Problem: From the following six letters: A, B, C, D, E, F, how many groups of 3 letters can be created if none of the letters from the original 6 are repeated in any group?
This is a permutation problem because none of the letters can be repeated. When the first letter of one of the permutation groups is chosen, there are only five remaining letters to choose from. Thus, the elements of the permutation group are filled sequentially.
n = 6 = total number of objects initially available for inclusion in each permutation group
x = 3 = number of objects that will sequentially fill the permutation group
The number of permutations of n = 6 different letters placed x = 3 at a time sequentially equally:
nPx = 6P3 = PERMUT(n,x) = PERMUT(6,3) = 120
Hand Calculation of Combination and Permutation
Problems
Go To
http://excelmasterseries.com/Excel_Statistical_Master/Combination s-Permutations.php
To View How To Solve Combination and Permutation Problems By Hand (No Excel)
(Is Your Internet Connection Turned On ?)
You'll Quickly See Why You Always Want To Use Excel To Solve Statistical Problems !
Chapter 3 - Correlation and Covariance
Analysis
Correlation and Covariance describe relationships between different variables. Both Correlation and Covariance describe whether variables move together in the same direction, move in opposite directions, or don't move in any related way at all.
Basic Explanation of Correlation and Covariance
Correlation and Covariance are very similar ways of describing
relationships between two variables. Correlation is a more well-known concept and more widely used. It will therefore be covered in the first half of this course module. Covariance will be covered in the second half.
The Difference Between Correlation and Covariance
A common question in statistics: What is the difference between correlation (the Spearman correlation) and covariance analysis.
First, let's discuss the commonality between correlation and covariance. Correlation and Covariance both describe relationships between 2 variables.
The Spearman Correlation is considered to be a standardized form of covariance.
Here are the differences between the Spearman Correlation and Covariance Analysis:
can be outside of that range.
In the covariance calculation, covariance values depend on the units of measure of X and Y. Covariance values of data sets using different sets of measure are not comparable. The Spearman correlation coefficient is not influenced by the units of measure when calculating correlation. The Spearman correlation can be used to compare the similarities of multiple data sets that use different units of measure or scale. The Spearman correlation solves the units-of-measure problem by normalizing the covariance to the product of the standard deviations of all variables being compared. The dimensions or units of measure are then taken out of the equation.
The Spearman correlation tells you how close or far two variables are from being independent from each other. It must be remembered that high Spearman correlation interpretation does not imply causality between the two variables. The covariance calculation and covariance analysis tells you how much two variables tend to change together.
Correlation Analysis
Positive Correlation vs. Negative Correlation
Positive Correlation
If two variables are "positively correlated," they move in the same direction. When one goes up, the other goes up as well. Two variables that are positively correlated have a correlation coefficient that is
between 0 and +1. The closer the correlation coefficient is to +1, the more exactly the two variables move together.. A correlation
coefficient between two variables of exactly +1.00 means that both variables move in lock-step with each other. A correlation coefficient between two variables of 0 indicates that there is no relationship between the movement of one variable and movement of the other variable.
Negative Correlation
If two variables are "negatively correlated," they move in opposite
directions. When one goes up, the other goes down. When one variable goes down, the other goes up. Two variables that are "negatively
correlated" have a correlation coefficient that is between -1 and 0. The closer the correlation coefficient is to -1, the more exactly the two variables move in opposite directions. A correlation coefficient between two variables of exactly -1.00 means that both variables move lock-step with each other in opposite directions. A correlation coefficient between two variables of 0 indicates that there is no relationship
between the movement of one variable and movement of the other variable.
Calculation of Correlation Coefficient
The correlation also describes how linear a relationship is between two variables. The Correlation Coefficient can have values between -1 and +1. Below is the formula for calculating the Correlation Coefficient. Excel does such a great job in calculating correlation and covariance that it is not necessary to memorize the formulas of covariance and correlation, but here they are, along with examples worked out in Excel:
Correlation of variables x and y from a known population = ρ ("rho")
ρ = ( Covariance of x and y) / ( Standard Deviation of x * Standard
Deviation of y )
ρ = Population Correlation Coefficient = σxy / ( σx * σy) (Covariance will be explained later in this module)
Correlation of variables x and y randomly sampled from an unknown population = r
(This is the normal situation)
r = ( Sample Covariance of x and y) / ( Sample Standard Deviation of x *
Sample Standard Deviation of y )
Excel Functions Used When Calculating Correlation Coefficient
CORREL (Highlighted block of cells of 2 variables)
= r = Sample Correlation between two variables x and y
The Excel Statistical function CORREL calculates the correlation between 2 variables. The only inputs needed in the CORREL formula are specifying the locations of the blocks of cells for each variable. This is done by highlighting each block of cells of values to be correlated after the function is inserted. CORREL is one of the statistical functions.
CORREL can be used to calculate the correlation between only two variables. The Excel Data Analysis tool Correlation, which is discussed after CORREL, can calculate correlations between multiple variables simultaneously.
Here is an example of CORREL in use: