Correlation Coefficient

Correlation Coefficient

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Correlation tests for a relationship between two variables.

Causation means that one thing "causes" another

Multiple Choice

A correlation shows that two things

• Are Related

• Cause Each Other to Happen

• Are Unrelated

Multiple Choice

This implies that one variable has a direct affect on the other variable.

• Correlation

• Causation

Study Time and Grades

As study hours increase, grades increase. There is a positive correlation between hours and grades.

So....Do hours spent studying cause higher grades?

Ice Cream Sales and Shark Attacks

As ice cream sales increase, shark attacks increase. As ice cream sales decrease, shark attacks decrease.

So....ice cream sales cause shark attacks?

​Correlation vs. Causation

​Causation means that x makes y happen. For example, the more cash you spend, the less cash you will have. This example has a correlation AND causation.

BUT… Correlation ≠ Causation.

Example: There may be a correlation between the number of glue sticks in a classroom and the number of students in a classroom. However, adding glue sticks to a classroom does not make more students be in the classroom.

Multiple Choice

Correlation or Causation?

The number of times you come to tutoring and your test scores improving

• Correlation

• Causation

Multiple Choice

Correlation or Causation?

The inches of snowfall and the number of heart attacks

• Correlation

• Causation

Multiple Choice

Correlation or Causation?

The speed you drive and the amount of fuel you use

• Correlation

• Causation

Multiple Choice

Which example shows CAUSATION? (Which one CAUSED the other to happen.)

• High social media usage and reduced grades.

• Car ran out of gas and being stranded on the side of the road.

• Recess time and number of friends.

Multiple Choice

Which example shows CAUSATION? (Which one CAUSED the other to happen.)

• Rainfall and time spent reading.

• Eating more butter and increased divorce rates.

• It started raining and the baseball game was delayed.

Multiple Choice

Which one of these examples is NOT an example of CAUSATION?

• Kryptonite causes Superman to lose all his powers.

• Temperatures of less than 32˚F cause water to freeze.

• Umbrellas cause rainstorms.

• Stepping on the gas pedal causes a car to move faster.

​Positive Association	​Negative Association	​No Correlation
​Both variables are increasing.	One variable is increasing, the other is decreasing.​	There is no relationship between the variables.​

​Types of Correlation

Multiple Choice

What is the type of correlation?

• Negative 
• no correlation
• positive 

Multiple Choice

What type of association does this graph have?

• positive
• negative
• none
• all of the above

Multiple Choice

Number of days absent from school vs. math average

• Positive Correlation

• Negative Correlation

• No Correlation

Multiple Choice

What type of association does this graph have?

• positive 
• negative
• none
• all of the above

Multiple Choice

Amount of Education and Amount of income

• Positive Correlation

• Negative Correlation

• No Correlation

Multiple Choice

Which scatter plot shows a linear relationship between x and y?

• Graph A

• Graph B

• Graph C

• Graph D

​Correlation Coefficient (r)

Correlation Coefficient ​(r) describes the strength and direction of the correlation.

​

​r values can not be less than -1 or greater than +1. This can be written as the inequality  -1 < r < 1

The closer to 0, the weaker the correlation. Very close to 0 is no correlation. The closer to -1 or 1, the stronger the correlation. -1 and 1 are perfect.

Correlation coefficient (r) formulas are used to find how strong a relationship is between data. The formulas return a value between -1 and 1, where:

• 1 indicates a strong positive relationship.

• -1 indicates a strong negative relationship.

• A result of zero indicates no relationship at all.

Multiple Choice

Average time spent listening to music per day and average time spent watching TV per day.  $$r=-.17$$  

• strong positive

• strong negative

• weak positive

• weak negative

Multiple Choice

Score on science exam and number of words written on the essay question.  $$r=.28$$  

• strong positive

• strong negative

• weak positive

• weak negative

Multiple Choice

Car weight and distance traveled using a full tank of gas.  $$r=-.86$$  

• strong positive

• strong negative

• weak positive

• weak negative

Multiple Choice

Number of steps taken per day and number of kilometers walked per day. $$r=.92$$  

• strong positive

• strong negative

• weak positive

• weak negative

Multiple Choice

Describe the relationship between the variables.

• Weak positive correlation
• Weak negative correlation
• Strong positive correlation
• Strong negative correlation

Multiple Choice

Test scores vs. shoe sizes

• Positive Correlation

• Negative Correlation

• No Correlation

Match

Match the following

Non-linear Correlation

No Correlation

Positive, Linear Correlation

Negative, Linear Correlation

• Non-linear Correlation

• No Correlation

• Positive, Linear Correlation

• Negative, Linear Correlation

Multiple Choice

A perfect positive correlation has a correlation coefficient of ____. 

• 1
• -1
• 0

Multiple Choice

A perfect negative correlation has a correlation coefficient of ____. 

• 1
• -1
• 0

Multiple Choice

Which graph depicts the strongest positive correlation?

Multiple Choice

Estimate the correlation coefficient for this scatterplot.

• A) r = -0.9

• B) r = 1

• C) r = 0.9

• D) r = -1

Multiple Choice

Estimate the correlation coefficient for this scatterplot.

• r = 1.2
• r = 0.89
• r = 0
• r = -0.89

To show the overall trend of a data set, we can use a ​Line of Best Fit.

​

A line of best fit should:

• Follow the trend of the data

• Split the data in half

• Help to estimate or predict values that are not already given in the set.

Line of Best Fit

​

Multiple Choice

Is this line an accurate line estimate of best fit for the data?

• Yes

• No

Multiple Choice

Which of the following lines represents the line of best fit for the scatter plot?

• A

• B

• C

• D

Multiple Choice

Which of the following lines represents the line of best fit for the scatter plot?

• A

• B

• C

• D

Multiple Choice

Which of the following lines represents the line of best fit for the scatter plot?

• A

• B

• C

• There is no line of best fit for this graph because there is no trend.

Multiple Choice

Select the statement that best describes the line.

• The line is the line of best fit.

• The line is NOT a line of best fit because it does not pass through two points.

• The line is NOT a line of best fit because it does not go through the middle of the data set.

• The line is NOT a line of best fit because it does not go through the origin.

​Outliers	​Clusters
​An outlier is a data point in a scatter plot that does not follow the trend of the data.	A group of data points that are closely grouped together.​

​outlier

cluster

Multiple Choice

What are the coordinates of the outlier?

• (1,9)

• (9,1)

• (2,2)

• y=x