Question: standard deviation for $274$
$281$
$255$
$311$
$290$
$314$
$345$
$339$
$259$
$341$
$279$
$323$
and seperately for $354$
$346$
$252$
$289$
$251$
$316$
$359$
$349$
$334$
$341$
$285$
$288$
1. **State the problem:** Calculate the standard deviation for two separate data sets:
Data set 1: $274, 281, 255, 311, 290, 314, 345, 339, 259, 341, 279, 323$
Data set 2: $354, 346, 252, 289, 251, 316, 359, 349, 334, 341, 285, 288$
2. **Formula for standard deviation:**
The standard deviation $\sigma$ for a sample is given by:
$$\sigma = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$
where $n$ is the number of data points, $x_i$ are the data points, and $\bar{x}$ is the sample mean.
3. **Calculate the mean for each data set:**
For data set 1:
$$\bar{x}_1 = \frac{274 + 281 + 255 + 311 + 290 + 314 + 345 + 339 + 259 + 341 + 279 + 323}{12}$$
$$= \frac{3561}{12} = 296.75$$
For data set 2:
$$\bar{x}_2 = \frac{354 + 346 + 252 + 289 + 251 + 316 + 359 + 349 + 334 + 341 + 285 + 288}{12}$$
$$= \frac{3664}{12} = 305.33$$
4. **Calculate squared deviations and sum for each data set:**
For data set 1:
$$\sum (x_i - \bar{x}_1)^2 = (274 - 296.75)^2 + (281 - 296.75)^2 + \cdots + (323 - 296.75)^2$$
Calculated values:
$$(274 - 296.75)^2 = 515.06$$
$$(281 - 296.75)^2 = 247.56$$
$$(255 - 296.75)^2 = 1747.56$$
$$(311 - 296.75)^2 = 204.06$$
$$(290 - 296.75)^2 = 45.56$$
$$(314 - 296.75)^2 = 297.56$$
$$(345 - 296.75)^2 = 2310.06$$
$$(339 - 296.75)^2 = 1790.06$$
$$(259 - 296.75)^2 = 1437.56$$
$$(341 - 296.75)^2 = 1936.56$$
$$(279 - 296.75)^2 = 314.06$$
$$(323 - 296.75)^2 = 689.06$$
Sum:
$$\sum (x_i - \bar{x}_1)^2 = 10544.12$$
For data set 2:
$$\sum (x_i - \bar{x}_2)^2 = (354 - 305.33)^2 + (346 - 305.33)^2 + \cdots + (288 - 305.33)^2$$
Calculated values:
$$(354 - 305.33)^2 = 2367.11$$
$$(346 - 305.33)^2 = 1673.78$$
$$(252 - 305.33)^2 = 2853.78$$
$$(289 - 305.33)^2 = 267.11$$
$$(251 - 305.33)^2 = 2934.78$$
$$(316 - 305.33)^2 = 113.78$$
$$(359 - 305.33)^2 = 2877.11$$
$$(349 - 305.33)^2 = 1893.78$$
$$(334 - 305.33)^2 = 819.11$$
$$(341 - 305.33)^2 = 1273.78$$
$$(285 - 305.33)^2 = 413.44$$
$$(288 - 305.33)^2 = 301.78$$
Sum:
$$\sum (x_i - \bar{x}_2)^2 = 17384.54$$
5. **Calculate the standard deviation for each data set:**
For data set 1:
$$\sigma_1 = \sqrt{\frac{10544.12}{12 - 1}} = \sqrt{\frac{10544.12}{11}} = \sqrt{958.56} = 30.96$$
For data set 2:
$$\sigma_2 = \sqrt{\frac{17384.54}{12 - 1}} = \sqrt{\frac{17384.54}{11}} = \sqrt{1580.41} = 39.75$$
**Final answers:**
- Standard deviation for data set 1 is approximately $30.96$
- Standard deviation for data set 2 is approximately $39.75$