Find deviations of $x$ and $y$ from their own means, then apply the correlation formula in deviation form.

$\bar x=4,\bar y=4$; $\sum(x-\bar x)(y-\bar y)=4$; $\sum(x-\bar x)^2=10$; $\sum(y-\bar y)^2=10$

$r=4/\sqrt{10\times10}=4/10=0.4$

Step 1 — Find the means. $\bar{x} = \dfrac{2+3+5+4+6}{5}=\dfrac{20}{5}=4$. $\bar{y}=\dfrac{3+4+2+5+6}{5}=\dfrac{20}{5}=4$.

Step 2 — Tabulate deviations, their product, and their squares.

$\sum(x-\bar x)(y-\bar y)=2+0-2+0+4=4$. $\sum(x-\bar x)^2=4+1+1+0+4=10$. $\sum(y-\bar y)^2=1+0+4+1+4=10$.

Step 3 — Apply the formula.

Independent check. Since $\sum(x-\bar x)^2=\sum(y-\bar y)^2=10$ here (a coincidence of this particular data), the denominator simplifies to exactly $10$, so $r$ is simply the cross-product sum divided by $10$ — re-confirming $r=4/10=0.4$ without needing a square-root calculation at all.

Missing a negative sign in an individual product (row 3 here contributes $-2$, not $+2$) will silently shift the final numerator — add the signed products carefully rather than adding magnitudes and guessing the overall sign. Also confirm both deviation columns individually sum to zero ($\sum(x-\bar x)=0$, $\sum(y-\bar y)=0$) as a quick check before proceeding.