A random sample of students was asked how many lectures they had attended in the previous week. 

Their responses where: 4, 3, 0, 4, 5, 3, 2, 5, 3, 1, 4, 5.

What is the sample standard deviation for the number of lectures attended, to 1 decimal place?

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?

---

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.

A random sample of students was asked how many lectures they had attended in the previous week. Their responses were: 4, 3, 0, 4, 5, 3, 2, 5, 3, 1, 4, 5. What is the sample standard deviation for the number of lectures attended, to 1 decimal place?

Learn how to find the sample standard deviation for a dataset of student lecture attendance. This example uses 12 data points with a step-by-step breakdown, covering essential statistical concepts like the mean and variance. Suitable for students in Grades 9-12.

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