Let's walk through the solution based on the provided steps.

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

The data set is: $58, 60, 74$.

The formula for the sample mean is: $\bar{x} = \frac{\sum{x}}{n}$ Where $n = 3$ (since there are 3 values) and $\sum{x}$ is the sum of the data points.

$\bar{x} = \frac{58 + 60 + 74}{3} = \frac{192}{3} = 64$

So, the sample mean $\bar{x}$ is 64.

Step 2: Calculate the deviations and squares of the deviations

For each data point, we calculate the deviation from the mean ($x - \bar{x}$) and the square of the deviation:

• Deviation of 58: $58 - 64 = -6$ Square of deviation: $(-6)^2 = 36$

• Deviation of 60: $60 - 64 = -4$ Square of deviation: $(-4)^2 = 16$

• Deviation of 74: $74 - 64 = 10$ Square of deviation: $(10)^2 = 100$

Step 3: Calculate the sample variance and sample standard deviation

The formula for the sample variance ($s^2$) is: $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$ Where $n = 3$ and $\sum (x - \bar{x})^2$ is the sum of the squared deviations.

$s^2 = \frac{36 + 16 + 100}{3 - 1} = \frac{152}{2} = 76$

So, the sample variance is 76.

Now, the formula for the sample standard deviation ($s$) is: $s = \sqrt{s^2} = \sqrt{76} \approx 8.7$

Thus, the sample standard deviation is approximately 8.7.

Do you have any questions or need further details? Here are 5 related questions to expand your understanding:

1. How would the sample mean change if another data point, 68, was added to the set?
2. What is the difference between sample standard deviation and population standard deviation?
3. How would the sample variance be affected if the data set consisted of larger numbers?
4. What is the significance of squaring the deviations when calculating standard deviation?
5. How can standard deviation help interpret data spread in real-world applications?

Tip: Squaring deviations ensures that both positive and negative deviations contribute equally to the variance.