CorrelationDefinition: In a distribution if the change in one variable effects a change in the other variable, the variable are said to be correlated(or there is a correlation between the variables) Let X and Y measure some characteristics of a particular system .To study the overall measure of the system it is necessary to measure the interdependence of X and Y.
Correlation If the quantities(X,Y) vary in such a way that change in one variable corresponds to change in the other variable then the variables X and Y are correlated.Types of Correlation: The important ways of classifying thecorrelation are:1. Positive and Negative2. Simple , Partial and Multiple3. Linear and non-Linear.
CorrelationMethods of Studying Correlation: The following are the important methods of ascertaining between two variables. Scatter diagram method Karl Pearson’s Co-efficient Spearman’s Rank Correlation Co-Efficient• Scatter Diagram Method: The simplest device forstudying correlation between two variables is a special type of dot chart.
CorrelationKarl Pearson’s Co-Efficient of Correlation: r = (∑ xy) / (N σxσ y ) where x x = (x-¯) y y= (y- ¯) σx =Standard Deviation of Series X. σ y =Standard Deviation of Series Y. r= The correlation Coefficient N=Number of Pairs of Observation.
Demerits• By applying Scatter diagram method we can getan idea about the direction of correlation and alsowhether it is high or low. But we cannot establish theexact degree of correlation between the variables as it ispossible by applying mathematical methods.• The Karl Pearson’s method is based on theassumption that the population being studied is normallydistributed. When it is known that the population is notnormal or when he shape the distribution is not known,there is need for a measure of correlation that involvesno assumption about the parameter of the population.
So Why Rank Correlation???? It is possible to avoid making any assumptionsabout the populations being studied by ranking theobservations according to size and basing thecalculations on the ranks rather than upon the originalobservations. It does not matter which way the items areranked, item number one may be the largest or it may bethe smallest. Using ranks rather than actual observationsgives the coefficient of rank correlations. This method of finding out co variability or thelack of it between two variables was developed by theBritish Psychologist Charles Edward Spearman in 1904.
Ranking A ranking is a relationship between a set of items such that, for any two items, the first is either ranked higher than, ranked lower than or ranked equal to the second. In mathematics, this is known as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects can have the same ranking. The rankings themselves are totally ordered. For example, materials are totally preordered by hardness, while degrees of hardness are totally ordered.
Rank Correlation “Rank correlation” is the study of relationshipsbetween different rankings on the same set of items. Itdeals with measuring correspondence between tworankings, and assessing the significance of thiscorrespondence. Spearman’s correlation coefficient isdefined as: r = 1-((6∑D2)/(N(N-1)2))Where r , denotes rank coefficient of correlation and Drefers to the difference of rank relation between paired Items in two series.
FeaturesThe rank method has principal uses:• The sum of the differences between two variables is zero.• Spearman’s rank correlation coefficient ρ is the Pearsonian correlation coefficient between the ranks.• The rank correlation can be interpreted in the same way as Karl Pearson’s correlation coefficient.
Features• Karl Pearson correlation coefficient assumes that thesample observations are drawn from a normalpopulation. Rank correlation coefficient is a distribution-free measure since no strict assumption is made aboutthe population from which it is drawn.• The values obtained for two formulae are different dueto the fact that when ranking is used some informationis hidden.• Spearman’s formula is the only formulae available tofind the correlation between qualitative characters.
Types of Rank Methods In the rank correlation we may have two types of problems:• Where ranks are given• Where ranks are not given• Where repeated ranks occurNote: If r = 1 then there is a perfect Positive correlation If r = 0 then the variables are uncorrelated If r=-1 then there is a perfect Negative Correlation
Where ranks are given :Where actual ranks are given to us the steps required forcomputing rank correlation are:• Take the difference of the two ranks, i.e., (R1-R2)and denotes these differences by D.• Square these difference and obtain the total• Apply the formula
Where ranks are not given• When we are given the actual data and not theranks, it will be necessary to assign the ranks. Rankscan be assigned by taking either highest values as 1 orthe lowest value as 1. But whether we start with thelowest value or the highest value we must follow thesame method in case of both the variables.
Steps To Find RC• Step 1: – Draw the table like
• Step 2: – Fill the data field with the given data
• Step 3: • Give the Rank for the data
• Step 4 – Find the difference d & d2
• Step 5: – Apply the formula: r= Where d= difference, n=no.of data