Question

using technology to calculate standard deviation henry recorded the number of miles he biked each day for a week. his miles were 25, 40, 35, 25, 40, 60, and 75. enter the data into the statistics calculator. what is the standard deviation of the miles henry biked to the nearest tenth?

Explanation:

Step1: Find the mean

The data set is \( 25, 40, 35, 25, 40, 60, 75 \). The number of data points \( n = 7 \).
The mean \( \bar{x}=\frac{25 + 40 + 35 + 25 + 40 + 60 + 75}{7}=\frac{290}{7}\approx41.4286 \)

Step2: Calculate squared differences

For each data point \( x_i \), calculate \( (x_i-\bar{x})^2 \):

• \( (25 - 41.4286)^2\approx(-16.4286)^2\approx269.9 \)
• \( (40 - 41.4286)^2\approx(-1.4286)^2\approx2.04 \)
• \( (35 - 41.4286)^2\approx(-6.4286)^2\approx41.32 \)
• \( (25 - 41.4286)^2\approx(-16.4286)^2\approx269.9 \)
• \( (40 - 41.4286)^2\approx(-1.4286)^2\approx2.04 \)
• \( (60 - 41.4286)^2\approx(18.5714)^2\approx344.9 \)
• \( (75 - 41.4286)^2\approx(33.5714)^2\approx1127.0 \)

Step3: Find the sum of squared differences

Sum \( = 269.9+2.04 + 41.32+269.9+2.04+344.9+1127.0 = 2057.1 \)

Step4: Calculate variance

Variance \( s^2=\frac{\text{Sum of squared differences}}{n - 1}=\frac{2057.1}{6}\approx342.85 \)

Step5: Calculate standard deviation

Standard deviation \( s=\sqrt{342.85}\approx18.5 \)

Answer:

\( 18.5 \)