Week 3 Discussion: Analyzing Correlation

Correlation

A correlation exists between two quantitative variables when a change in one variable is associated with a

change in the second variable. It measures the direction and strength of the relationship.

Types of Correlation

Positive correlation: both variables tend to increase or decrease together. On the scatterplot, the points tend to move in an upward direction. Negative correlation: the two variables tend to change in opposite directions with one increasing and the other decreasing. On the scatterplot, the points tend to move in a downward direction. No correlation: There is no apparent relationship between the two variables. On the scatterplot, the points tend to have no visible pattern and are scattered. Nonlinear correlation: the two variables are related but the relationship does not appear to follow a straight-line pattern. On the scatterplot, the points tend to show a parabolic or cubic pattern. We will study only linear correlation.

For most purposes, it is sufficient to state whether the correlation is strong, weak, or nonexistent by looking at

the scatterplot. However, sometimes it is useful to be more precise and associate a numerical value to

measure the strength using the correlation coefficient which is denoted by the letter r.

Properties of the Correlation Coefficient, r

• The correlation coefficient is a unitless number that measures the strength of the correlation and has a value between -1 and +1.

• If there is no correlation between the two variables meaning the points on the scatterplot are random, the correlation coefficient will be close to 0.

• If there is a positive correlation, the correlation coefficient will be positive, greater than 0 to 1. A correlation coefficient closer to one will be clustered together in an upward sloping straight-line pattern.

• If there is a negative correlation, the correlation coefficient is negative, less than 0 to -1. A correlation coefficient closer to -1 will be clustered together in a downward sloping straight-line pattern.

• Like the mean and standard deviation, the correlation coefficient is not resistant to outliers.

• Changing the units of measurement on the two variables does not change the correlation coefficient because correlation does not have units of measure.

The figure below shows the strength of the correlation and how it relates to the scatterplot. Notice that a

perfect correlation of +1 in the first graphic shows the points along a straight line. The last graphic show a

Correlation

perfect negative correlation where the points are along a negative straight line. In the center, notice the

correlation coefficient of 0, no correlation shows a random pattern. The other graphics represent different

correlation coefficients with varying linear patterns.

There are no hard and fast rules on what numbers mean weak, moderate or strong correlations; however,

here is a rule of thumb:

0 < 0 .4 weak

0.4 < 0.6 moderate

0.6 – 1 strong, where the numbers can be positive or negative.

Example: For the following relationships, determine whether the correlation would be positive, negative or

none.

a) The age and weight of a baby.

b) The amount of free time you have and the number of sports that you play.

c) The price of a shirt and the color of the buttons.

d) The number of karats in a diamond and the price.

Solution: a) As a baby ages it gains weight. Since both variables increase together, the correlation is

positive.

b) If you increase the number of sports that you play, the amount of free time that you have decreases. This

would be a negative correlation.

c) The price of a shirt has no relationship to the color of the buttons so this would have no correlation.

d) As the size of a diamond increases, the price also increases so this would be a positive correlation.

Example: Drilling down beneath a lake in Alaska yields chemical evidence of past changes in climate. Biological silicon, left by the skeletons of single-celled creatures called diatoms, is a measure of the abundance of life in the lake. A rather complex variable based on the ratio of certain isotopes relative to ocean water gives an indirect measure of moisture, mostly from snow. As we drill down, we look further into the past. Here are data from 2300 to 12,000 years ago.

(a) Make a scatterplot of silicon (response) against isotope (explanatory). Ignoring the outlier, describe the

direction, form, and strength of the relationship. The researchers say that this and relationships among other variables they measured are evidence for cyclic changes in climate that are linked to changes in the sun's activity. (b) The researchers single out one point: The open circle in the plot is an outlier that was excluded in the correlation analysis." Circle this outlier on your graph. What is the correlation with and without this point?

Solution: a) Enter the data into Stat Crunch with the isotope in column 1 and silicon in column 2. Select data, Graph, Scatterplot and var1 as x and var2 as y. Click Compute. The scatterplot is shown on the left. b) The outlier is in the upper right corner and represents the ordered pair (-19.37, 337) and is shown with a red circle on the scatterplot. In StatCrunch select Stat, Regression, Simple linear, var1 as x and var2 as y and click Compute. The correlation including the outlier is -0.329

Delete the point (-19.37, 337) and repeat the steps given above. If we remove that ordered pair, the correlation coefficient is -0.815.

Example: An auctioneer decided to record his opening recommended bid and the winning bid for several items. Does the correlation coefficient suggest that there is a linear correlation between these two variables?

Opening bid 1500 500 500 400 300

Winning bid 650 175 125 275 125

Solution: Using the TI-84 calculator, enter the opening bid in L1 and the winning bid in L2. Refer to page 44 of the TI-84 technology manual in the D2L classroom for instructions. The correlation coefficient is 0.9469. This confirms that there is a strong correlation between the opening bid and winning bid.

Example: A food industry group asked 3368 people to guess the number of calories in each of several common foods. The table below gives the averages of their guesses and the correct number of calories. (a) We think that how many calories a food actually has helps explain people's guesses of how many calories it has. With this in mind, make a scatterplot of these data. (b) Find the correlation r Explain why your r is reasonable based on the scatterplot. (c) The guesses are all higher than the true calorie counts. Does this fact influence the correlation in any way? How would r change if every guess were 100 calories higher? (d) The guesses are much too high for spaghetti and snack cake. Circle these points on your scatterplot. Calculate r for the other eight foods, leaving out these two points. Explain why r changed in the direction that it did.

Solution: a) Enter the data into Stat Disk with guessed calories as the first column and correct calories as the

second column. Then follow the directions on page 13 of the Stat Disk User’s Manual for the scatterplot. The results are shown to the left. There appears to be a strong positive correlation between the two variables. b) Follow the directions on page 13 to find the correlation

coefficient. As given to the left. Note that the correlation coefficient is 0.8245 which does fall in the strong, positive category. This agrees with the assessment using the scatterplot.

c) The correlation would be unaffected by adding 100 to each data value. d) The two outliers are circled in red on the scatterplot. Removing these two data value gives the correlation coefficient shown to the left as 0.9837. The correlation is increased because the scatter around the green line is reduced when the outliers are deleted.