Interesting! Correlation = 1 means that for one variable, there is a precise relationship with another variable. There is no cause and effect or ongoing relationship. It's a one-off association. If a variable is constant, it ceases to be a variable and has variance equal to zero, which means the correlation coefficient becomes undefined. This is because for the coefficient to be calculated, there is division by the sum of variances, both of which are zero, and you can't divide by zero. So there is no such thing as a correlation coefficient between two constant variables

It should, yes.

it depends a bit on how you fit your data, but always remember you cannot really extend your correlation beyond your measured range.

If you fit data with a line fit function would look like this:

Y = 2X + 3, then yes, if

X1=3

X2=3

X3=3

Then Y1=Y2=Y3=9

In a perfect correlation, this is what you expect.

Amir Nakar Hi, read my answer above. A constant variable has no variance and therefore is not a variable, and correlation therefore is invalid since it is determined by dividing by variance, which in this case is zero. So a constant variable is never correlated with another variable

Yes it's true, for the calculation of the correlation coefficients in the statistical methods, covariance and variances are to be computed first, then formula for correlation coefficients is applied for its estimation.

Andrew Paul McKenzie Pegman

How about the following variable values:

Var1 values: 1,2,3,3,3,4,5,6

Var2 values: 2,4,6,6,6,8,10,12

Correlation between Var1, Var2?

r = 1

in the situation asked about, when you observe the third, fourth and fifth measurements - the variable behaves in a constant fashion.

This kind of thing happens very often in natural science

Hope this clarifies

Amir Nakar repeated values are different from a variable being constant and having no variance. That was the situation Gautam Kumar Das was describing :)

Correlation coefficient value 1 indicates the variables to be highly dependent... It is very rare to see independent variables to have correlation coefficient value 1 and even in dependent variables.. the number of observation is very much important in this case because the value may reduce to less than one if the number of observation increase... If the value decreases to less than 1 then we can not expect of a constant value for a variable with respect to a constant value of other.

Perfect correlation is deterministic, and it is a mathematical function. Eg: Y=2X.

It is possible when a third factor (covariate) influences and is not considered, correlation is not causation.

Remember it is for the data the relationship holds and cannot be extrapolated or generalized to new data.

Direction of Correlation coefficient is also important in explaining result; it may be

Perfect Positive correlation or Perfect Negative correlation.

Secondly, review your variables whether that appeals to estimate correlation analysis. For example it is useless to measure 'sale of a soft drink brand with Cholera disease'.

Correlation estimated for the number of tigers in the two tiger zones counted with the camera traps installed by the forest department, 73 tigers in one tiger zone where 1156 Cuddeback camera traps were installed, and 23 tigers in another tiger zone where 272 Cuddeback camera traps were installed for the tiger census, but the said cameras were installed in two different periods, and the two tiger zones are situated adjacent to each other, for this situation, tigers might migrate from one zone to another in the two different phases of the census operations that would make anomalous in the count, so that the number of tigers of two zones are tried to correlate considering x and y variables, where n is 1428, total number of cameras installed separately in two zones, and it results 1 in the analysis of correlation coefficients, after R adjusted, that's a problematic situation for the interpretation, may I interpret that no tiger did not migrate from one habitat to another in the two phases of the census operation, or simply the estimation for these variables have no appeal i.e. irrelevant for determination of correlation coefficients ?

The degree of association between two continuous variables is the correlation coefficient (r). For interpreting the meaning of correlation coefficient, independence, and complete association in a positive or negative aspect is represented by numbers 0, +1 and -1, with intermediate numbers. Hence, 1 indicates that the two variables are either positively or negatively correlated and when r=1, it indicates that the two observed values were highly correlated.

Then the number of two variables co-exists in their own stands, without movement, migration, overlap, or cross each other, they remain constant as they were before

Assume Gaussian distribution variables x_1 and x_2 with std deviations s_1 and s_2. If the variables are perfectly correlated (rho = 1), then the cross-correlation is,

s_12 = (s_1s_2)^0.5. __ eq(1)

Then the joint distribution of (x_1, x_2) will have the covariance matrix,

s(1,2) = [s_1^2, s_1s_2...

s_2s_1, s_2^2] ____ eq(2)

with the cross-correlation derived in the first paragraph. It can be seen that for rho = 1, the covariance matrix s(1,2) is singular and the distribution is essentially an impulse function.

In other words, This can only happen when the jointly distributed variables exhibit a linear relationship,

x_1 = a*x_2 ____ eq(3)

To prove this, the std deviation s_1 from the above equation can be written as

s_1 = a * s_2 _____ eq(4)

which gives the correlation s_12 as,

s_12 = sqrt(a) * s_2 ____ eq(5).

for a = (s_1/s_2), the condition for perfect correlation (rho = 1) is satisfied.

Thus, eq(3) is correct. According to eq (5), if the std deviation of the random variables are unchanged, x_2 will stay constant as long as x_1 is constant.