# 12.3 The regression equation  (Page 4/8)

 Page 4 / 8

The correlation coefficient, r , developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y .

The correlation coefficient is calculated as

$r=\frac{n\Sigma \left(xy\right)-\left(\Sigma x\right)\left(\Sigma y\right)}{\sqrt{\left[n\Sigma {x}^{2}-{\left(\Sigma x\right)}^{2}\right]\left[n\Sigma {y}^{2}-{\left(\Sigma y\right)}^{2}\right]}}$

where n = the number of data points.

If you suspect a linear relationship between x and y , then r can measure how strong the linear relationship is.

## What the value of r Tells us:

• The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
• The size of the correlation r indicates the strength of the linear relationship between x and y . Values of r close to –1 or to +1 indicate a stronger linear relationship between x and y .
• If r = 0 there is absolutely no linear relationship between x and y (no linear correlation) .
• If r = 1, there is perfect positive correlation. If r = –1, there is perfect negativecorrelation. In both these cases, all of the original data points lie on a straight line. Of course,in the real world, this will not generally happen.

## What the sign of r Tells us

• A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation) .
• A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation) .
• The sign of r is the same as the sign of the slope, b , of the best-fit line.

## Note

Strong correlation does not suggest that x causes y or y causes x . We say "correlation does not imply causation."

The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate r . The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

## The coefficient of determination

The variable r 2 is called the coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:

• ${r}^{2}$ , when expressed as a percent, represents the percent of variation in the dependent (predicted) variable y that can be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.
• 1 – ${r}^{2}$ , when expressed as a percentage, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.

Consider the third exam/final exam example introduced in the previous section

• The line of best fit is: ŷ = –173.51 + 4.83x
• The correlation coefficient is r = 0.6631
• The coefficient of determination is r 2 = 0.6631 2 = 0.4397
• Interpretation of r 2 in the context of this example:
• Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line.
• Therefore, approximately 56% of the variation (1 – 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best-fit regression line. (This is seen as the scattering of the points about the line.)

## Chapter review

A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of y and the estimated value of y . The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.

The correlation coefficient r measures the strength of the linear association between x and y . The variable r has to be between –1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r 2 , is equal to the square of the correlation coefficient. When expressed as a percent, r 2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line.

Use the following information to answer the next five exercises . A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Draw a scatter plot of the data.

Use regression to find the equation for the line of best fit.

ŷ = 2.23 + 1.99 x

Draw the line of best fit on the scatter plot.

What is the slope of the line of best fit? What does it represent?

The slope is 1.99 ( b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another \$1.99 million in pay each year.

What is the y -intercept of the line of best fit? What does it represent?

What does an r value of zero mean?

It means that there is no correlation between the data sets.

When n = 2 and r = 1, are the data significant? Explain.

When n = 100 and r = -0.89, is there a significant correlation? Explain.

Yes, there are enough data points and the value of r is strong enough to show that there is a strong negative correlation between the data sets.

what is statistics
statistics is the collection and interpretation of data
Enhance
the science of summarization and description of numerical facts
Enhance
Is the estimation of probability
Zaini
mr. zaini..can u tell me more clearly how to calculated pair t test
Haai
do you have MG Akarwal Statistics' book Zaini?
Enhance
Haai how r u?
Enhance
maybe .... mathematics is the science of simplification and statistics is the interpretation of such values and its implications.
Miguel
can we discuss about pair test
Haai
what is outlier?
outlier is an observation point that is distant from other observations.
Gidigah
what is its effect on mode?
Usama
Outlier  have little effect on the mode of a given set of data.
Gidigah
How can you identify a possible outlier(s) in a data set.
Daniel
The best visualisation method to identify the outlier is box and wisker method or boxplot diagram. The points which are located outside the max edge of wisker(both side) are considered as outlier.
Akash
@Daniel Adunkwah - Usually you can identify an outlier visually. They lie outside the observed pattern of the other data points, thus they're called outliers.
Ron
what is completeness?
I don't get the example
ways of collecting data at least 10 and explain
Example of discrete variable
Gbenga
I am new here, can I get someone to guide up?
alayo
dies outcome is 1, 2, 3, 4, 5, 6 nothing come outside of it. it is an example of discrete variable
jainesh
continue variable is any value value between 0 to 1 it could be 4digit values eg 0.1, 0.21, 0.13, 0.623, 0.32
jainesh
hi
Kachalla
what's up here ... am new here
Kachalla
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me
solve what?
Ambo
mean
Vanarith
What is the end points of a confidence interval called?
lower and upper endpoints
Bheka
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
Sarbaz
6.5
phoenix
11
Shakir
7.5
Ron
why is always lower class bundry used
Caleb
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
Diamatu
Abdul
if putting y=3x examine that correlation coefficient between x and y=3x is 1.
what is permutation
how to construct a histogram
You have to plot the class midpoint and the frequency
Wydny
ok so you use those two to draw the histogram right.
Amford
yes
Wydny
ok can i be a friend so you can be teaching me small small
Amford
how do you calculate cost effectiveness?
George
Hi everyone, this is a very good statistical group and am glad to be part of it. I'm just not sure how did I end up here cos this discussion just popes on my screen so if I wanna ask something in the future, how will I find you?
Bheka
To make a histogram, follow these steps: On the vertical axis, place frequencies. Label this axis "Frequency". On the horizontal axis, place the lower value of each interval. ... Draw a bar extending from the lower value of each interval to the lower value of the next interval.
Divya
I really appreciate that
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
yes of course must have use f test and also use t test individually multple coefficients
rishi
Alright
umar
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
akeem
Is R square cannot analysis linear regression of X vs Y relationship?
Mok
To be an economist you have to be professional in maths
umar
hi frnds
Shehu