To solve the problem of calculating the sample standard deviation of the given data, let’s go step by step.

The dataset is: $6, 9, 14, 10, 5, 8, 11$

Step 1: Calculate the Mean

The mean ($\bar{x}$) is given by: $\bar{x} = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

$\bar{x} = \frac{6 + 9 + 14 + 10 + 5 + 8 + 11}{7} = \frac{63}{7} = 9$

Step 2: Calculate the Deviations and Their Squares

For each data point ($x_i$), calculate $(x_i - \bar{x})^2$:

$(6 - 9)^2 = (-3)^2 = 9$ $(9 - 9)^2 = 0^2 = 0$ $(14 - 9)^2 = 5^2 = 25$ $(10 - 9)^2 = 1^2 = 1$ $(5 - 9)^2 = (-4)^2 = 16$ $(8 - 9)^2 = (-1)^2 = 1$ $(11 - 9)^2 = 2^2 = 4$

The squared deviations are: $9, 0, 25, 1, 16, 1, 4$

Step 3: Calculate the Variance

The formula for sample variance ($s^2$) is: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

Here, $n = 7$, so $n - 1 = 6$. Calculate the sum of the squared deviations: $\sum (x_i - \bar{x})^2 = 9 + 0 + 25 + 1 + 16 + 1 + 4 = 56$

Thus: $s^2 = \frac{56}{6} = 9.3333$

Step 4: Calculate the Standard Deviation

The standard deviation ($s$) is the square root of the variance: $s = \sqrt{9.3333} \approx 3.05$

Final Answer:

The sample standard deviation is: $\boxed{3.05}$

Let me know if you'd like further clarifications or need additional assistance.

5 Related Questions:

1. What is the difference between population standard deviation and sample standard deviation?
2. How does the standard deviation change if one data point is removed from the set?
3. What are the uses of standard deviation in real-life applications?
4. How can variance be interpreted in statistical terms?
5. What are alternative methods to compute standard deviation manually?

Tip:

Always double-check whether you need a sample or population standard deviation. The formula differs by using $n$ vs. $n-1$.