[!TLDR] Class midpoints as $x_i$; weighted mean, then the shortcut identity $\sigma^2=\frac1N\sum f_ix_i^2-\bar x^2$. [!ANSWER] Mean $=37$ kg, Va …

Class midpoints: $x_i:15,25,35,45,55$; frequencies $f_i:2,5,12,7,4$; $N=2+5+12+7+4=30$. $\sum f_ix_i=15(2)+25(5)+35(12)+45(7)+55(4)=30+125+420+315+220=1110$, so $\bar x=1110/30=37$. Using the shortcut identity: $\sum f_ix_i^2=15^2(2)+25^2(5)+35^2(12)+45^2(7)+55^2(4)=450+3125+14700+14175+12100=44550$, giving $\sigma^2=44550/30-37^2=1485-1369=116$. (Direct-method cross-check: deviations $x_i-37$ are $-22,-12,-2,8,18$, squares $484,144,4,64,324$; weighted sum $2(484)+5(144)+12(4)+7( …

Find the class midpoints and frequency-weighted mean; compute the variance either directly ($\sigma^2=\frac1N\sum f_i(x_i-\bar x)^2$) or via the shortcut id …

Arithmetic slips in computing $x_i^2$ for larger midpoints (e.g. $55^2=3025$, not $2525$) are the most c …