Calculate the sample standard deviation of the quantitative data shown. Round to two decimal places.
x
24
25
28
19
23
15

sample standard deviation = 

To calculate the sample standard deviation of the given data, we will follow these steps:

1. **Find the mean (average) of the dataTo calculate the sample standard deviation of the given data, we will follow these steps:

1. **Find the mean (average) of the **
   \[
   \text{Mean} (\bar{x}) = \frac{\sum{x}}{n}
   \]
   where \( n \) is the number of data points.

2. **Calculate the variance:**
   \[
   \text{Variance} (s^2) = \frac{\sum{(x_i - \bar{x})^2}}{n - 1}
   \]
   where \( x_i \) represents each data point.

3. **Calculate the sample standard deviation:**
   \[
   s = \sqrt{s^2}
   \]

Now, let's calculate it.

### 1. Calculate the mean
\[
\bar{x} = \frac{24 + 25 + 28 + 19 + 23 + 15}{6} = \frac{134}{6} = 22.33
\]

### 2. Calculate the variance
\[
\begin{align*}
(x_1 - \bar{x})^2 & = (24 - 22.33)^2 = 2.79 \\
(x_2 - \bar{x})^2 & = (25 - 22.33)^2 = 7.11 \\
(x_3 - \bar{x})^2 & = (28 - 22.33)^2 = 32.11 \\
(x_4 - \bar{x})^2 & = (19 - 22.33)^2 = 11.11 \\
(x_5 - \bar{x})^2 & = (23 - 22.33)^2 = 0.44 \\
(x_6 - \bar{x})^2 & = (15 - 22.33)^2 = 53.78 \\
\end{align*}
\]

Sum of squared deviations:
\[
\sum{(x_i - \bar{x})^2} = 2.79 + 7.11 + 32.11 + 11.11 + 0.44 + 53.78 = 107.34
\]

Variance:
\[
s^2 = \frac{107.34}{6 - 1} = \frac{107.34}{5} = 21.47
\]

### 3. Calculate the sample standard deviation
\[
s = \sqrt{21.47} \approx 4.63
\]

So, the sample standard deviation is approximately **4.63**.

Would you like further details or have any questions?

Here are five related questions:

1. How do you calculate the mean for a different dataset?
2. What is the difference between sample standard deviation and population standard deviation?
3. How does the sample size affect the standard deviation?
4. Can you explain why we divide by \(n - 1\) instead of \(n\) when calculating the sample variance?
5. How would the standard deviation change if the data had a larger spread?

**Tip:** Always use \( n-1 \) in the denominator when calculating sample variance to account for the bias in estimating population variance from a sample.

Calculate the sample standard deviation of the quantitative data shown. Round to two decimal places. x: 24, 25, 28, 19, 23, 15

Learn how to calculate the sample standard deviation for the data set 24, 25, 28, 19, 23, 15. This problem involves key statistical concepts such as mean, variance, and standard deviation, suitable for students in Grades 10-12.

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