How the Standard Deviation Calculation Works

Standard deviation is a measure of the amount of variation or dispersion in a set of data. It tells you how spread out the data is around the mean. A small standard deviation means the data points are close to the mean, while a large standard deviation indicates that the data points are spread out over a wide range. To calculate the standard deviation, follow these steps:

1. Collect your data points.
2. Calculate the mean (\( \mu \)) of the dataset. The mean is the sum of all data points divided by the number of data points.
• Formula: \( \mu = \frac{\sum x_i}{n} \)
3. For each data point (\( x_i \)), subtract the mean and square the result (the squared difference).
• Formula: \( (x_i - \mu)^2 \)
4. Calculate the mean of the squared differences.
• Formula: \( \frac{\sum (x_i - \mu)^2}{n} \) (for a population) or \( \frac{\sum (x_i - \mu)^2}{n-1} \) (for a sample).
5. Take the square root of the mean of the squared differences. This value is the standard deviation.
• Formula: \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \) (for a population) or \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}} \) (for a sample).

Standard deviation is a useful tool to assess the spread of data and is commonly used in fields such as finance, science, and quality control. It provides insight into how reliable the mean is and how much variation there is in the data.

Extra Tip

When the data is normally distributed, approximately 68% of data points will fall within one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will fall within three standard deviations. This is known as the **68-95-99.7 rule** or **Empirical Rule**.

Example: Suppose you have the following dataset of test scores: 50, 60, 70, 80, 90, 100, 110.

• First, calculate the mean \( \mu \):
• \( \mu = \frac{50 + 60 + 70 + 80 + 90 + 100 + 110}{7} = 80 \)
• Next, calculate the squared differences from the mean for each data point:
• \( (50 - 80)^2 = 900 \)
• \( (60 - 80)^2 = 400 \)
• \( (70 - 80)^2 = 100 \)
• \( (80 - 80)^2 = 0 \)
• \( (90 - 80)^2 = 100 \)
• \( (100 - 80)^2 = 400 \)
• \( (110 - 80)^2 = 900 \)
• Now, calculate the mean of the squared differences:
• \( \frac{900 + 400 + 100 + 0 + 100 + 400 + 900}{7} = 400 \)
• Finally, take the square root of the mean of the squared differences:
• \( \sigma = \sqrt{400} = 20 \)

So, the standard deviation of this dataset is 20, meaning that the data points deviate, on average, by 20 units from the mean (80).

Population vs. Sample Standard Deviation

If you are calculating the standard deviation for a **population**, use \( n \) (the total number of data points) in the denominator when calculating the mean of the squared differences.

If you are calculating the standard deviation for a **sample**, use \( n - 1 \) in the denominator to account for the fact that a sample may not perfectly represent the population.

Standard Deviation Formula for Sample:

For a sample of data points, the formula is:

\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]

Where \( \bar{x} \) is the sample mean, \( x_i \) is each data point, and \( n \) is the sample size.

Interpreting Standard Deviation

In general, a large standard deviation indicates that the data points are spread out over a wide range, while a small standard deviation indicates that the data points are clustered closely around the mean.

Example of interpretation: If a class of students has test scores with a standard deviation of 2, it means that most students’ scores are within 2 points of the mean. If another class has a standard deviation of 10, the students’ scores are much more spread out around the mean.