To calculate the sample standard deviation, we will use the formula for sample standard deviation:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

Where:

• $x_i$ are the data points,
• $\bar{x}$ is the sample mean,
• $n$ is the number of data points.

Step 1: Find the sample mean ($\bar{x}$)

The data points are: $4, 3, 0, 4, 5, 3, 2, 5, 3, 1, 4, 5$

First, find the sum of the data points: $\text{Sum} = 4 + 3 + 0 + 4 + 5 + 3 + 2 + 5 + 3 + 1 + 4 + 5 = 39$

Next, calculate the mean: $\bar{x} = \frac{\text{Sum}}{n} = \frac{39}{12} = 3.25$

Step 2: Calculate each squared difference from the mean

Now, calculate the difference between each data point and the mean, square it, and sum them up.

$(4 - 3.25)^2 = 0.5625, \quad (3 - 3.25)^2 = 0.0625, \quad (0 - 3.25)^2 = 10.5625$ $(4 - 3.25)^2 = 0.5625, \quad (5 - 3.25)^2 = 3.0625, \quad (3 - 3.25)^2 = 0.0625$ $(2 - 3.25)^2 = 1.5625, \quad (5 - 3.25)^2 = 3.0625, \quad (3 - 3.25)^2 = 0.0625$ $(1 - 3.25)^2 = 5.0625, \quad (4 - 3.25)^2 = 0.5625, \quad (5 - 3.25)^2 = 3.0625$

Summing these squared differences: $\text{Sum of squared differences} = 0.5625 + 0.0625 + 10.5625 + 0.5625 + 3.0625 + 0.0625 + 1.5625 + 3.0625 + 0.0625 + 5.0625 + 0.5625 + 3.0625 = 27.194$

Step 3: Apply the formula

Now we divide by $n - 1$ (where $n = 12$): $\frac{27.194}{12 - 1} = \frac{27.194}{11} = 2.472$

Finally, take the square root to find the sample standard deviation: $s = \sqrt{2.472} \approx 1.6$

Conclusion

The sample standard deviation is approximately 1.6.

Would you like further details or have any questions?

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Here are five related questions you could explore:

1. How does sample standard deviation differ from population standard deviation?
2. What are the effects of adding or removing a data point on the sample standard deviation?
3. How would the calculation change if one data point were an outlier, like 10?
4. How does standard deviation relate to variance?
5. How is the sample mean different from the population mean in terms of usage?

Tip: Always check whether you need sample or population standard deviation, as the formula changes slightly.