Let's go through the steps to calculate the mean, standard deviation, and variance for this data. Since the data was initially considered a sample, we'll first compute it as a sample, and then I'll show the population standard deviation.

1. Mean Calculation

The formula for the mean is:

$\text{Mean} = \frac{\sum x}{n}$

Where:

• $\sum x$ is the sum of the data points.
• $n$ is the number of data points.

Data: $x = [17.8, 15, 12.2, 11.6, 44.8, 14.7, 33.4, 33, 34.7]$

Sum of data: $\sum x = 17.8 + 15 + 12.2 + 11.6 + 44.8 + 14.7 + 33.4 + 33 + 34.7 = 227.2$

Number of data points: $n = 9$

Mean: $\text{Mean} = \frac{227.2}{9} = 25.36 \quad (\text{rounded to two decimal places})$

2. Variance and Standard Deviation (Sample)

The formula for sample variance is:

$s^2 = \frac{\sum (x_i - \text{Mean})^2}{n - 1}$

Where $s^2$ is the sample variance, $x_i$ are the individual data points, and $n$ is the sample size.

Step 1: Calculate the squared deviations
For each $x_i$, we subtract the mean and then square the result: $(17.8 - 25.36)^2 = (-7.56)^2 = 57.14$ $(15 - 25.36)^2 = (-10.36)^2 = 107.33$ $(12.2 - 25.36)^2 = (-13.16)^2 = 173.18$ $(11.6 - 25.36)^2 = (-13.76)^2 = 189.29$ $(44.8 - 25.36)^2 = (19.44)^2 = 377.15$ $(14.7 - 25.36)^2 = (-10.66)^2 = 113.63$ $(33.4 - 25.36)^2 = (8.04)^2 = 64.64$ $(33 - 25.36)^2 = (7.64)^2 = 58.43$ $(34.7 - 25.36)^2 = (9.34)^2 = 87.33$

Step 2: Sum of squared deviations: $\sum (x_i - \text{Mean})^2 = 57.14 + 107.33 + 173.18 + 189.29 + 377.15 + 113.63 + 64.64 + 58.43 + 87.33 = 1227.12$

Step 3: Calculate sample variance: $s^2 = \frac{1227.12}{9 - 1} = \frac{1227.12}{8} = 153.39$

Step 4: Sample standard deviation: $s = \sqrt{153.39} = 12.39 \quad (\text{rounded to two decimal places})$

3. Population Standard Deviation

Now that we know the data is from a population, the formula for population standard deviation is:

$\sigma = \sqrt{\frac{\sum (x_i - \text{Mean})^2}{n}}$

Step 1: Population variance: $\sigma^2 = \frac{1227.12}{9} = 136.34$

Step 2: Population standard deviation: $\sigma = \sqrt{136.34} = 11.68 \quad (\text{rounded to two decimal places})$

Summary of Results:

• Mean = 25.36
• Sample Standard Deviation = 12.39
• Sample Variance = 153.39
• Population Standard Deviation = 11.68

Would you like further details or explanations on any of these steps? Here are 5 related questions you might find interesting:

1. How does sample standard deviation differ from population standard deviation?
2. What is the role of degrees of freedom in the calculation of variance?
3. How would the variance change if we added more data points to the sample?
4. Can the mean, variance, and standard deviation tell us about the shape of the data distribution?
5. What is the effect of outliers on the mean and standard deviation?

Tip: When dealing with population data, always divide by $n$ in variance calculations, while for sample data, divide by $n-1$ to account for the correction.