It means that you have chosen that the error will not be more than 5%. so the results will be Ok and accepted. They are within the error.

Where do you see a contradiction?

Bilal,

I agree with Jochen. There is no contradiction here. It simply means that you have sufficient sample size and therefore small enough standard errors for .31 to be significantly different from zero -- which actually doesn't require an especially large sample for that magnitude.

Pat

There is no contradiction here.

1) Suppose a model of the following:

R (radon) = a*T (temperature) + b + E, where E is a random variable (noise, or unaccounted disturbances)

If the variance of E is high compared to the contribution of T (determined by a and range of T values), then correlation coefficient will not be high. This simply means that the correlation between R and T is real, but not highly predictive due to the high contribution of noise.

2) Another possibility is the contribution of other regressors you have not taken into consideration:

R = aT + bX1 + cX2 + ... (actually, in the case above, we lump all those Xi into E)

If X1, X2,.. are measurable and NOT random but changed in a biased fashion, then since correlation coefficient only considers the collective change in T and R together, but neglect the changes in X1, X2, correlation coefficient may be come out to be smaller than expected. There may be a higher dependency between T and R, but clouded by the affect of X1, X2 on R. The remedy to this problem is to determine the "partial correlation".

Dear Bilal, I agree with all the previous comments, but I am even confused by the question. I don't know why you would call a correlation of .3 close to zero. That to me is not a trivial correlation at all. It means there may be a lot of error or one or more unaccounted explanatory variables involved, but when there is a correlation of .3, that is not close to 0 in my opinion. As others have mentioned, it normally would not take a very large sample size to get a standard error small enough to have that correlation have a p-value where it is reasonably unlikely to come from a population where the relationship between the two variables is 0. Your p-value suggests that chance is very small indeed. Bob

Thanks to all for response to my query. Please guide me, whether my discussion according to given results is correct or not. please made correction on my attached file.

Bilal, you major problem is the scientific interpretation of your data, and this can't be done by others, espesially not be non-experts and not by statisticians. I do neither know your research question nor the do I have the scientific background. However, I would guess that temperature and humidity should anyway be related, so I wonder why this was tested. I can't judge what effect size (r) can be considered relevant. Further I don't know if the assumption of liner relationships are sensible.

By what you wrote in the beginning, I would rather ask: how and by how much does the Radon concentration change depending on temperature and humitity? Since both predictors likely correlate, it might even be better to use only one of them. Is there a physical/chemical model behind (how should the temp. or humidity influence the Radon concentration?; how are temp and humidita connected?) I'd find this kind of quantitative analyses based on knowledge about the underlying physical and/or chemical releationships much more instructive than simply giving measurs of the "tightness of the (linear) correlation" (as done by calculating r).

If you stay with this simple correlation analysis, then I'd suggest to give the partial correlations of Radeon with one predictor after partializing out the other.

The answer to your problem might be simpler than you would imagine. Remember, statistical significance is based in part on the size of one's sample. .31 is not "puny" and if your sample size is large enough your standard error will be small, hence a statistically significant correlation. With a large enough sample size, anything can be statistically significant. Also, you MUST do a scatterplot. If your sample size is small, skewness and outliers can affect effect sizes, which in turn affect p values.

Agree: scatterplot! scatterplot! scatterplot! (drawit!- understand it! - interpret it!)

Bilal, let me share with you my observation on the matter. First, a weak-to-moderately weak relationship maybe a substantial predictive value in the case of what you are studying. Second, significant relationship means this observed magnitude of relationship in your samples would more likely be observed in the populations they represent. On the basis of this magnitude (-0.305 to -0.492) and significance (magnitude more likely observable in the population), do you think your result is of practical value in connection to what you are studying?

I believe the (highly significant) correlation between taking aspirin and not having a heart attack is .04 - that's the correlation, not the p-value!

I would not describe a correlation coefficient of 0.31 as "close to 0" but as "moderately strong." The magnitude of a coefficient and it's statistical significance are two different concepts. The significance level simply gives you the probability that a coefficient of this magnitude could be due to pure chance. In a larger sample a smaller coefficient may turn out significant, while in a small sample even a relative large coefficient may fail to reach significance. There is no contradiction between the magnitude of the correlation coefficient you found and the fact that it is statistically significant.

Sorry everybody, I answered before I read the other responses and thus I only restated what others stated much better.

Thomas, just a note: you wrote "The significance level simply gives you the probability that a coefficient of this magnitude could be due to pure chance." - I would avoid such kind of explanation. What is "chance"? -- it is a probability! The whole sentence makes no sense, and it is at best a circular statement. Values do not happen to be as they are "by chance". There is a reason why these values are like they are. The point is: we simply do not *know* why they are like this. Thus saing that a value is "by chance" like this we mean that we can not give a reason why it must/should be like this. If we can (quantitatively) compare our degree of expectation of different possible values by probabilities.

The correct statement would have been: "The significance level simply gives you the probability that a coefficient of this magnitude (or greater) would be observed under the null hypothesis" (where this null hypothesis is: "no correlation").

Jochen, I agree with you in incorrect usage of the word "chance", however I disagree with your deterministic approach. :) I guess a better description may be obtained using the concept (or phrase) of "inherent variability" instead of chance. Hence, p-value gives us the probability that the outcome will be observed only due to inherent variability in the system, i.e. everything has worked the way it is modeled, and/or the way it has been experienced previously under normal operating conditions. However, your rejection, I guess, would still apply to this definition, and you would say there SHOULD be a reason for the inherent variability, but we simply do not know them. And to this, I would kindly disagree. Only in a total deterministic universe, we would be justified in making this claim. We might be living in one, but this would be a philosophical position, not a scientific one.

Yes, Burak, nice response, and I essentially agree. Especially to the last thing you wrote: it is a matter of philosophy, not of science! As far as scienence is concerned, the only thing we can say is that we do not know the reasons of the variability, and therefore we model the outcome by probabalistic models. In some cases we might have an idea of how a part of the observed variabilty could possibly be explained by facts we possibly could know (or measure/observe). In other cases we might not even have an idea. But whatever the reasons and possibilities were: right now we simply can not explain the variability, and it is outside of science to discuss things that lack any explanation. Determinism, indeterminism, objective truth, ... these are all philosophical concepts, not scientific concepts. This is evident when we start to recognize that "time", "space", "energy" are already simply *models* that are (extremely) useful to explain certain observations, but these are no existing things as-is, as described in the physical theories where they are part of. Now if we think that "time" for instance is a model, all derivations of (in)determinism are related to this model, are only valid within this model and is nothing that is defined by itself. However, I have to admit that it is very difficult to think about anything without using "time" as inherent mental concept. Some parts of quantum mechanics give useful descriptions without using the "time"-concept what is typically very difficult to interpret for our "time-addicted" brains.

Peace upon you

This question need clear and brief answer not deep details.

I think correlation is a confused measurement. So before explain the results I suggest to send your data to take a look and then we can talk about results.

I highly recommend looking at The Earth is Round (P < 0.05) by Jacob Cohen. His central aim is to critique null hypothesis testing, but along the way he articulates the logic of NHT, p-values, and t-values with exceptional clarity and brevity.

it can also be helpful to calculate the percentage of variance that accounts for the variation between the variables. This can be done by squaring the r value. apologies if this was already mentioned.

..actually Randall its called coefficient of determination.

I would not say that a correlation of 0.30 is close to zero, this is because of the range of values that the correlation between the two variables can take. I would say for example that a correlation coefficient could be 0.03 or less, is indeed close to zero. For the figure posted i would say that it is low.

In the case of simple linear regression, for the sake of an example. A correlation of 1.0 and -1.0 (perfect as it is called) implies that all the points lie on the regression line. To the extent that the magnitude of the correlation decreases, the data points will have the tendency to deviate from the regression line, but we should notice some degree of trend. In the case extreme of zero correlation you, you would observe in the scatter diagram a cloud of points where you can not see any trend.

The significance of a correlation is closely associated with the sample size and certainly with the variability in both variables.

most of the relevant aspects have been already answered. I like particularly the posts by Thomas and Jochen. Remember the following:

- a regression slope measures the sensibility of y to x (model 1 regression)

- a correlation coefficient measures the predictability of y given x

- these two are different. In fact you can have a regression line with a slope close to 0 and still, all observations fall exactly on top of the line, in which case Pearson's coefficient is 1. On the other hand, you can have a regression with a steep slope and still with wide data scatter around it, and thus moderate Pearson's coefficient. In both cases the significance is high (p

It means that you have chosen that the error will not be more than 5%. so the results will be Ok and accepted. They are within the error.

Issam, I see your anwer can be interpreted in a "right" and in a "wrong" way. What is meant (correctly) with the 5% is the type-I error RATE. The 5% are not the individual error of the experiment. You have no possibility to assess the actual error. The result may be drastically wrong, so the individual result may be far worse than a 5% error. If is is about rejecting a null, a false rejection is a full error, 100% so to say. One just does not know whether or not the rejection was correct. The only thing is that the long run rate of similar tests will not produce more than 5% of such 100% wrong conclusions. I added this comment because many people seem to confuse "errors" with "error rates", what are entirely different concepts, and this confusion often leads to harmfully wrong interpretations of results.

Your results shows that both of the variables radon and air pressure have impact other than random variation but there may be some other factors those also have impact on your variables but are not included in your study therefore high noise or random variation decreased the correlation but correlation is not near to zero but medium correlation: Coefficient, r

Strength of Association is given as;

Small

.1 to .3

-0.1 to -0.3

Medium

.3 to .5

-0.3 to -0.5

Large

.5 to 1.0

-0.5 to -1.0.

Thanks hope this explanation will be acceptable by all teachers.