MATH 1107Elementary Statistics

Lecture 6Scatterplots, Association

and Correlation

• Take a look at the data on the right.

• In the US, Is there a relationship between: – Household income and

violent crime?– Household income and Death

rate?• If there is a relationship, is it

direct or inverse?• If there is a significant

relationship, can we use it to predict the likelihood of crime or death rate in a particular state?

Median Household

IncomeViolent Crime

Death Rate

25,891 872 9.641,969 660 3.829,593 671 823,893 577 10.535,173 1,120 732,716 579 6.541,059 495 8.635,739 621 8.627,456 1,207 10.428,889 733 742,171 258 627,784 281 7.631,707 977 8.728,663 508 828,880 278 9.6

MATH 1107 – Scatterplot, Association and Correlation

Median Household Income and Violent Crime

0

200

400

600

800

1,000

1,200

1,400

20,000 25,000 30,000 35,000 40,000 45,000

Income

Viol

ent C

rime

MATH 1107 – Scatterplot, Association and CorrelationUsing scatterplots, we can evaluate the direction, the form, and the strength of the relationship:

Median Household Income and Death Rate

3.0

5.0

7.0

9.0

11.0

13.0

20,000 25,000 30,000 35,000 40,000 45,000Median Income

Deat

h Rat

e

MATH 1107 – Scatterplot, Association and CorrelationEvaluate the direction, the form, and the strength of the relationship:

Specifically, here are the things to look for in a scatterplot:

1. Direction – Does the direction of the data move NW to SE? This indicates a negative relationship. Does the data move from the SW to the NE? This is a positive relationship. Does the data look like a cloud? This indicates no relationship.2.Form – Does the data demonstrate a linear pattern? A curved pattern? 3. Strength – Is the data really tight? This indicates a strong correlation. Is the data really spreadout? This indicates a weak correlation.4. Are there any outliers? Or groups of outliers?

MATH 1107 – Scatterplot, Association and Correlation

A few additional points about scatterplots:

1. The dependent variable – typically the one that you want to better understand or predict – is plotted on the y-axis and the independent variable is plotted on the x-axis.

2. We typically only use scatterplots with quantitative data. Question – why?

3. Rescale the scatterplot to only include the scales necessary to see the relationships. However, if the axes do not start at zero, this should be noted at the bottom of the page. Question – why?

4. Changing the units of a scatterplot should not change the relationship (i.e., lbs to kg)

MATH 1107 – Scatterplot, Association and Correlation

Correlations can be measured as a number between 0 and 1 (absolute value). Where, numbers closer to 0 indicate less correlation and numbers closer to 1 indicate more correlation.

A basic interpretation of the correlation coefficient is – x% of the change in variables 1 and two move together.

Correlation coefficients are really not expressed in meaningful units.

And…remember that correlation coefficients are related to LINEAR relationships!

MATH 1107 – Scatterplot, Association and Correlation

Lets examine some scatterplots and correlation coefficients in EXCEL. Go to the EXCEL datasets on the website.

MATH 1107 – Scatterplot, Association and Correlation