Question: standard deviation for offensive and defensive Offense:\tDefense:\n354\t274\n346\t281\n252\t255\n289\t311\n251\t290\n316\t314\n359\t345\n349\t339\n334\t259\n341\t341\n285\t279\n288\t323
1. **State the problem:** Calculate the standard deviation for the given offense and defense data sets.
2. **Recall the formula for standard deviation:**
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}$$
where $\mu$ is the mean of the data and $N$ is the number of data points.
3. **Calculate the mean for Offense:**
$$\mu_{offense} = \frac{354 + 346 + 252 + 289 + 251 + 316 + 359 + 349 + 334 + 341 + 285 + 288}{12}$$
$$= \frac{3664}{12} = 305.3333$$
4. **Calculate the mean for Defense:**
$$\mu_{defense} = \frac{274 + 281 + 255 + 311 + 290 + 314 + 345 + 339 + 259 + 341 + 279 + 323}{12}$$
$$= \frac{3561}{12} = 296.75$$
5. **Calculate squared deviations for Offense:**
$$\sum (x_i - \mu_{offense})^2 = (354 - 305.3333)^2 + (346 - 305.3333)^2 + \ldots + (288 - 305.3333)^2$$
$$= 2373.78 + 1673.78 + 2841.78 + 264.44 + 2917.78 + 113.78 + 2877.78 + 1857.78 + 828.44 + 1275.11 + 414.78 + 301.44 = 17252.67$$
6. **Calculate squared deviations for Defense:**
$$\sum (x_i - \mu_{defense})^2 = (274 - 296.75)^2 + (281 - 296.75)^2 + \ldots + (323 - 296.75)^2$$
$$= 522.56 + 247.56 + 1765.56 + 205.56 + 11.06 + 299.06 + 2336.06 + 1790.06 + 1416.56 + 1180.56 + 313.56 + 689.06 = 10767.6$$
7. **Calculate standard deviation for Offense:**
$$\sigma_{offense} = \sqrt{\frac{17252.67}{12}} = \sqrt{1437.72} = 37.92$$
8. **Calculate standard deviation for Defense:**
$$\sigma_{defense} = \sqrt{\frac{10767.6}{12}} = \sqrt{897.3} = 29.95$$
**Final answer:**
- Standard deviation for Offense is approximately $37.92$
- Standard deviation for Defense is approximately $29.95$