1.03 Numerical data

Recall that data can be either numerical or categorical. While column (or bar) graphs 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 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.
$45-49$45−49	$50-54$50−54	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 the 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 $45-50$45−50, the class centre is calculated as follows:

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

 

Since 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

 

A histogram is not a bar graph

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

• Histograms show the distribution of data values, whereas a bar graph is used to compare frequencies.
• Histograms are used for numerical data, whereas bar graphs are used for categorical data.
• A histogram has a numerical scale on both axes, while a bar graph only has a numerical scale on the vertical axis (unless the categories are ordinal, which would not be a continuous scale e.g. year, rating).
• The columns in a bar graph could be reordered, 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 reordered.

Histogram		Bar graph

 

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

Example 1

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?

 

1. Think: We want to look at the $70\le t<75$70≤t<75 class interval.
   Do: The class interval contains $40$40 countries.
2. Think: Both the $50\le t<55$50≤t<55 and $55\le t<60$55≤t<60 class intervals need to be considered.
   Do: There are $5$5 countries in the $50\le t<55$50≤t<55 class interval, and $10$10 countries in the $55\le t<60$55≤t<60 class 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

 

Dot plots

Dot plots are a graphical way of displaying the distribution of numerical or categorical data on a simple scale with dots representing the frequency of outcomes. They are best used for small to medium size sets of data and are good for visually highlighting how the data is spread and whether there are any gaps in the data or outliers.

In a dot plot, each individual value is represented by a single dot, displayed above a horizontal line. When data values are identical, the dots are stacked vertically. The graph appears similar to a pictograph or column graph with the number of dots representing the total count.

• To correctly display the distribution of the data, the dots must be evenly spaced in columns above the line
• The scale or categories on the horizontal line should be evenly spaced
• A dot plot does not have a vertical axis
• The dot plot should be appropriately labelled

 

Practice questions

Question 4

 

Stem plot

A stem plot, or stem and leaf plot, is used for organising and displaying numerical data. It is appropriate for small to moderately sized data sets. The graph is similar to a column graph on its side, an advantage of a stem plot over a column graph is the individual scores are retained and further calculations can be made accurately.

In a stem plot, the right-most digit in each data value is split from the other digits, to become the leaf. The remaining digits become the stem.

The values in a stem plot are generally arranged in ascending order (from lowest to highest) from the centre out. To emphasise this, it is often called an ordered stem plot.

The data values $10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58$10,13,16,21,26,27,28,35,35,36,41,41,45,46,49,50,53,56,58 are displayed in the stem plot below.

• The stems are arranged in ascending order, to form a column, with the lowest value at the top
• The leaf values are arranged in ascending order from the stem out, in rows, next to their corresponding stem
• A single vertical line separates the stem and leaf values
• There are no commas or other symbols between the leaves, only a space between them
• In order to correctly display the distribution of the data, the leaves must line up in imaginary columns, with each data value directly below the one above
• A stem plot includes a key that describes the way in which the stem the leaf combine to form the data value

 

Practice questions

QUESTION 5

 

 

Back-to-back stem and leaf plot

Back-to-back stem and leaf plots allow for the display of two data sets at the same time. These types of plots are a great way to make comparisons between data sets.

Reading a back-to-back stem and leaf plot is very similar to a regular stem and leaf plot. The "stem" is used to group the scores and each "leaf" indicates the individual scores within each group. The "stem" is a column and the stem values are written downwards in that column. The "leaf" values are written across in the rows corresponding to the "stem" value. In a back-to-back stem and leaf plot, one set of data is displayed on the left and one set of data is written on the right. The "leaf" values are still written in ascending order from the stem outwards.

 

Practice questions

QUESTION 6