Q 612. Correlation is an indicator or relationship between two variables (X and Y). While potentially the coefficient of correlation can range from -1 to +1, could there be a situation where it is either -1 or +1 i.e. Perfect Correlation? Provide examples to support your answer.

 

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Positive correlation

If 2 Parameters are directly proportional (if one increases the other as well increases, if one parameter decreases the other as well decreases) then they are said to have a positive correlation.

 

Example

Savings Vs financial security

Overtime worked and extra income

Salary Vs work satisfaction

Increased moisture Vs crop production

 

Perfect positive correlation

Perfect positive correlation means that if the first variable moves by some x% then second variable also moves by same amount in the same direction.  With the help of one variable we can predict the other variable.

 

Example

Length of the square to the perimeter of the square.

 

Radius of the circle to the perimeter of the circle.

 

Over time Hours to Over time Amount

 

Temperature to Pressure Co-relation

 

 

 

Negative correlation

If 2 Parameters are inversely proportional (if one parameter increases the other decrease or if one parameter decreases the other parameter increases) then they are said to have a negative correlation.

 

Examples

 

Colder the nights in winter and higher the energy bills(gas bill)

Higher transportation speed and decreased travel time

Increased exercise and fewer medical expenses

Higher loan payments and lower total interest amount owed

Increased absenteeism and lower overall income

 

Perfect Negative correlation

Perfect Negative correlation means that if the first variable moves by say x% then second variable must also move by the same amount in the opposite direction. Nothing more nothing less. With the help of one variable we can predict the other variable.

 

Examples

Air Lines Oil Price to Profit

 

 

Balls left in 50 Over Cricket Match

 

Could there be a situation where it is either -1 or +1 i.e. Perfect Correlation?

In theory Yes, its possible (see above examples). In practical its very rare, almost no as there will always be a measurement errors , Instruments uncertainty no matter how accurate your instrument is.

 

Example 

if you find a perfect square, when you measure length of the square and measure all the lengths for calculating perimeter. You are measuring with say measurement tape there will be always be chance of a measurement error and instrument uncertainty.

 

Correlation coefficient is an indicator which confirms the availability of a linear relationship between two variables (say “X” and “Y” variables). Measured correlation coefficient (signified by "r") falls between -1 to +1 and consider as unit free.

Formula :

n= number of data pairs

xi & yi are average of x & y variables.

X-bar and y-bar are average of x and y measurements.

σx and σy are standard deviations of x and y respectively.

There are three major conditions as stated below-

Condition-1: “ if r value is greater than zero”

If we find r value greater than zero between x and y variables, it indicates positive correlation between both variables. Positive correlation is nothing but proportional relationship which determines incrementation in x variable will increase Y variable as well.

 

Condition-2: “ if r value is less than zero”

There will be inversely proportional relationship between x and y variable if computed r-value is less than zero. Negative relationship takes place when x independent variable increase, the value of dependent variable decreases and vice versa.

 

Condition-3: “ if r value is zero”

If coefficient value is close to or equal to zero means, there is no relation between both variables.

 

Magnitude of correlation will depend on how close the r-value towards 1. Below table can help out to understand how strong our calculated correlation is:

 

 

Perfect correlation either “+1” or “-1” is difficult to find out practically because there are always common cause and environment variation available in system. This ideal condition can be possible in case of changing both variables with constant interval values.

Let us understand for perfect positive correlation (r=1):

Increment in environment temperature by 2 degree centigrade will increase electricity units (kwh) by 30 .  In that case all data points will be on linear line as shown in below graph and coefficient will be “+1”.

 

Correlation of coefficient helps to establish relation between predicated and actual value in statistical experiment. Calculated value explain the exactness between predicted value and exact value. It assess strength of association between data variables.

 

Karl Pearson's person method is most common which measures the strength and direction of linear relationship between two variables.

 

If x & y are the two variables of discussion, then the correlation coefficient can be calculated using the formula

 

 

There are two type of correlation  i.e. Positive or negative relation.

 

Perfect correlation of coefficient means where 100% of the time, variables in questions move together by exact same percentage and direction.

 

Perfect correlation is use in Super Market or e- Commerce platform where Time spent by customer in the Store or website can be established perfect correlation with money spent by customers.

 

Also similarly in real life fuel efficiency of car and money spend per kiliometer by car user for fuel charges can be calculated. 

 

Sometime researcher can misguide report users through misguided analysis as it only show degree of association but not confirm agreement.

 

for example internet explorer vs. Murder rate. It can lead to erroneous conclusion. 

 

The coefficient of correlation is hardly perceived as a perfect +1 or -1. 

The obvious reason is the complexity of the formula used might not bring exact +1 or -1.

 

The other reason might be that i can think one of my ownself. I take clonezapam(anti anxiety medicine or alpazolam) of potencity 0.25mg every night. I get a sleep of 8 hrs. So one tablet and one night sleep of 8 hrs may look like a perfect correlation. However, if i increase the potencity of 1 tab to 0.5mg, i will still sleep 8 hrs, but i think the sleep will be more sound and will make me feel  more fresh the next morning. Now although it is still one tablet and one night sleep, but i cant exactly measure how much out of 100pc, was 0.25mg effective or was 0.5mg effective. And vice versa for negative correlation. Perfect example could be how much anxiety goes down by taking 0.5mg instead of 0.25mg.

 

On the flip side, i also think that artificially we can achieve perfect correlation as used by the infertility clinics. However, if we need to analyse the depth, we will have to use several procedures to study the composition of an embryo to have it achieved perfect correlation. 

 

So in my own thoughts, perfect correlation could be visible only in picture, but the reality behind is far complex which might be near to perfect correlation but not absolute correlation.

Indeed, a precise linear relationship between two variables is indicated by a perfect correlation, which might have a correlation coefficient of either -1 or +1. All the data points in these situations fall precisely on a straight line.

The following scenarios can lead to perfect correlation:

Children's Height and Age: You would anticipate a strong positive association between a child's age and height among a group of children of the same age. Children typically grow taller as they get older. The correlation coefficient in this scenario would be extremely near to +1.

Time and Distance at a Constant Speed: When an object moves at a constant speed, its distance travelled, and its time taken are exactly equal. Distance and time would have a perfect positive correlation in this case.

Temperature in Fahrenheit and Celsius: There is a linear relationship between the two measures of temperature. The formula is F = (9/5) C + 32. Plotting the Fahrenheit and Celsius temperatures in this instance yields a perfect positive connection.

Height and Shoe Size in a Population with a Uniform Age Range: We can anticipate a strong positive correlation between height and shoe size in a group of individuals who are of the same age. Larger feet are typically found in taller people. The correlation coefficient in this scenario would be extremely near to +1.

Negative Correlation in Perfect Competition: In economics, the quantity requested, and the price of an item have a perfect negative correlation when there is perfect competition in the market. The quantity demanded falls as the good's price rises and vice versa. The correlation coefficient in this instance would be -1.

Number of Correct Answers on a Math Test and Math Scores: You would anticipate a perfect positive connection between the total score and the number of correct answers if all students took the identical math test.

While these examples show instances in which perfect correlation is possible, it is important to remember that real world data rarely exhibits perfect correlation because of measurement error, natural variability, and other factors that can affect the relationship between variables.

All the answers are very interesting. Some of the examples quoted for perfect correlation uses well established formulae, where of course there is only 1 input variable (e.g. perimeter of a square), but again the fact is that in real life there is hardly any situation where there will be only 1 factor affecting the output. Plus it was correctly identified that there will be some measurement system errors while measuring the length of the perfect square.

 

There are also a few answers where we have forgotten to check for X and Y variables (e.g. Temp conversion). Yes, correlation can be done between any 2 variables, however it will not be meaningful. Read more about it at the below link

https://www.benchmarksixsigma.com/forum/topic/35495-correlation-and-causation/

 

The best answer has been provided by Adil Khan. Well done!