15.12 Investigating data with scatter plots

When we have bivariate data, we want to determine what sort of relationship the two variables have. As the independent variable (x) changes notice how the dependent variable (y) tends to change. Just by observation, we may notice the following:

• A simple relationship: if the distribution of points appears to follow a trend either linear or non-linear depending on if the points appear to follow the shape of a line or not.

  Consider being given an x-value that doesn't correspond to any data point we have. Does the data set give us an idea of what y-value that point should have to fit in with the rest of the data? If yes there might be a relationship. If not, there might be no relationship between the variables.

• Outliers: in a scatterplot, any data points that are very different from the other data points will be quite obvious especially if the rest of the points appear to have a relationship.

Even when two variables have a relationship, it may not be a causal relationship. We cannot say for sure that a change in the value of x causes y to change or that the value of y causes a corresponding value of x even when a relationship is apparent. It may be that both x and y have a relationship with some other hidden variable, which creates an indirect relationship between x and y.

Examples

Example 1

The scatter plot shows the relationship between sea temperature and the amount of healthy coral.

a

Which variable is the dependent variable?

A

Sea temperature

B

Level of healthy coral

Worked Solution

Create a strategy

The dependent variable is placed on the vertical axis and is affected by the independent variable.

Apply the idea

The level of healthy coral is determined by sea temperature and is on the vertical axis, making it the dependent variable. So, the correct answer is B.

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b

Which variable is the independent variable?

A

Sea temperature

B

Level of healthy coral

Worked Solution

Create a strategy

An independent variable is a variable that stands alone and is not changed by the other variables you are measuring.

Apply the idea

From the previous problem, we know that the level of healthy coral is the dependent variable, so this means the sea temperature is the independent variable. The correct answer is A.

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Example 2

The following scatterplot shows the height and weight of six students:

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\text{Height (cm)}

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\text{Weight (kg)}

a

How tall is the student who weighs 42kg?

Worked Solution

Create a strategy

Look for the weight on the vertical axis to find the point and see which height it is above on the horizontal axis.

Apply the idea

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\text{Height (cm)}

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\text{Weight (kg)}

By finding 42 on the vertical axis, then moving right to the point and down to the horizontal axis, we can see that the corresponding value on the horizontal axis is 149.

The student weighs 149\text{ cm.}

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b

How tall is the tallest student?

Worked Solution

Create a strategy

The tallest student is represented by the point furthest to the right, since the height increases to the right.

Apply the idea

The point furthest to the right has coordinates (156,58). The height is the first coordinate, so the tallest person is 156\text{ cm}.

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Example 3

The price of ten houses are graphed against the house's land area on the following graph:

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\text{Area (m}^2)

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\text{Price}

Describe the relationship between land area and house price in the data.

Worked Solution

Create a strategy

Describe what happens to the dependent variable (price) as the independent variable (area) increases.

Apply the idea

The area increases from left to right. We can see from the graph that the price increases (rises) from left to right.

So as the land area increases, the house price increases.

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