Understanding Standard Deviation
Standard deviation is a measure that quantifies the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
The Formula
The formula for the population standard deviation ($\sigma$) is:
$$\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$$
Where:
- $\bar{x}$ is the mean of the data set.
- $x_i$ is each individual value.
- $n$ is the number of values.
Step-by-Step Solution
Given the data set: 100, 150, 200, 250, 300
Step 1: Find the mean ($\bar{x}$)
$$\bar{x} = \frac{100 + 150 + 200 + 250 + 300}{5} = \frac{1000}{5} = 200$$
Step 2: Calculate the squared deviations $(x_i - \bar{x})^2$
- $(100 - 200)^2 = (-100)^2 = 10,000$
- $(150 - 200)^2 = (-50)^2 = 2,500$
- $(200 - 200)^2 = (0)^2 = 0$
- $(250 - 200)^2 = (50)^2 = 2,500$
- $(300 - 200)^2 = (100)^2 = 10,000$
Step 3: Find the sum of squared deviations
$$\sum = 10,000 + 2,500 + 0 + 2,500 + 10,000 = 25,000$$
Step 4: Divide by $n$ (Variance)
$$Variance = \frac{25,000}{5} = 5,000$$
Step 5: Take the square root
$$\sigma = \sqrt{5,000} \approx 70.71$$
So, the standard deviation of the provided data is approximately 70.71.