From the data given below,state which group is more variable,$A$ or $B$?
Marks $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ Group $A$ $9$ $17$ $32$ $33$ $40$ $10$ $9$ Group $B$ $10$ $20$ $30$ $25$ $43$ $15$ $7$
| Marks | $10-20$ | $20-30$ | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ |
| Group $A$ | $9$ | $17$ | $32$ | $33$ | $40$ | $10$ | $9$ |
| Group $B$ | $10$ | $20$ | $30$ | $25$ | $43$ | $15$ | $7$ |
(B) To determine which group is more variable,we compare their standard deviations. Since the means of both groups are equal,the group with the higher standard deviation is more variable.
For Group $A$:
Mean $\bar{x}_A = 44.6$,Variance $\sigma_A^2 = 227.84$,Standard Deviation $\sigma_A = \sqrt{227.84} \approx 15.09$.
For Group $B$:
Mean $\bar{x}_B = 44.6$,Variance $\sigma_B^2 = 243.84$,Standard Deviation $\sigma_B = \sqrt{243.84} \approx 15.61$.
Since $\sigma_B > \sigma_A$,Group $B$ is more variable.
For Group $A$:
Mean $\bar{x}_A = 44.6$,Variance $\sigma_A^2 = 227.84$,Standard Deviation $\sigma_A = \sqrt{227.84} \approx 15.09$.
For Group $B$:
Mean $\bar{x}_B = 44.6$,Variance $\sigma_B^2 = 243.84$,Standard Deviation $\sigma_B = \sqrt{243.84} \approx 15.61$.
Since $\sigma_B > \sigma_A$,Group $B$ is more variable.
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