribution table, we need to decide on the number of class intervals. Although there is no fixed rule, using between $5$5 and $10$10 class intervals usually produces good results for most data sets. 

We know that the lowest life expectancy in the data is $52.9$52.9 years (Lesotho in southern Africa), while the highest is $84.2$84.2 (Japan). These values indicate that the scale on the horizontal axis of our histogram should be from at least $50$50 to $85$85 years. It seems appropriate to have class intervals of width $5$5 years, which means we will have $7$7 class intervals in total. We'll use the variable $t$t to represent average life expectancy in the table below:

class interval	frequency
$50\le t<55$50≤t<55	$5$5
$55\le t<60$55≤t<60	$10$10
$60\le t<65$60≤t<65	$25$25
$65\le t<70$65≤t<70	$26$26
$70\le t<75$70≤t<75	$40$40
$75\le t<80$75≤t<80	$49$49
$80\le t<85$80≤t<85	$28$28

This compact table now represents all $183$183 life expectancy values.

With our frequency distribution table complete, we are ready to create a histogram:

Use the histogram to answer the following questions:

1. How many countries have an average life expectancy greater than or equal to $70$70 but less than $75$75 years?
2. How many countries have an average life expectancy lower than $60$60 years? 

Solution

1. The $70\le t<75$70≤t<75 year class interval contains $40$40 countries.
2. Both the $50\le t<55$50≤t<55 and $55\le t<60$55≤t<60 class intervals need to be considered. 
   There are $5$5 countries in the $50\le t<55$50≤t<55 year interval.
   There are $10$10 countries in the $55\le t<60$55≤t<60 year interval.
   So there are $5+10=15$5+10=15 countries with an average life expectancy lower than $60$60 years.

 

 

Practice questions

Question 2

Question 3