Plot the pairs mentally (or on paper) and judge the direction of the cloud first; then compute $r$ to check that judgement numerically.

Step 1 — Visual judgement (scatter diagram reasoning). Reading down both rows together: every time $x$ increases from one month to the next, $y$ also increases (2→50, 4→55, 6→65, 8→70, 10→85 — no reversal anywhere). A scatter diagram of these five points would therefore show a cloud of points running consistently from lower-left to upper-right — the classic picture of a strong positive linear relationship, with the points expected to lie fairly close to a single upward-sloping line (since the increases are fairly steady, not erratic).

Step 2 — Verify numerically. Find the means. $\bar x=\dfrac{2+4+6+8+10}{5}=6$. $\bar y=\dfrac{50+55+65+70+85}{5}=\dfrac{325}{5}=65$.

Step 3 — Tabulate deviations, products and squares.

Step 4 — Apply Pearson's formula.

Since a valid correlation coefficient can never exceed $1$ in magnitude, this signals an arithmetic slip — re-checking the cross-product column: $(-4)(-15)=60$, $(-2)(-10)=20$, $(0)(0)=0$, $(2)(5)=10$, $(4)(20)=80$; total $=60+20+0+10+80=170$, which is correct as tabulated, so the error must be in a squared term. Re-checking $\sum(y-\bar y)^2$: $225+100+0+25+400=750$, not $630$ as first written (the last term, $20^2=400$, not $280$) — correcting this: $\sum(y-\bar y)^2=750$.

Step 5 — Recompute with the corrected sum.

$r = \dfrac{170}{\sqrt{40}\sqrt{750}} = \dfrac{170}{\sqrt{30000}} \approx \dfrac{170}{173.2} \approx 0.981$ …

Always sanity-check a computed $r$ against the mandatory range $-1\le r\le1$ before reporting it — a value outside that range, as happened at Step 4 here, is an unambiguous signal that a squared term or cross-product was mis-computed …

Trusting a scatter diagram's visual impression alone without the numerical check (or vice versa, trusting a computed $r$ without a sanity check on its permitted range) misses exactly the kind of arithmetic slip this worked so …