Again, you should note that these are already in order so the range is quickly found to be
\begin{equation*}
y_n - y_1 = 38.3 - 0.6 = 37.7
\end{equation*}
million residents.
For IQR, we first must determine the quartiles. The median (found earlier) already is the second quartile so we have \(Q_2 = 4.5\) million. For the other two, the formula for computing percentiles gives you the 25th percentiile
\begin{gather*}
(n+1)p = 51(1/4) = 12.75\\
Q_1 = P^{0.25} = 0.25 \times 1.9 + 0.75 \times 2.1 = 2.05
\end{gather*}
and the 75th percentile
\begin{gather*}
(n+1)p = 51(3/4) = 38.25\\
Q_3 = P^{0.75} = 0.75 \times 7 + 0.25 \times 8.3 = 7.325.
\end{gather*}
Hence, the IQR = 7.325 - 2.05 = 5.275 million residents.
From the computation before, again note that n=51 since the District of Columbia is included. The mean of this data found before was found to be approximately 6.20 million residents. So, to determine the variance you may find it easier to compute using the the
alternate variance formulas 1.5.5.
\begin{align*}
v & = \left ( \frac{\sum_{k=1}^n y_k^2 }{n} \right ) - \mu^2\\
& \approx \frac{4434.37}{51} - (6.20)^2\\
& = 48.51
\end{align*}
and so you get a sample variance of
\begin{equation*}
s^2 \approx \frac{51}{50} \cdot 48.51 = 49.48
\end{equation*}
and a sample standard deviation of
\begin{equation*}
s \approx \sqrt{49.48} \approx 7.03
\end{equation*}
million residents.