# Sample size and correlation

As a rule of thumb, statisticians consider an r-value of 0.

7 to be strong.

Let’s take a look at this example.

A survey is conducted on two-thirds of the student cohort of a certain level and we observe that there is no strong correlation between the responses to any pair of the questions.

Based on the 116 survey responses, we observe that the correlation coefficient ranges from 0.

11 to 0.

59.

Next I want to find out if there are differing results with smaller sample sizes.

All survey respondents (N=116)I then performed bootstrapping and selected random samples of 50 respondents ten times from the total pool of survey respondents.

Even though the sample size is now smaller, there are strong correlations observed for bootstrapped sample 6 (school v math, school v humanities, math v science) and sample 10 (school v math).

So in the event that we actually only polled the sample of respondents in bootstrapped sample 6 (to represent the whole population), we would have made a conclusion that there is a strong correlation between those variables.

And this shows that having a large sample size doesn’t mean that we are more likely to observe stronger correlations; where in this case, a larger sample size actually weakens the correlation.

This goes to show that what matters more is understanding the homogeneity of the population and how we perform the sampling (i.

e.

whether the sample is randomly selected and representative of the stratification of the population).

A smaller sample with high homogeneity will display a greater correlation coefficient than a large sample with low homogeneity (high heterogeneity).

So if we choose to focus on a population that is homogeneous, we might not need a large sample size to reflect the correlation.

Of course, if we want to be conservative, we can adjust the threshold of which we consider a strong correlation, while considering the confidence interval of correlation coefficient.

There are online calculators and also a package in R that can be used for computing for the confidence interval of the correlation coefficient.

It takes into account the observed sample correlation coefficient, sample size and confidence level (typically 0.

95).

Here, I would also like to reference a paper on “At what sample size do correlations stabilize?”, for which results indicate that in typical scenarios the sample size should approach 250 for stable estimates.

This would be a trivial solution in this case as it means I have to poll the entire population (the student cohort is <250).

In other words, we should try to obtain a larger sample whenever possible.

The comment from Chris Draheim in a thread, “What is the minimum sample size to run Pearson’s R?”, on ResearchGate also highlights the instability of small samples: “I wouldn’t trust any correlation without at least 50 or 60 observations, with 80–100 being around where I feel comfortable.

From my experience with pilot data and analyzing subsets of datasets or presenting data on an ongoing study, correlations with 20–40 subjects can be markedly different than when you have 80–100, I’ve even seen correlations between two tasks going from -.

70 to +.

40 when the observations were doubled.

it’s also important to identify outliers, even with larger sample sizes an outlier or two can have a large effect on the magnitude of the correlation, since this is least squares after all.

”Separately, it would be interesting to study how the profile of bootstrapped sample 6 differs from the rest to show such differences in the correlation analysis.

Correlation coefficients for bootstrapped sample 1 and bootstrapped sample 2Correlation coefficients for bootstrapped sample 3 and bootstrapped sample 4Correlation coefficients for bootstrapped sample 5 and bootstrapped sample 6Correlation coefficients for bootstrapped sample 7 and bootstrapped sample 8Correlation coefficients for bootstrapped sample 9 and bootstrapped sample 10References:https://www.

surveysystem.

com/correlation.

htmhttps://www.

checkmarket.

com/blog/how-to-estimate-your-population-and-survey-sample-size/https://www.

r-bloggers.

com/how-to-calculate-confidence-intervals-of-correlations-with-r/Originally published at https://projectosyo.

wixsite.

com.

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