Correlation

While understanding the behaviour of the variables, measuring the relationship between the variables also becomes crucial. The coefficient explains the strength of the relationship between any two or more variables.

Correlation is simplest tool used to measure the relationship. It is actually the covariance of standardized variables. Or it is the average of standardized covariance.

Assumptions 

a.       Linearity of data

b.      Homoscedasticity – equal error variance at any point along the linear relationship

c.       No outliers – A large difference between r and rho is a sign of its presence.

d.      Measurement error reduces systematic covariance and hence lowers r leading to attenuation.

e.      Unrestricted variance in the variables.

f.        Similar distributions (type) of the variables

g.       Normality of variables and errors.

Based on the type of variables, there are different types of correlation methods:
1. Pearson's r Correlation: Interval vs Interval

2. Spearman's Rho: Ordinal vs Ordinal or Interval vs Ordinal

3. Polyserial Correlation: Interval vs Ordinal (when the distribution between the interval variable and a latent continuous variable underlying the ordinal variable is bivariate normal)

4. Polychoric Correlation: Ordinal vs Ordinal (when the distribution between the two latent continuous variables underlying the two ordinal variables is bivariate normal)

5. Biserial Correlation: Interval vs Dichotomous with bivariate normality assumption. This can be greater than 1.

6. Point Biserial Correlation: Interval vs Dichotomous

7. Rank Biserial Correlation: Ordinal vs Dichotomous with bivariate normality assumption
Chi-Square Test is used to test the bivariate normality. If p<0.05, there is bivariate normality.

The table below is a pictorial representation

Measures of central tendency - Suitable application

The measure of central tendency is a single value that represents the whole set of observations. Given below is 5 measures of central tendency and their suitable application in real-life.

1) Mean - It is the average of all the observations. It is applicable to continuous data following normal distribution, i.e. the outliers or extreme values in the data set do not affect the average significantly.

Eg: Average score of a class of students, average height of a group of members.

                             
2) Median - It is the middle most observation in the data set when it is arranged in ascending order. That is, the measure is based on position. It is applicable to continuous and discrete data not following normal distibution. That is, when there are outliers that can affect the centralization, the median is observed. It can be mostly related to revenue.

Eg: Average income of a group of associates, Average turnover of a company.


3) Mode - It is the value that occurs most number of times in the data set. It is applicable to discrete or continuous data concerning frequency or common occurance. In some cases, the data set may have two modes or no mode.

Eg: Movie with average popularity (measured from the number of tickets sold), average earthquake measurement in an area,


4) Geometric mean - It is the nth root of the product of n values. The geometric mean of two numbers, 6 and 8 is the square root of (6x8 = 48). It is used in the case of observations related to time.

Eg: Average rate of return in investment in a year is calculated by taking the root of the product of each year's rate of return.


5) Harmonic mean: It is the reciprocal of the arithmetic mean obtained from taking the values in reciprocal form. It is used when the values of a variable depend on other factors or determinants. That is, this measure is used to harmonize the values and then find their mean.

Eg:
a) Calculating the average speed on a car in a distance of 40 kms. from four different readings as:
100 km/hr in the first 10 kms
90 km/hr in the second 10 kms
110 km/hr in the third 10 kms
105 km/hr in the final 10 kms

b) Calculating the average score of a student based on the scores in written test and interview. This helps to  eliminate two types of students
i) who score well in written test but score poorly in interview
ii) who score poorly in written test but score well in interview.

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Test to measure relationship between variables

Based on the scale of the variables, different statistical methods are available to determine the relationship or association between the variables. Suitable graphs also provide more clarity on the relationship.

Nominal vs Nominal:
Chi-square, Phi, Cramer 's V                                                                          (Clustered Bar graph)

Nominal vs Ordinal :
Chi-square, Phi (2X2, 2X3, 3X2 tables), Cramer's V (> 3X3 table)           (Clustered Bar graph)

Nominal vs Interval / Ratio :
Point bi-serial correlation                                                                             (Scatter plot, Bar chart)


Ordinal vs Ordinal :
Spearman's Rho, Kendall's Tau                                                  (Scatter Plot, Clustered Bar Graph)

Ordinal vs Interval / Ratio :
Spearman's Rho, Kendall's Tau, Point bi-serial correlation                                      (Scatter Plot)

Interval/Ratio vs Interval/Ratio :
Pearson r                                                                                                             (Scatter Plot)

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Understanding Level of Measurement

Level of Measurement: It provides a classification used to describe the nature of the information contained within the variable. The classification level of the variable will determine the types of calculations that can be performed using the variable.


Nominal Scale: The level of measurement involving numeric or alphanumeric responses which can be grouped. But the groups are distint and cannot be compared. Eg. Red, Green, Yellow, Male, Female

Ordinal Scale: The level of measurement involving numeric or alphanumeric responses which can be grouped. But the groups can be arranged in an order. In other words, the distance between the numeric responses is fixed but meaningless. Eg. Rating, Ranking

Interval Scale: The level of measurement involving numeric responses with decimal point. In other words, the distance between the responses is not fixed but meaningful. Eg. Temperature.

Ratio Scale: The level of measurement involving numeric responses with absolute zero. In other words, the responses represent intervals but are not expressed below the zero value. Eg. Height, Weight, Coverage value in Market Research.

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Understanding Variables - clarity on data

To understand the variable type is very much important in all the statistical analysis. Given below is the classification table on the statistical variables.

Description

Variables:
Quantitative variables: Variables containing numerical responses. Eg. Temperature, Age, Monthly Income etc.

Qualitative variables: Variables containing numerice or alphanumeric responses. Based on the responses, the data can be categorised. Hence it is also called as Categorical variable. Eg. Gender, Employee id, Department, Favorite colour etc.

Continuous variables: Variables containing numeric responses with decimal point. Eg. Temperature, Marks, Cost of products etc.

Discrete variables: Variables containing numeric responses as integers i.e. without decimal point. Eg. Age, Number of defects, Number of products purchased or sold.

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