
The
Statistics Calculator
Statistical
Analysis Tests At Your Fingertips

Correlation Types
Correlation is a measure of
association between two variables. The
variables are not designated as dependent
or independent. The two most popular
correlation coefficients are: Spearman's
correlation coefficient rho and Pearson's
productmoment correlation coefficient.
When calculating a correlation
coefficient for ordinal data, select
Spearman's technique. For interval or
ratiotype data, use Pearson's technique.
The value of a correlation coefficient
can vary from minus one to plus one. A
minus one indicates a perfect negative
correlation, while a plus one indicates a
perfect positive correlation. A
correlation of zero means there is no
relationship between the two variables.
When there is a negative correlation
between two variables, as the value of
one variable increases, the value of the
other variable decreases, and vise versa.
In other words, for a negative
correlation, the variables work opposite
each other. When there is a positive
correlation between two variables, as the
value of one variable increases, the
value of the other variable also
increases. The variables move together.
The standard error of a correlation
coefficient is used to determine the
confidence intervals around a true
correlation of zero. If your correlation
coefficient falls outside of this range,
then it is significantly different than
zero. The standard error can be
calculated for interval or ratiotype
data (i.e., only for Pearson's
productmoment correlation).
The significance (probability) of the
correlation coefficient is determined
from the tstatistic. The probability of
the tstatistic indicates whether the
observed correlation coefficient occurred
by chance if the true correlation is
zero. In other words, it asks if the
correlation is significantly different
than zero. When the tstatistic is
calculated for Spearman's rankdifference
correlation coefficient, there must be at
least 30 cases before the tdistribution
can be used to determine the probability.
If there are fewer than 30 cases, you
must refer to a special table to find the
probability of the correlation
coefficient.
Example
A company wanted
to know if there is a significant
relationship between the total number of
salespeople and the total number of
sales. They collect data for five months.
Variable
1 
Variable
2 
207 
6907 
180 
5991 
220 
6810 
205 
6553 
190 
6190 

Correlation
coefficient = .921
Standard error of the coefficient = ..068
ttest for the significance of the
coefficient = 4.100
Degrees of freedom = 3
Twotailed probability = .0263
Another Example
Respondents to a
survey were asked to judge the quality of
a product on a fourpoint Likert scale
(excellent, good, fair, poor). They were
also asked to judge the reputation of the
company that made the product on a
threepoint scale (good, fair, poor). Is
there a significant relationship between
respondents perceptions of the company
and their perceptions of quality of the
product?
Since both
variables are ordinal, Spearman's method
is chosen. The first variable is the
rating for the quality the product.
Responses are coded as 4=excellent,
3=good, 2=fair, and 1=poor. The second
variable is the perceived reputation of
the company and is coded 3=good, 2=fair,
and 1=poor.
Variable
1 
Variable
2 
4 
3 
2 
2 
1 
2 
3 
3 
4 
3 
1 
1 
2 
1 

Correlation
coefficient rho = .830
ttest for the significance of the
coefficient = 3.332
Number of data pairs = 7
Probability must
be determined from a table because of the
small sample size.
Regression
Simple regression is used to examine
the relationship between one dependent
and one independent variable. After
performing an analysis, the regression
statistics can be used to predict the
dependent variable when the independent
variable is known. Regression goes beyond
correlation by adding prediction
capabilities.
People use regression on an intuitive
level every day. In business, a
welldressed man is thought to be
financially successful. A mother knows
that more sugar in her children's diet
results in higher energy levels. The ease
of waking up in the morning often depends
on how late you went to bed the night
before. Quantitative regression adds
precision by developing a mathematical
formula that can be used for predictive
purposes.
For example, a medical researcher
might want to use body weight
(independent variable) to predict the
most appropriate dose for a new drug
(dependent variable). The purpose of
running the regression is to find a
formula that fits the relationship
between the two variables. Then you can
use that formula to predict values for
the dependent variable when only the
independent variable is known. A doctor
could prescribe the proper dose based on
a person's body weight.
The regression line (known as the least
squares line) is a plot of the
expected value of the dependent variable
for all values of the independent
variable. Technically, it is the line
that "minimizes the squared
residuals". The regression line is
the one that best fits the data on a
scatterplot.
Using the regression equation, the
dependent variable may be predicted from
the independent variable. The slope of
the regression line (b) is defined as the
rise divided by the run. The y intercept
(a) is the point on the y axis where the
regression line would intercept the y
axis. The slope and y intercept are
incorporated into the regression
equation. The intercept is usually called
the constant, and the slope is referred
to as the coefficient. Since the
regression model is usually not a perfect
predictor, there is also an error term in
the equation.
In the regression equation, y is
always the dependent variable and x is
always the independent variable. Here are
three equivalent ways to mathematically
describe a linear regression model.
y = intercept + (slope x)
+ error
y = constant + (coefficientx)
+ error
y = a + bx + e
The significance of the slope of the
regression line is determined from the
tstatistic. It is the probability that
the observed correlation coefficient
occurred by chance if the true
correlation is zero. Some researchers
prefer to report the Fratio instead of
the tstatistic. The Fratio is equal to
the tstatistic squared.
The tstatistic for the significance
of the slope is essentially a test to
determine if the regression model
(equation) is usable. If the slope is
significantly different than zero, then
we can use the regression model to
predict the dependent variable for any
value of the independent variable.
On the other hand, take an example
where the slope is zero. It has no
prediction ability because for every
value of the independent variable, the
prediction for the dependent variable
would be the same. Knowing the value of
the independent variable would not
improve our ability to predict the
dependent variable. Thus, if the slope is
not significantly different than zero,
don't use the model to make predictions.
The coefficient of determination
(rsquared) is the square of the
correlation coefficient. Its value may
vary from zero to one. It has the
advantage over the correlation
coefficient in that it may be interpreted
directly as the proportion of variance in
the dependent variable that can be
accounted for by the regression equation.
For example, an rsquared value of .49
means that 49% of the variance in the
dependent variable can be explained by
the regression equation. The other 51% is
unexplained.
The standard error of the estimate for
regression measures the amount of
variability in the points around the
regression line. It is the standard
deviation of the data points as they are
distributed around the regression line.
The standard error of the estimate can be
used to develop confidence intervals
around a prediction.
Example
A company wants
to know if there is a significant
relationship between its advertising
expenditures and its sales volume. The
independent variable is advertising
budget and the dependent variable is
sales volume. A lag time of one month
will be used because sales are expected
to lag behind actual advertising
expenditures. Data was collected for a
six month period. All figures are in
thousands of dollars. Is there a
significant relationship between
advertising budget and sales volume?
Indep.
Var. 
Depen.
Var 
4.2 
27.1 
6.1 
30.4 
3.9 
25.0 
5.7 
29.7 
7.3 
40.1 
5.9 
28.8 

Model: y =
9.873 + (3.682x) + error
Standard error of the estimate = 2.637
ttest for the significance of the slope
= 3.961
Degrees of freedom = 4
Twotailed probability = .0149
rsquared = .807
You might make a
statement in a report like this: A simple
linear regression was performed on six
months of data to determine if there was
a significant relationship between
advertising expenditures and sales
volume. The tstatistic for the slope was
significant at the .05 critical alpha
level, t(4)=3.96, p=.015. Thus, we reject
the null hypothesis and conclude that
there was a positive significant
relationship between advertising
expenditures and sales volume.
Furthermore, 80.7% of the variability in
sales volume could be explained by
advertising expenditures.
How to Order Statistics Calculator
