To find the variance and standard deviation, we will use the computational variance formula. The computational variance formula is:

$\text{Variance} \ (\sigma^2) = \frac{1}{n} \left( \sum x_i^2 \right) - \left( \frac{1}{n} \sum x_i \right)^2$

Where:

• $n$ is the number of data points,
• $x_i$ represents each value in the data set,
• $\sum x_i$ is the sum of all the values,
• $\sum x_i^2$ is the sum of the squares of each value.

Step-by-step solution:

Step 1: Count the number of data points (n)

The data set is: $10, 15, 15, 17, 17, 17, 18, 18, 19, 22, 25, 27, 33, 35, 39, 50, 75, 76$ There are 18 data points, so $n = 18$.

Step 2: Calculate $\sum x_i$

$\sum x_i = 10 + 15 + 15 + 17 + 17 + 17 + 18 + 18 + 19 + 22 + 25 + 27 + 33 + 35 + 39 + 50 + 75 + 76 = 538$

Step 3: Calculate $\sum x_i^2$

$\sum x_i^2 = 10^2 + 15^2 + 15^2 + 17^2 + 17^2 + 17^2 + 18^2 + 18^2 + 19^2 + 22^2 + 25^2 + 27^2 + 33^2 + 35^2 + 39^2 + 50^2 + 75^2 + 76^2$

Let's compute the squared values:

10^2 & = 100, \\ 15^2 & = 225, \\ 15^2 & = 225, \\ 17^2 & = 289, \\ 17^2 & = 289, \\ 17^2 & = 289, \\ 18^2 & = 324, \\ 18^2 & = 324, \\ 19^2 & = 361, \\ 22^2 & = 484, \\ 25^2 & = 625, \\ 27^2 & = 729, \\ 33^2 & = 1089, \\ 35^2 & = 1225, \\ 39^2 & = 1521, \\ 50^2 & = 2500, \\ 75^2 & = 5625, \\ 76^2 & = 5776 \end{align*}$$ Now sum all the squared values: $$\sum x_i^2 = 100 + 225 + 225 + 289 + 289 + 289 + 324 + 324 + 361 + 484 + 625 + 729 + 1089 + 1225 + 1521 + 2500 + 5625 + 5776 = 21099$$ #### Step 4: Apply the computational variance formula $$\text{Variance} \ (\sigma^2) = \frac{1}{18} \times 21099 - \left( \frac{1}{18} \times 538 \right)^2$$ First, calculate $$\frac{1}{18} \times 21099$$: $$\frac{21099}{18} = 1172.17$$ Next, calculate $$\frac{1}{18} \times 538$$: $$\frac{538}{18} = 29.89$$ Now square $$29.89$$: $$29.89^2 = 893.72$$ Finally, subtract to get the variance: $$\text{Variance} \ (\sigma^2) = 1172.17 - 893.72 = 278.45$$ #### Step 5: Calculate the standard deviation The standard deviation is the square root of the variance: $$\text{Standard deviation} \ (\sigma) = \sqrt{278.45} = 16.69$$ ### Final answers: - **Variance** = 278.45 - **Standard deviation** = 16.69 Would you like further details on this? Here are 5 related questions: 1. How do you compute variance for a sample instead of a population? 2. What is the significance of variance in statistical analysis? 3. Can you explain the difference between standard deviation and variance? 4. How do outliers affect the variance and standard deviation? 5. Can you show how to find the mean absolute deviation (MAD) for this data? **Tip**: When calculating variance for a sample, use $$n-1$$ in the denominator to correct for bias in small samples.