# Scatter Plots

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## Scatter Plots – Definition

In everyday life, we often see graphs that show how two things are related, like how much exercise people do and how much water they drink. In maths, we use scatter plots to find patterns in this type of data. Scatter plots provide a simple yet powerful way to visualise and analyse the relationship between two variables. Whether in the classroom or real-world applications, they help us understand trends, make predictions, and identify unusual patterns. By plotting individual data points on a graph, scatter plots enable us to quickly discern whether a relationship between variables exists, and if so, what kind of relationship it is.

A scatter plot is a type of graph used in statistics to show the relationship between two different sets of data. On a scatter plot, each point represents a pair of values.

### Scatter Plots – Variables

Scatter plots are essential tools in statistics and data analysis. They help us see if there is a relationship between two variables, also known as bivariate data, such as height and weight, or study time and test scores. In these plots, we often deal with two types of variables: independent and dependent.

Variable Type Description Position in Scatter Plot Example
Independent Variable The variable that you change or control in an experiment. Typically plotted on the x-axis. Amount of time spent studying.
Dependent Variable The variable that depends on the independent variable and what you measure in the experiment. Usually plotted on the y-axis. Test scores in a study about study time.

Bivariate Data: This term refers to when you look at two variables together to see how they relate. For example, you might compare rainfall amounts with how well crops grow. Each point on a scatter plot shows one set of these two things, which helps us see if they might affect each other.

Understanding the roles of independent and dependent variables in scatter plots is essential for correctly interpreting the data. These plots are mainly used to examine the effect of the independent variable (like rainfall) on the dependent variable (like crop growth). This understanding is especially important in fields such as science, economics and social research, where predicting trends and analysing variable relationships is key.

## How to Graph a Scatter Plot

Let's create a scatter plot comparing the number of hours of sleep a student got with the grade they received on their latest maths test.

Step 1: Choose and Define Two Variables

For our scatter plot, we will compare:

• x-axis (Independent Variable): Number of hours of sleep
• y-axis (Dependent Variable): Test grade (out of 100)

Step 2: Draw and Label Axes Create a horizontal line (x-axis) and a vertical line (y-axis) on graph paper or in a graphing tool.

• Label the x-axis as "Hours of Sleep."
• Label the y-axis as "Test Grade (%)."

Step 3: Choose an Appropriate Interval

Before plotting the data, it's important to choose suitable intervals for the axes. This will help in accurately placing and reading the data points.

• For the x-axis (Hours of Sleep), consider the range of hours you want to include. For example, you might choose an interval of 1 hour and range from 0 to 12 hours.
• For the y-axis (Test Grade), choose an interval that makes sense for test scores. You might use an interval of 10% for grades ranging from 0 to 100%.

Choosing the right intervals will make your scatter plot more readable and your data easier to interpret.

Step 4: Plot Points

The data in the table can be translated into coordinates (x,y).

Plot each coordinate on the graph where the x-value (hours of sleep) and y-value (test grade) intersect.

### Constructing a Scatter Plot – Guided Practice

It’s your turn to create a scatter plot, you will need a piece of graph paper and a pencil to try it yourself.

You're curious if warmer weather leads to more ice cream sales. Using data from the past week, plot a scatter plot with temperature (in °C) on one axis and ice cream sales (in £) on the other to investigate this.

Choose and define the two variables.
Using graph paper, draw and label the axes accordingly. Determine the best interval to use based on the information given.
On your graph, plot the coordinates of each data set.
What is association?
What are clusters?
What are outliers?

Scatter plots not only show relationships between two variables but also reveal the nature of these relationships. There are two primary types of trends that scatter plots can illustrate: linear and non-linear.

A linear trend in a scatter plot shows a straight-line relationship between the variables. This means as one variable increases or decreases, the other variable changes at a constant rate.

Real-World Example: A linear trend could be seen in a scatter plot comparing the speed of an internet connection to the time it takes to download a large file. Generally, as internet speed increases, the download time decreases consistently.

A non-linear trend indicates that the relationship between the variables changes at different rates. This trend is represented by a curved line on the scatter plot.

Real-World Example: An example of a non-linear trend could be the relationship between speed and fuel efficiency in cars. Initially, as speed increases, fuel efficiency improves, but after reaching an optimal speed, further speed increases might decrease efficiency.

Understanding these trends is crucial for interpreting scatter plots accurately. It allows us to make more nuanced predictions and understand complex relationships in data, which is especially important in fields like environmental science, economics and engineering.

## Scatter Plots – Real-World Application

Scatter plots are incredibly useful in various real-world situations, particularly for making predictions. A common application is in understanding consumer behaviour based on environmental factors.

Consider a situation where a local business wants to estimate the number of beachgoers based on the day's temperature. They collect data over several weeks to analyse the trend and make predictions.

Temperature (°C) Beach Attendance
21 120
24 200
26 180
29 210
31 190
34 220

Prediction: At 32°C, predicting beach attendance becomes more nuanced due to the non-linear trend. The business might expect attendance to be around 200, considering the fluctuations observed at similar temperatures.

Scatter plots and their line of best fit in these scenarios are valuable for their ability to reveal complex patterns and trends that are not immediately obvious, aiding in more accurate predictions and better decision-making.

## Constructing Scatter Plots – Exercises

Grab some graph paper and try the following scatter plot problems on your own!

Using the data set {(2,4), (3,6), (4,7), (5,7), (6,8), (7,10)}, create a scatter plot. Then, describe the pattern you see and identify any outliers or clusters.
For the data {(1,10), (2,8), (3,6), (4,5), (5,3), (6,1)}, make a scatter plot, describe its pattern and check for outliers or clusters.
Plot these data points on a scatter plot: {(3,2), (4,4), (5,5), (6,5), (7,5), (8,20)}. Describe the overall pattern and identify any outliers or clusters.
Create a scatter plot using the weekly data of hours spent on social media (x) and total hours of sleep (y): {(10, 56), (15, 52), (20, 49), (25, 43), (30, 39), (35, 35)}. Describe any patterns and identify outliers or clusters, considering the impact of social media on sleep.
Using data from a local coffee shop, plot a scatter plot with the temperature outside (x, in °C) and the number of hot chocolates sold (y): {(5, 120), (10, 110), (15, 80), (20, 60), (25, 30), (30, 20)}. Analyse the pattern and look for any outliers or clusters in the context of weather and hot chocolate sales.

## Scatter Plots – Summary

Key Learnings from this Text:

• Scatter plots display the relationship between bivariate data (two variables).
• They help identify patterns, associations, outliers and clusters in data.
• Positive association shows an upward trend, negative association shows a downward trend and no association indicates a random pattern.
• Scatter plots can show either linear trends, where data points form a straight line, or non-linear trends, where the data points create a curved pattern.
• Scatter plots are valuable tools in statistics and real-world data analysis.

## Scatter Plots – Frequently Asked Questions

What is a scatter plot?
Why are scatter plots used?
How do you create a scatter plot?
What does a positive association on a scatter plot indicate?
Can scatter plots show negative associations?
What does it mean if there's no clear pattern in a scatter plot?
How can you identify outliers on a scatter plot?
What are clusters in a scatter plot?
Can scatter plots be used for prediction?
Are scatter plots only used in maths?
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