Let's break down the computation **step by step** for the formulas shown in your image, explaining each part clearly: --- ### **Step 1: Understanding the Formula for Standard Deviation** The formula for the sample standard deviation \( s \) is: \[ s = \sqrt{\frac{n\sum x^2 - (\sum x)^2}{n(n-1)}} \] Where: - \( n \) = number of data points - \( \sum x^2 \) = sum of the squares of the data points - \( (\sum x)^2 \) = square of the sum of the data points --- ### **Step 2: Plugging in the Values** From the problem: - \( n = 15 \) - \( \sum x^2 = 3974 \) - \( \sum x = 232 \) Substitute these values into the formula: \[ s = \sqrt{\frac{15 \times 3974 - (232)^2}{15 \times (15-1)}} \] --- ### **Step 3: Calculate the Numerator** Calculate each part of the numerator: - \( 15 \times 3974 = 59610 \) - \( (232)^2 = 53824 \) So, \[ \text{Numerator} = 59610 - 53824 = 5786 \] --- ### **Step 4: Calculate the Denominator** Calculate the denominator: - \( 15 \times (15-1) = 15 \times 14 = 210 \) --- ### **Step 5: Divide Numerator by Denominator** \[ \frac{5786}{210} \approx 27.552 \] --- ### **Step 6: Take the Square Root** \[ s = \sqrt{27.552} \approx 5.25 \] --- ### **Step 7: Coefficient of Standard Deviation (CSD)** The formula for CSD is: \[ CSD = \frac{s}{\bar{x}} \] Where \( \bar{x} \) is the mean. Given: - \( s = 5.25 \) - \( \bar{x} = 15.47 \) So, \[ CSD = \frac{5.25}{15.47} \approx 0.34 \] --- ## **Final Answers** - **Standard Deviation (\( s \))**: \( 5.25 \) - **Coefficient of Standard Deviation (CSD)**: \( 0.34 \) --- ### **Summary Table** | Step | Calculation | Result | |------|-------------|-------------| | 1 | \( 15 \times 3974 \) | 59610 | | 2 | \( 232^2 \) | 53824 | | 3 | Subtract Numerator | 5786 | | 4 | Denominator | 210 | | 5 | Divide | 27.552 | | 6 | Square Root | 5.25 | | 7 | \( \frac{5.25}{15.47} \) | 0.34 | Let me know if you want a breakdown of the calculation for the mean (\( \bar{x} \)) as well!