Find the means, tabulate deviations, and apply Pearson's formula.

$\bar x=15,\bar y=21$; $\sum(x-\bar x)(y-\bar y)=215$; $\sum(x-\bar x)^2=250$; $\sum(y-\bar y)^2=188$.

$r=215/\sqrt{250\times188}=215/216.8\approx0.992$.

Step 1 — Find the means. $\bar x = \dfrac{5+10+15+20+25}{5}=15$. $\bar y = \dfrac{12+18+20+25+30}{5}=\dfrac{105}{5}=21$.

Step 2 — Tabulate deviations, products and squares.

Step 3 — Apply the formula.

Independent check (direct method). $\sum xy = 5(12)+10(18)+15(20)+20(25)+25(30) = 60+180+300+500+750=1790$. $\sum x^2=25+100+225+400+625=1375$, $\sum y^2=144+324+400+625+900=2393$. Numerator: $5(1790)-(75)(105)=8950-7875=1075$. Denominator: $\sqrt{5(1375)-5625}\sqrt{5(2393)-11025} = \sqrt{1250}\sqrt{940} \approx 35.36\times30.66\approx1084.3$. $r \approx 1075/1084.3 \approx 0.991$ — matches the deviation-method result of $\approx0.992$ closely (the tiny gap is rounding), confirming a very strong positive correlation.

Arithmetic slips in squaring larger deviations (e.g. $(-10)^2$ or $9^2$) are more likely with bigger numbers like these — it is worth re-checking each squared term individually rather than trusting a running mental total, since a single wrong square noticeably shifts $r$.