8.2 Estimation by clustering

$\text{27}+\text{48}+\text{31}+\text{52}$ .

27 and 31 cluster near 30. Their sum is about $\text{2}\cdot \text{30}=\text{60}$ .

48 and 52 cluster near 50. Their sum is about $\text{2}\cdot \text{50}=\text{100}$ .

Thus, $\text{27}+\text{48}+\text{31}+5\text{2}$ is about $\begin{array}{ccc}(2\cdot 30)+(2\cdot 50)& =& 60+100\hfill \\ & =& 160\hfill \end{array}$

In fact, $\text{27}+\text{48}+\text{31}+\text{52}=\text{158}$ .

$\text{88}+\text{21}+\text{19}+\text{91}$ .

88 and 91 cluster near 90. Their sum is about $\text{2}\cdot \text{90}=\text{180}$ .

21 and 19 cluster near 20. Their sum is about $\text{2}\cdot \text{20}=\text{40}$ .

Thus, $\text{88}+\text{21}+\text{19}+\text{91}$ is about $\begin{array}{ccc}(2\cdot 90)+(2\cdot 20)& =& 180+40\hfill \\ & =& 220\hfill \end{array}$

In fact, $\text{88}+\text{21}+\text{19}+\text{91}=\text{219}$ .

$\text{17}+\text{21}+\text{48}+\text{18}$ .

17, 21, and 18 cluster near 20. Their sum is about $\text{3}\cdot \text{20}=\text{60}$ .

48 is about 50.

Thus, $\text{17}+\text{21}+\text{48}+\text{18}$ is about $\begin{array}{ccc}(3\cdot 20)+50& =& 60+50\hfill \\ & =& 110\hfill \end{array}$

In fact, $\text{17}+\text{21}+\text{48}+\text{18}=\text{104}$ .

$\text{61}+\text{48}+\text{49}+\text{57}+\text{52}$ .

61 and 57 cluster near 60. Their sum is about $\text{2}\cdot \text{60}=\text{120}$ .

48, 49, and 52 cluster near 50. Their sum is about $3\cdot \text{50}=\text{150}$ .

Thus, $\text{61}+\text{48}+\text{49}+\text{57}+\text{52}$ is about $\begin{array}{ccc}(2\cdot 60)+(3\cdot 50)& =& 120+150\hfill \\ & =& 270\hfill \end{array}$

In fact, $\text{61}+\text{48}+\text{49}+\text{57}+\text{52}=\text{267}$ .

$\text{706}+\text{321}+\text{293}+\text{684}$ .

706 and 684 cluster near 700. Their sum is about $\text{2}\cdot \text{700}=\text{1,400}$ .

321 and 293 cluster near 300. Their sum is about $\text{2}\cdot \text{300}=\text{600}$ .

Thus, $\text{706}+\text{321}+\text{293}+\text{684}$ is about $\begin{array}{ccc}(2\cdot 700)+(2\cdot 300)& =& \mathrm{1,400}+600\hfill \\ & =& \mathrm{2,000}\hfill \end{array}$

In fact, $\text{706}+\text{321}+\text{293}+\text{684}=\text{2,004}$ .