Dear Nur Amira,

The assumptions for Pearson correlation coefficient are as follows: level of measurement, related pairs, absence of outliers, normality of variables, linearity, and homoscedasticity.

Level of measurement refers to each variable. For a Pearson correlation, each variable should be continuous.  If one or both of the variables are ordinal in measurement, then a Spearman correlation could be conducted instead.

Related pairs refers to the pairs of variables. Each participant or observation should have a pair of values. So if the correlation was between weight and height, then each observation used should have both a weight and a height value.

Absence of outliers refers to not having outliers in either variable. Having an outlier can skew the results of the correlation by pulling the line of best fit formed by the correlation too far in one direction or another.  Typically, an outlier is defined as a value that is 3.29 standard deviations from the mean, or a standardized value of less than ±3.29.

Linearity and homoscedasticity refer to the shape of the values formed by the scatterplot. For linearity, a “straight line” relationship between the variable should be formed.  If a line were to be drawn between all the dots going from left to right, the line should be straight and not curved.  Homoscedasticity refers to the distance between the points to that straight line. The shape of the scatterplot should be tube-like in shape. If the shape is cone-like, then homoskedasticity would not be met.

I hope it will help you.

All the best !

A relation between two variables is spurious because we don't have two things in isolation. Therefore, I recommend not to test the significance of the simple relation between two variables (INTERVAL AND OR RATIO) . It is highly recommended to use partial and semi partial correlation coefficients.

If you insist on testing the significance of the relation between two variables measured  at interval and or ratio levels and if you don't have outliers then you have to satisfy the bivariate normality assumption needed to satisfy the assumption underlying the  the statistical test of significance.

Correlations analysis  is different from MLR.   You  can avoid specification error within  multiple regression by  building a model based on

Theory

Previous research

And logical basis

The purpose of multiple regression is  explanation and prediction. You are able to think  causally and study the effect of one variable on  the other keeping the effects of the other variables (brought explicitly into the model)  constant by using statistical control.  We know we deal with uncertainty and theory in incomplete and that is the reason we introduce the error term to take care of the variables we don't know   Moreover  we have assumption underlying the error term  and we test the violation of these assumptions.  etc.....

As compared  to pearson correlation coefficient  between just two variables.  You have the notmality assumption if you want to test the significance of this coefficient   but in my  opinion you beteer test the partial and semi partial correlation coefficient that are used in MLR. A simple relation is spurious and it is recommended to go back to theory and previous research and test the partial and semi partial

I think the answer depends on whether you are asking about estimation or inference.  You may find Chapter 6 in Thom Baguley's book Serious Stats helpful.  See the Google Books link below.  HTH.

https://books.google.ca/books?id=ObUcBQAAQBAJ&pg=PA211&lpg=PA211&dq=%22Statistical+inference+with+the+product-moment+correlation+coefficient%22&source=bl&ots=-eiUhGjD2o&sig=d4UWXjr4ZcR9br5NaAgWaW94qQM&hl=en&sa=X&ved=0ahUKEwiumdKB24fOAhXI5oMKHWnvA1EQ6AEIHTAA#v=onepage&q=%22Statistical%20inference%20with%20the%20product-moment%20correlation%20coefficient%22&f=false

Dear Nur

The assumptions included normal distribution of the variables and the variables must be scale type.

In the case of non-normality or ordinal variables, you can use Spearman correlation . 

Good Luck 

to the discussion above: I think that spurious correlation is only relevant for compositional data. 

The assumptions for Pearson correlation coefficient are as follows: level of measurement, related pairs, absence of outliers, normality of variables, linearity, and homoscedasticity.

Level of measurement refers to each variable. For a Pearson correlation, each variable should be continuous.  If one or both of the variables are ordinal in measurement, then a Spearman correlation could be conducted instead.

Related pairs refers to the pairs of variables. Each participant or observation should have a pair of values. So if the correlation was between weight and height, then each observation used should have both a weight and a height value.

Absence of outliers refers to not having outliers in either variable. Having an outlier can skew the results of the correlation by pulling the line of best fit formed by the correlation too far in one direction or another.  Typically, an outlier is defined as a value that is 3.29 standard deviations from the mean, or a standardized value of less than ±3.29.

Linearity and homoscedasticity refer to the shape of the values formed by the scatterplot. For linearity, a “straight line” relationship between the variable should be formed.  If a line were to be drawn between all the dots going from left to right, the line should be straight and not curved.  Homoscedasticity refers to the distance between the points to that straight line. The shape of the scatterplot should be tube-like in shape. If the shape is cone-like, then homoskedasticity would not be met.

Each variable should be continous, pair of values, absence of outliers in either variable and linearity are assumptions for pearson correlation analysis.

Assumptions for the Pearson correlation coefficient include: level of measurement, related pairs, absence of outliers, normality of variables, linearity, and homoscedasticity.