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# Line Graphs

Julie has recorded the number of ice creams she sold in her truck depending on the temperature outside. She wants to identify if there is a possible connection between the two variables. What do you think she could do to analyse this correlation? Well, it is always helpful to view the comparison between a set of objects visually through a graph.

However, we now need to think about the type of graph we could potentially use to plot these two variables against each other. Any guesses? Let me give you a hint: line graphs! Throughout this topic, we aim to explore the concept of a line graph and analyse its behaviour. The main objective here is to observe whether any significant relationships arise between the given variables we are investigating. So, let's begin!

## Definition of a Line Graph

A line graph is a tool used to graphically display information, linking two variables where one is usually plotted on the x-axis and the other on the y-axis. Each point is plotted based on its x- and y-values. All the points are then connected together through straight line segments. Below is an example of a line graph used to display the number of rainy days in Dubai per year.

Line graph displaying the number of rainy days in Dubai per year between 2008 and 2012, StudySmarter Originals

Say we are told to find the number of rainy days in a given year. In order to do this, we would simply need to find the y value of the plotted line on that given year.

For example, if we were looking to find the number of rainy days in 2011, we can see that the line for that year arrives at a y-value of 40. Hence, the number of rainy days that year would be 40.

A line graph may sometimes be referred to as a line chart. They are often used to illustrate patterns that may arise between a pair of data sets and make predictions about future data values. One data set is usually dependent on the other.

A line graph is a type of graph that displays data that varies over time. It is made of a set of points that are joined by a straight line.

## Components of a Line Graph

Let us return to our previous line graph above. There are seven vital components we need to consider when observing or plotting a given line graph.

Components of a graph, StudySmarter Originals

 No. Component Description 1 Axes There are 2 axes to take into account hereHorizontal axis: also known as the x-axis or independent axis.Vertical axis: also known as the y-axis or dependant axis.These axes intersect at the origin (0, 0). 2 Title The title explains what the graph is about. 3 Labels The horizontal label at the bottom and the vertical label on the side indicate the type of data displayed. 4 Scales The horizontal scale at the bottom and the vertical scale on the side represent the amount or number of items. 5 Points The points are denoted by an ordered pair of coordinates, (x, y) 6 Lines The line connecting a pair of points illustrates an estimated value between them. 7 Slope The slope of a line demonstrates its steepness and is used to compare the change in magnitude between a pair of successive points on a graph (e.g. a steeper slope infers a larger difference in magnitude between two points)

## Types of Line Graphs

There are three main different types of line graphs, namely:

1. Simple line graph

2. Multiple line graph

3. Compound line graph

Each of these graphs has its own way of calculating key statistical figures in relation to the data they contain within them. Before we explore each type above, let us learn how to interpret a given line graph.

### How to Read a Line Graph

Say we are given a line graph. We are told to decipher the information illustrated by this graph. In this case, we can carry out a 5-step process to do this.

1. Observe the title of the graph;

2. Recognise the axis labels;

3. Analyse the slope between each pair of consecutive points. Make note of any significant trends;

4. Determine the exact figures by looking at the corresponding data points;

5. Evaluate the mean, median, mode and range.

With that in mind, let us now the three types of line graphs mentioned above in turn.

### Simple line graph

A simple line graph is a graph showing one data set only. This means that only one line is plotted on the graph. Let us go back to our example mentioned at the beginning of this topic.

Below is a line graph displaying the number of ice creams sold by Julie's truck at different temperatures. This is the title of the line graph.

Line graph showing the number of ice creams sold by Julie's truck at different temperatures, StudySmarter Originals

The horizontal axis (or x-axis) shows the temperature for a particular day. This is the independent axis. The vertical axis (or y-axis) shows the number of ice creams sold. These values are dependent on the temperature.

Let us now break down the contents of this graph. We shall first observe the slope between each consecutive pair of points. Notice that there is indeed a pattern here: the greater the temperature, the more ice creams are sold. Therefore, if we were to extend the data set for the horizontal axis, we would expect a continuous positive slope.

Say we are told to find the total number of ice creams sold between these 5 temperatures. Here, we would simply sum up all the y-values (number of ice creams sold) corresponding to each x-value (temperature). Thus, the total number of ice creams sold is:

12 + 47 + 55 + 72 + 89 = 275

Let us now attempt to find the mean, median, mode and range of this graph. Note that the mean, median, mode and range of a graph are known as the central tendency in statistics!

Recall the definitions of the central tendencies listed above.
• Mean: This is the average result from our sample and is calculated by adding all the numbers in our data set and dividing that total (or sum) by the number of values.
• Median: This is the middle number upon listing them from the smallest to the largest number.
• Mode: This is the number with the highest frequency in our data set.
• Range: This is the difference between the largest and smallest number in our data set.
 Central Tendency Calculation Mean As the total number of ice creams sold is equal to 275 and there are 5 different values of temperatures, the mean is,Thus, Julie sells an average of 55 ice creams each day. Median Listing the number of ice creams sold in ascending order, we Hence, the median is 55 ice creams, which were sold on the day when the temperature was 20 Celsius. Mode The mode is equal to 89 ice creams, which were sold on the day when the temperature was 30 Celsius. Range The smallest number of ice creams sold is 12 while the largest number of ice creams sold is 89. The difference is,Therefore, the range is 77 ice creams.

### Multiple line graph

A multiple line graph consists of a diagram with more than one plotted line.

Let us bring back the graph displaying the number of ice creams sold by Julie's truck at different temperatures, but this time between the years 2016 and 2017. We shall assume that the previous graph represents the number of ice creams sold by Julie in the year 2016. The line graphs for 2016 and 2017 are displayed in a singular graph below and are represented by the blue line and pink line respectively.

Line graph showing the number of ice creams sold by Julie's truck at different temperatures between 2016 and 2017, StudySmarter Originals

Deciphering this graph is similar to the simple line graph case. However, the only difference here is that we would have to analyse more than one data set and deal with more numbers. Since we are dealing with more values, be careful not to make careless mistakes! Always list your data set so that your calculations are accurate.

Let us first take note of any distinct patterns on the graph above. There are two patterns here:

1. The greater the temperature, the more ice creams are sold. This is the same trend we found in our previous example.

2. The number of ice creams sold in 2017 is greater than the number of ice creams sold in 2016 for each temperature value. This is because each pink point happens to be plotted above each corresponding blue point. This essentially means that Julie received more business in 2017 compared to 2016.

Now let us find the total number of ice creams sold between these 5 temperatures for each year. From the previous example, we know that the total number of ice creams sold in 2016 is 275. We can apply the same method to find the total number of ice creams sold in 2017.

21 + 57 + 64 + 78 + 100 = 320

Say we are also told to find the total number of ice creams sold in both years. To do this, we simply need to add the total number of ice creams sold in 2016 and the total number of ice creams sold in 2017. This yields,

275 + 320 = 595

We will now evaluate the mean, median, mode and range of this graph.

 Central Tendency Calculation Mean The average number of ice creams sold in 2016 is 55, as we have calculated before. The average number of ice creams sold in 2017 is 64 sinceThe average number of ice creams sold in both years is found by dividing the sum of the average number of ice creams sold in 2016 and the average number of ice creams sold in 2017 by 2. Rounding this off to the nearest whole number, we find that the average number of ice cream sold in both years is 60. Median In 2016, the median is 55 ice creams, which were sold on the day when the temperature was 20 Celsius. In 2017, the median is 64 ice creams, which were sold on the day when the temperature was 20 Celsius. Mode In 2016, the mode is equal to 89 ice creams, which were sold on the day when the temperature was 30 Celsius. In 2017, the mode is equal to 100 ice creams, which were sold on the day when the temperature was 30 Celsius. Range In 2016, the range is 77 ice creams, as we have calculated before. The smallest number of ice creams sold in 2017 is 21 while the largest number of ice creams sold is 100. The difference is,Therefore, the range in 2017 is 79 ice creams.

### Compound line graph

A compound line graph shows the layers of data showing what proportions make up the total values. These graphs are also known as stacked area graphs. The way these graphs are read is by finding the difference between the highest part of a specific area and the lowest part of that same area, at the same x-coordinate.

Let us show this with an example. The graph below shows the number of vehicles sold by a motor company in Dundee throughout 2018. The graph shows the quantity for 3 types of vehicles: cars, trucks and motorcycles. The graph is also divided into 4 sections (called quartiles) throughout the year: Quarter 1, Quarter 2, Quarter 3 and Quarter 4.

Number of vehicles sold by a motor company in Dundee in 2018, StudySmarter Originals

To analyse this type of graph, we simply need to identify the upper and lower bounds of each quartile. For example, say we are told to find the number of motorcycles sold in the 4th Quarter. Observe the section in which this quarter is displayed. The number of motorcycles sold in the 4th Quarter has an upper bound of 26 and lower bound of 21. The number of motorcycles sold is simply the difference between these two values.

26 – 21 = 5

Thus, 5 motorcycles were sold in the 4th Quarter of 2018. In the same manner, try working out the number of trucks and cars sold in the 2nd Quarter and 3rd Quarter. Remember to take note of the upper and lower bounds for each case!

## How to Plot a Line Graph

We shall now move on to drawing line graphs. This follows a 5-step procedure, as listed below.

1. Construct the x and y-axes. Label each axis clearly. Ensure that the axis scales are divided into even intervals;

2. Come up with a brief and informative title for your graph. Make sure that it relays the purpose of your graph;

3. Plot the data on the graph with respect to the corresponding x and y-values given;

4. Connect each pair of successive points with a straight line segment;

5. If you are comparing several data sets, make a key to distinguish the line graph for each sample using different colours. It is often advised to only have a maximum of 4 data sets in one line graph to avoid confusion.

Let's show this with an example.

A farm wants to observe the number of offspring a rabbit produces over the years 2008 and 2011. The table below shows the number of offspring for each year.

 Year Number of Offspring 2008 16 2009 27 2010 5 2011 19

Create a line graph to show this data.

Solution

Let us first create the axes for this graph and label them. Furthermore, a concise title for this graph can be: Number of Offspring Produced Between 2008 and 2011.

Example 1, Part 1, StudySmarter Originals

Let us now plot our given data to illustrate our complete line graph.

Example 1, Part 2, StudySmarter Originals

Just like any other graph, line graphs also have their pros and cons. The table below describes the advantages and disadvantages of line graphs.

 Advantages Disadvantages Able to show contrast and patterns over a given timeframe. Confusion may arise if too many lines are plotted within the same axes. Creates clutter as well. Simple to understand and efficient. It may be difficult to plot a wide range of data over a line graph. Able to point out anomalies found across a sample. Inaccuracies may occur if inappropriate increments for each axis scale are used. Able to plot more than one line within the same axis. This allows comparisons to be made. Only ideal for representing data that have total figures such as whole numbers. Plotting values involving fractions and decimals can be challenging.

## More Examples Involving Line Graphs

Let us end this discussion with two more worked examples applying line graphs.

Sarah decided to record the number of books she read each month between the years 2020 and 2021. She constructed a table to keep track of how much she has read over these two years.

 Month 2020 2021 January 3 6 February 4 3 March 4 2 April 5 1 May 2 2 June 1 2 July 5 4 August 2 5 September 3 6 October 2 5 November 4 3 December 4 1

Construct a line graph that displays this data. In which year did she read the most books?

Solution

Let us first come up with a simple title. We shall call it: Number of books Sarah read per month. Now let us assemble the axes of this graph.

Example 2, Part 1, StudySmarter Originals

We shall now plot our given data to illustrate our complete line graph.

Example 2, Part 1, StudySmarter Originals

To find the total number of books Sarah read each year, we simply add the number of books read each month throughout the years 2020 and 2021 respectively.

Total number of books read in 2020: 39

Total number of books read in 2021: 40

Thus, Sarah read more books in 2021 compared to 2020.

The line graph below shows the number of oranges harvested by one of Samuel's orange trees between the months of July and December.

Example 3, StudySmarter Originals

Looking at the figure above, answer the following questions:

1. How many oranges did Samuel harvest between the months of July and December?
2. What trend do you observe in this graph?
3. What is the average number of oranges harvested between these months?
4. In which month did Samuel yield the most oranges?

Solution

Question 1: To find the total number of oranges, we simply need to add the yield of each month as described on the y-axis of the line graph above. Summing these values up, we obtain

52 + 61 + 66 + 45 + 32 + 29 = 285

Thus, a total of 285 oranges were harvested between the months of July and December.

Question 2: Based on the graph above, we find that there is an increase in yield between the months of July and September. However, between the months of September and December, there is a decrease in yield.

Question 3: To find the average (or mean) we simply divide the total number of oranges produced between the months of July and December and divide it by 6 since we are analysing Samuel's harvest over a period of 6 months.

Rounding this off to the nearest whole number, we find that the average number of oranges produced by Samuel's orange tree is 48.

Question 4: Here, we want to observe the highest point of the graph. We find that the peak of this line graph is situated in the month of September when Samuel harvested a total of 66 oranges.

## Line graphs - Key takeaways

• Line graphs are used to represent trends in a set of data.
• There are three types of line graphs:
• Simple line graphs: only one data set (one line) is present
• Multiple line graphs: contain multiple data sets within a graph
• Compound line graphs: have layers adding to the total values measured with two or more fields explored
1. Observe the title of the graph;

2. Recognise the axis labels;

3. Analyse the slope between each pair of consecutive points. Make note of any significant trends;

4. Determine the exact figures by looking at the corresponding data points;

5. Evaluate the mean, median, mode and range.

• Drawing line graphs:

1. Construct the x and y-axes. Label each axis clearly. Ensure that the axis scales are divided into even intervals;

2. Come up with a brief and informative title for your graph. Make sure that it relays the purpose of your graph;

3. Plot the data on the graph with respect to the corresponding x and y-values given;

4. Connect each pair of successive points with a straight line segment;

5. If you are comparing several data sets, make a key to distinguish the line graph for each sample using different colours. It is often advised to only have a maximum of 4 data sets in one line graph to avoid confusion.

A line graph is a type of graph that displays data that varies over time. It is made of a set of points that are joined by a straight line.

Line graphs are used to represent trends in a set of data.

Simple line graph

1. Simple line graph
2. Multiple line graph
3. Compound line graph
1. Construct the x and y-axes. Label each axis clearly. Ensure that the axis scales are divided into even intervals;

2. Come up with a brief and informative title for your graph. Make sure that it relays the purpose of your graph;

3. Plot the data on the graph with respect to the corresponding x and y-values given;

4. Connect each pair of successive points with a straight line segment;

5. If you are comparing several data sets, make a key to distinguish the line graph for each sample using different colours. It is often advised to only have a maximum of 4 data sets in one line graph to avoid confusion.

## Final Line Graphs Quiz

Question

What are the three types of line graphs?

Simple, multiple, compounding

Show question

Question

What is a single line graph?

A line graph which only has one line

Show question

Question

What is the difference between a simple and multiple line graph?

A simple line graph only has one line and a multiple line graph has two or more

Show question

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