To compute the standard deviation of a set of numbers, follow these steps:
1. Compute the mean (average) of the numbers.
2. Subtract the mean and square the result for each number.
3. Compute the mean of the squared differences.
4. Take the square root of the mean of squared differences.
Let’s apply these steps to compute the standard deviation of the given weights:
Weights: wi = [50, 60, 70, 80, 90]
1. Compute the mean (average) of the weights:
$\bar{w} = \frac{1}{n} \sum_{i=1}^{n} w_i = \frac{50 + 60 + 70 + 80 + 90}{5} = \frac{350}{5} = 70$
2. Subtract the mean and square the result for each weight:
$$\begin{aligned}
(w_i - \bar{w})^2 &= [(50-70)^2, (60-70)^2, (70-70)^2, (80-70)^2, (90-70)^2] \\
&= [400, 100, 0, 100, 400]\end{aligned}$$
3. Compute the mean of the squared differences:
$\text{MSD} = \frac{1}{n} \sum_{i=1}^{n} (w_i - \bar{w})^2 = \frac{400 + 100 + 0 + 100 + 400}{5} = \frac{1000}{5} = 200$
4. Take the square root of the mean of squared differences:
$\text{SD}(\sigma) = \sqrt{\text{MSD}} = \sqrt{200} \approx 14.1421$
The standard deviation of the given weights is approximately 14.1421 kg.