Question

The heights (in cm) of 8 students are recorded as 162, 163, 160, 164, 160, 170, 161, 164. The standard deviation of the data is closest to:

• A. 3.04
• B. 3.14
• C. 3.22
• D. 3.32

Hint:

For calculating the standard deviation, always find the mean first, then compute the squared differences from the mean for each data point, and finally, take the square root of the average of those squared differences.

Answer 1

The Correct Option is A

The standard deviation \( \sigma \) of a data set is given by the formula: \[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2} \] Where: - \( x_i \) are the data points, - \( \bar{x} \) is the mean of the data, - \( n \) is the number of data points. First, find the mean \( \bar{x} \): \[ \bar{x} = \frac{162 + 163 + 160 + 164 + 160 + 170 + 161 + 164}{8} = \frac{1344}{8} = 168 \] Next, calculate the squared differences from the mean and sum them: \[ (162 - 168)^2 = 36, \quad (163 - 168)^2 = 25, \quad (160 - 168)^2 = 64, \quad (164 - 168)^2 = 16, \quad (160 - 168)^2 = 64,\] \[\quad (170 - 168)^2 = 4, \quad (161 - 168)^2 = 49, \quad (164 - 168)^2 = 16 \] Sum of squared differences: \[ 36 + 25 + 64 + 16 + 64 + 4 + 49 + 16 = 274 \] Now, calculate the standard deviation: \[ \sigma = \sqrt{\frac{274}{8}} \approx 3.04 \] Thus, the correct answer is \( 3.04 \).