In the context of the confidence interval formula you provided:

$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$

$s$ represents the sample standard deviation. It is a measure of the spread or dispersion of the data points in the sample around the sample mean $\bar{x}$.

How to Calculate $s$ (Sample Standard Deviation):

Given a sample of $n$ data points $x_1, x_2, \ldots, x_n$, the sample standard deviation $s$ is calculated using the following formula:

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

Explanation of the Formula:

1. $\bar{x}$: This is the sample mean, calculated as:

   $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

2. $x_i$: Each individual data point in the sample.

3. $\sum_{i=1}^{n} (x_i - \bar{x})^2$: This represents the sum of the squared differences between each data point and the sample mean. It shows how much each data point deviates from the mean.

4. $n-1$: This is the degrees of freedom, used instead of $n$ to correct for the bias in the estimation of the population standard deviation from a sample. It's especially important in small samples.

Why is $s$ Important?

The sample standard deviation $s$ is crucial because it provides an estimate of the variability in the data, and it is used to calculate the standard error of the mean, which appears in the confidence interval formula. The standard error is given by:

$\frac{s}{\sqrt{n}}$

This represents how much the sample mean $\bar{x}$ would vary from the true population mean if we were to take multiple samples.

Would you like further clarification on this topic or more examples?

Follow-up Questions:

1. How does the sample standard deviation differ from the population standard deviation?
2. Why do we use $n-1$ (degrees of freedom) instead of $n$ when calculating the sample standard deviation?
3. How does increasing the sample size $n$ affect the sample standard deviation and the confidence interval width?
4. Can you provide a practical example of calculating the sample standard deviation $s$ using a small dataset?
5. How does the sample standard deviation influence the margin of error in the confidence interval?

Tip:

Always double-check the calculation of the sample standard deviation, especially in small samples, as errors in this value can significantly impact the confidence interval and the conclusions drawn from the data analysis.