What is the sample standard deviation for the following numbers? 275,294,293,354,314,353,310,360,342,315,310,296

To calculate the sample standard deviation, follow these steps:

### Step 1: Find the mean of the data.
\[
\text{Mean} (\mu) = \frac{\text{Sum of all values}}{\text{Number of values}}
\]
The values are: 275, 294, 293, 354, 314, 353, 310, 360, 342, 315, 310, 296.

\[
\text{Mean} = \frac{275 + 294 + 293 + 354 + 314 + 353 + 310 + 360 + 342 + 315 + 310 + 296}{12}
\]
\[
\text{Mean} = \frac{3836}{12} = 319.67
\]

### Step 2: Subtract the mean from each data point and square the result.
\[
(275 - 319.67)^2 = (-44.67)^2 = 1994.77
\]
\[
(294 - 319.67)^2 = (-25.67)^2 = 659.45
\]
\[
(293 - 319.67)^2 = (-26.67)^2 = 711.11
\]
\[
(354 - 319.67)^2 = (34.33)^2 = 1178.19
\]
\[
(314 - 319.67)^2 = (-5.67)^2 = 32.11
\]
\[
(353 - 319.67)^2 = (33.33)^2 = 1111.11
\]
\[
(310 - 319.67)^2 = (-9.67)^2 = 93.45
\]
\[
(360 - 319.67)^2 = (40.33)^2 = 1626.51
\]
\[
(342 - 319.67)^2 = (22.33)^2 = 498.83
\]
\[
(315 - 319.67)^2 = (-4.67)^2 = 21.83
\]
\[
(310 - 319.67)^2 = (-9.67)^2 = 93.45
\]
\[
(296 - 319.67)^2 = (-23.67)^2 = 560.71
\]

### Step 3: Find the sum of these squared differences.
\[
\text{Sum of squared differences} = 1994.77 + 659.45 + 711.11 + 1178.19 + 32.11 + 1111.11 + 93.45 + 1626.51 + 498.83 + 21.83 + 93.45 + 560.71 = 8581.52
\]

### Step 4: Divide by \( n - 1 \) (since this is a sample, not the entire population).
\[
\frac{8581.52}{12 - 1} = \frac{8581.52}{11} = 780.14
\]

### Step 5: Take the square root of the result.
\[
\sqrt{780.14} = 27.93
\]

Thus, the **sample standard deviation** is approximately **27.93**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How is sample variance related to sample standard deviation?
2. What is the difference between population and sample standard deviation?
3. Why do we divide by \( n - 1 \) instead of \( n \) for sample standard deviation?
4. How does outlier data affect the standard deviation?
5. Can standard deviation be negative?

**Tip:** A small standard deviation indicates that the data points are close to the mean, while a large standard deviation indicates that the data points are spread out over a wider range.

Learn how to calculate the sample standard deviation for the data set: 275, 294, 293, 354, 314, 353, 310, 360, 342, 315, 310, 296. This guide walks through the calculation of the mean, variance, and standard deviation step by step, suitable for Grades 10-12 statistics students.

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