question_answer

Cost of living Index for some period is given in the following frequency distribution:
Index	1500 ? 1600	1600 ? 1700	1700 ? 1800	1800 ? 1900	1900 ? 2000	2000 ? 2100	2100 ? 2200
No. of weeks	3	11	12	7	9	8	2
Find the mode and median for above data.

Answer:

Index	Number of weeks \[({{f}_{i}})\]	\[c{{f}_{i}}\]
1500 ? 1600	3	3
1600 ? 1700	11  \[{{f}_{0}}\]	14
1700 ? 1800	12  \[{{f}_{1}}\]	26
1800 ? 1900	7   \[{{f}_{2}}\]	33
1900 ? 2000	9	42
2000 ? 2100	8	50
2100 ? 2200	2	52
	\[\sum{{{f}_{i}}=52}\]
\[n=52\]
\[\frac{n}{2}=\frac{52}{2}=26\]
\[\therefore \] Median class is 1700 ? 1800
Median \[=l+\frac{\frac{n}{2}-cf}{f}\times h\]
\[=1700+\frac{26-14}{12}\times 100\]
\[\frac{n}{2}=1700+\left( \frac{12}{12}\times 100 \right)=1800\]
Maximum frequency is 12
\[\therefore \] Modal class is 1700 ? 1800
Mode \[=l+\frac{{{f}_{1}}-{{f}_{0}}}{2{{f}_{1}}-{{f}_{0}}-{{f}_{2}}}\times h\]
\[=1700+\frac{12-11}{24-11-7}\times 100\]
\[=1700+\frac{1}{6}\times 100\]
\[=1700+16.67\]
\[=1716.67\]