6. Correlation

Johnny van Doorn

University of Amsterdam

11 September 2026

In this lecture we discuss:

The t-distribution
- Sampling distributions, continued
- Standard error and confidence intervals
Correlation
- From variance to covariance to \(r\)
- In JASP
Significance of a correlation
- Converting \(r\) to a \(t\)-statistic
Control for a third variable
- Partial correlation

Reading: Chapter 7 (§7.1–7.8), 5.7 (Scatterplots), 2.8–2.9 (Standard error & confidence intervals)

Not exam material: §7.2.5, §7.4.3–7.4.5 (Spearman, Kendall, point-biserial), §7.6 (comparing correlations).

Challenger

The statistics

cor.test(damagedTemp, damagedOr)$p.value

[1] 0.1731224

cor.test(temperature, damage)$p.value

[1] 0.001043532

	\(r\)	\(p\)
7 damaged flights	-0.58	.173
All 23 flights	-0.64	.001

Data: challenger.csv, from the Presidential Commission on the Space Shuttle Challenger Accident (1986, Vol. 1: 129-131), via Dalal, Fowlkes & Hoadley (1989).

T-distribution

Three distributions

What is the difference between

Population distribution
Sample distribution
Sampling distribution

Gosset

In probability and statistics, Student’s t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

In the English-language literature it takes its name from William Sealy Gosset’s 1908 paper in Biometrika under the pseudonym “Student”. Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples, for example the chemical properties of barley where sample sizes might be as low as 3.

Source: Wikipedia

Population distribution

set.seed(123)
layout(matrix(c(2:6,1,1,7:8,1,1,9:13), 4, 4))

n  <- 50    # Sample size
df <- n - 1 # Degrees of freedom

mu    <- 100
sigma <- 15

IQ <- seq(mu-45, mu+45, 1)

par(mar=c(4,2,2,0))  
plot(IQ, dnorm(IQ, mean = mu, sd = sigma), type='l', col="black", main = "Population Distribution", 
     lwd = 3, bty= "n", axes = FALSE, xlab = c(60, 140))
axis(1)

n.samples <- 12

for(i in 1:n.samples) {
  
  par(mar=c(2,2,2,0))  
  hist(rnorm(n, mu, sigma), main="Sample Distribution", xlab = c(60, 140),
       cex.axis=.5, col=rainbow(12)[i], cex.main = .75, las = 1, axes = FALSE)
axis(1)
  
}

Population distribution

A sample

Let’s take one sample from our normal population.

x <- rnorm(n, mu, sigma); x

 [1] 116.11018  99.58980  99.50004  77.25899 111.85578  96.83899  90.14886
 [8]  78.81961  95.50356  87.26408  94.04454  81.73600 125.31384  99.75996
[15] 116.12418  60.97450  93.20203  89.86777  81.65611 123.19914  78.77077
[22] 104.77585 112.69654 102.67285  86.87117 114.11749 102.55882  84.04753
[29]  79.17926 131.30076  89.82245  72.16643 107.99889 104.65345  79.69248
[36]  70.85565  98.25546 117.09094 109.54186  92.60594  87.48718 104.06600
[43] 102.36030 109.44568  94.06303 113.49031  87.53783  95.04183 111.11222
[50] 114.84957

hist(x, main = "Sample distribution", col = rainbow(12)[6], breaks = 15, las =1)
text(125, 7, bquote(bar(x) ~ " = " ~ .(round(mean(x),2))), cex = 2)

A sample

More samples

More samples, keeping \(\bar{x}\) each time.

n.samples     <- 1000
mean.x.values <- vector()

for(i in 1:n.samples) {
  x <- rnorm(n, mu, sigma)
  mean.x.values[i] <- mean(x)
}

Sampling distribution

The means have their own distribution.

hist(mean.x.values, 
     col  = rainbow(12),
     main = "Sampling distribution of the mean", 
     xlab = "1,000 sample means")

Sampling distribution

Standard Error and
Confidence Intervals

That spread has a name.

Standard Error

95% confidence interval

\[SE = \frac{\text{Standard deviation}}{\text{Square root of sample size}} = \frac{s}{\sqrt{n}}\]

Lowerbound = \(\bar{x} - 1.96 \times SE\)
Upperbound = \(\bar{x} + 1.96 \times SE\)

Standard Error

T-distribution

So if the population is normaly distributed (assumption of normality) the t-distribution represents the deviation of sample means from the population mean (\(\mu\)), given a certain sample size (\(df = n - 1\)).

The t-distibution therefore is different for different sample sizes and converges to a standard normal distribution if sample size is large enough.

The t-distribution is defined by:

\[\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!\]

where \(\nu\) is the number of degrees of freedom and \(\Gamma\) is the gamma function.

Source: wikipedia

The t-distribution family

Fatter tails for small \(n\) (few degrees of freedom); converges to the standard normal as \(n\) grows.

\(\alpha\) dictates the rejection region

Same logic as the binomial, now continuous: stricter \(\alpha\), region further into the tails.

Applet: a continuous sampling distribution

The same reasoning as the binomial applet, now for a continuous outcome.

Play around with this app to see how the t-distribution, \(\alpha\), and power hang together

Power in practice

Strive for 80%
Based on known effect size
Calculate number of subjects needed
Use JASP (Power module), G*Power, or SPSS to calculate

Alpha Power

R-Psychologist

Correlation

Pearson Correlation

In statistics, the Pearson correlation coefficient, also referred to as the Pearson’s r, Pearson product-moment correlation coefficient (PPMCC) or bivariate correlation, is a measure of the linear correlation between two variables X and Y. It has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation. It is widely used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.

Source: Wikipedia

Pearson Correlation

\[r_{xy} = \frac{{COV}_{xy}}{S_xS_y}\] Where \(S\) is the standard deviation and \(COV\) is the covariance.

\[{COV}_{xy} = \frac{\sum_{i=1}^N (x_i - \bar{x})(y_i - \bar{y})}{N-1}\]

Plot correlation

\[(x_i - \bar{x})(y_i - \bar{y})\]

Load data

data <- read.csv(datasetsPath("2026/exam-anxiety.csv"), check.names = FALSE)
data <- data[, c("Time spent revising", "Exam performance", "Exam anxiety")]
ssrTable(data)

Covariance

\[{COV}_{xy} = \frac{\sum_{i=1}^N (x_i - \bar{x})(y_i - \bar{y})}{N-1}\]

mean.exam    <- mean(exam_grade, na.rm=TRUE)
mean.anxiety <- mean(anxiety,    na.rm=TRUE)

delta.exam    <- exam_grade - mean.exam
delta.anxiety <- anxiety    - mean.anxiety

prod <- delta.exam * delta.anxiety

covariance <- sum(prod) / (N - 1)
covariance

[1] -196.554

Correlation

\[r_{xy} = \frac{{COV}_{xy}}{S_xS_y}\]

correlation <- covariance / ( sd(exam_grade) * sd(anxiety) ); correlation

[1] -0.4409934

Correlation

\[r_{xy} = \frac{{COV}_{xy}}{S_xS_y}\] \[{COV}_{xy} = \frac{\sum_{i=1}^N (x_i - \bar{x})(y_i - \bar{y})}{N-1}\]

cor(exam_grade, anxiety) # correlation

[1] -0.4409934

cor(z.exam,     z.anxiety) # correlation of z-scores

[1] -0.4409934

# covariance of z-scores
sum(z.exam * z.anxiety ) / (N - 1)

[1] -0.4409934

Plot correlation

Significance testing using \(t\)

A test statistic with a known probability distribution (the t-distribution).

It’s a standardized measure of how closely our observed statistic matches the value claimed by \(H_0\):

\[t = \frac{\text{estimate} - H_0 \text{ value}}{\text{SE}(\text{estimate})}\]

Significance of a correlation

We convert \(r\) to a \(t\)-statistic because \(t\) has a sampling distribution: the \(t\)-distribution with df = N-2:

\[t_r = \frac{r \sqrt{N-2}}{\sqrt{1 - r^2}}\]

\[{df} = N - 2\]

Hypotheses for \(t\)

\[ \begin{aligned} H_0 &: t_r = 0 \\ H_A &: t_r \neq 0 \\ H_A &: t_r > 0 \\ H_A &: t_r < 0 \\ \end{aligned} \]

\(r\) to \(t\)

df  <- N-2
t.r <- ( correlation*sqrt(df) ) / sqrt(1-correlation^2)

t.r

[1] -4.938025

df

[1] 101

One-sided vs two-sided

The hypotheses above translate into where we shade the rejection region:

Additional text here

Same \(\alpha\), split over two tails or spent on one.

Visualize

Locate in \(t\)-distribution

\[P(|t| \geq 4.94 \mid H_0) < .001\]

Back to Challenger

Flights	r	N	df	t	p
7 damaged flights	-0.58	7	5	-1.59	0.173
All 23 flights	-0.64	23	21	-3.80	0.001

Nearly the same \(r\); but \(\sqrt{N-2}\) causes the t-statistics to differ substantially.

Partial correlation

A third variable

All three are related:

Performance and anxiety: \(r = -0.44\)
Performance and revision: \(r = 0.40\)
Anxiety and revision: \(r = -0.71\)

How much of the association is unique to anxiety and performance?

Shared variance

Plain \(r\) takes every overlap with anxiety: \(A + C = 19.4\%\)
Partial \(r\) keeps only \(A\), against what revision leaves: \(\frac{A}{A + E} = \frac{5.1}{5.1 + 79.1} = 6.1\%\)

Partial correlation

\[\LARGE{r_{xy \cdot z} = \frac{r_{xy} - r_{xz} r_{yz}}{\sqrt{(1 - r_{xz}^2)(1 - r_{yz}^2)}}}\]

Or do anxious students simply revise less?

cor.exam.anxiety    <- cor(exam_grade, anxiety)
cor.exam.revise     <- cor(exam_grade, revise)
cor.anxiety.revise  <- cor(anxiety,    revise)

data.frame(cor.exam.anxiety, cor.exam.revise, cor.anxiety.revise)

  cor.exam.anxiety cor.exam.revise cor.anxiety.revise
1       -0.4409934       0.3967207         -0.7092493

numerator   <- cor.exam.anxiety - (cor.exam.revise * cor.anxiety.revise)
denominator <- sqrt( (1-cor.exam.revise^2)*(1-cor.anxiety.revise^2) )

partial.correlation <- numerator / denominator

partial.correlation

[1] -0.2466658

Holding revision constant, the link shrinks.

# Back to figure 7.9: A + C, then A / (A + E)
round(c(cor.exam.anxiety, partial.correlation)^2, 3)

[1] 0.194 0.061

Significance of partial correlation

Locate in t-distribution

df <- N - 3

t.pr <- ( partial.correlation*sqrt(df) ) / sqrt(1-partial.correlation^2)
t.pr

[1] -2.545307

p-value

JASP

Closing

Next Week

The linear model
- Predictions
- Assumptions
- Fun

Recommended Exercises

Exercise 7.1
- Note: pets.jasp can be downloaded from the JASP data library
Exercise 7.6
Exercise 7.7

Labcoat Leni 7.1

Chamorro-Premuzic et al. (2008): Why do you like your lecturers?

Do students like lecturers who resemble themselves?

stu_*: the student’s own five personality traits (NEO-FFI)
lec_*: how much they want each trait in a lecturer, from −5 to +5

Data + results (.jasp) · Results (.html)

Contact

6. Correlation

Challenger

The statistics

T-distribution

Three distributions

Gosset

Population distribution

Population distribution

A sample

A sample

More samples

Sampling distribution

Sampling distribution

Standard Error andConfidence Intervals

Standard Error

Standard Error

T-distribution

The t-distribution family

\(\alpha\) dictates the rejection region

Applet: a continuous sampling distribution

Power in practice

Alpha Power

Correlation

Pearson Correlation

Pearson Correlation

Plot correlation

Plot correlation

Plot correlation

Load data

Covariance

Correlation

Correlation

Plot correlation

Significance testing using \(t\)

Significance of a correlation

Hypotheses for \(t\)

\(r\) to \(t\)

One-sided vs two-sided

Visualize

Back to Challenger

Partial correlation

A third variable

Shared variance

Partial correlation

Significance of partial correlation

p-value

JASP

Closing

Next Week

Recommended Exercises

Labcoat Leni 7.1

Contact

Standard Error and
Confidence Intervals