Question

1. find the standard deviation by completing the table.

n = the number of data points = 16
\\(\sum x = 1248\\)
\\(\bar{x} = \frac{\sum x}{n} = 78\\)
\\(\

$$\begin{array}{c|c|c|c}x & x - \\bar{x} & (x - \\bar{x})^2 \\\\ \\hline 68 & -10 & 100 \\\\ 98 & 20 & 400 \\\\ 86 & 8 & 64 \\\\ 67 & -11 & 121 \\\\ 70 & -8 & 64 \\\\ 82 & & \\\\ 75 & & \\\\ 50 & & \\\\ 100 & & \\\\ 78 & & \\\\ 77 & & \\\\ 72 & & \\\\ 75 & & \\\\ 80 & & \\\\ 83 & & \\\\ 87 & & \\\\ \\end{array}$$

\\)

Explanation:

Step1: Calculate \( x - \bar{x} \) for each \( x \)

Given \( \bar{x} = 78 \), for each \( x \) value, subtract 78. For example:

• \( x = 82 \): \( 82 - 78 = 4 \)
• \( x = 75 \): \( 75 - 78 = -3 \)
• \( x = 50 \): \( 50 - 78 = -28 \)
• \( x = 100 \): \( 100 - 78 = 22 \)
• \( x = 78 \): \( 78 - 78 = 0 \)
• \( x = 77 \): \( 77 - 78 = -1 \)
• \( x = 72 \): \( 72 - 78 = -6 \)
• \( x = 75 \): \( 75 - 78 = -3 \)
• \( x = 80 \): \( 80 - 78 = 2 \)
• \( x = 83 \): \( 83 - 78 = 5 \)
• \( x = 87 \): \( 87 - 78 = 9 \)

Step2: Calculate \( (x - \bar{x})^2 \) for each \( x \)

Square the results from Step 1:

• \( x = 82 \): \( 4^2 = 16 \)
• \( x = 75 \): \( (-3)^2 = 9 \)
• \( x = 50 \): \( (-28)^2 = 784 \)
• \( x = 100 \): \( 22^2 = 484 \)
• \( x = 78 \): \( 0^2 = 0 \)
• \( x = 77 \): \( (-1)^2 = 1 \)
• \( x = 72 \): \( (-6)^2 = 36 \)
• \( x = 75 \): \( (-3)^2 = 9 \)
• \( x = 80 \): \( 2^2 = 4 \)
• \( x = 83 \): \( 5^2 = 25 \)
• \( x = 87 \): \( 9^2 = 81 \)

Step3: Sum all \( (x - \bar{x})^2 \)

First, sum the given \( (x - \bar{x})^2 \) values: \( 100 + 400 + 64 + 121 + 64 = 749 \)
Then sum the new ones: \( 16 + 9 + 784 + 484 + 0 + 1 + 36 + 9 + 4 + 25 + 81 = 1459 \)
Total sum: \( 749 + 1459 = 2208 \)

Step4: Calculate Variance

Variance \( s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} \), where \( n = 16 \). So \( s^2 = \frac{2208}{15} = 147.2 \)

Step5: Calculate Standard Deviation

Standard Deviation \( s = \sqrt{147.2} \approx 12.13 \)

Answer:

The standard deviation is approximately \( 12.13 \) (after completing the table and performing the calculations as above).