Impara Standard Deviation

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One of the most important measurements is standard deviation.

Note

Standard deviation is similar to variance because it is the square root of variance.

Therefore, the formulas will differ for the population ( $\sigma_p$ ) and sample ( $\sigma_s$ ).

\sigma_p = \sqrt{\frac{\sum^N_{i=1}(x_i-\mu)^2}{N}}, \quad \sigma_s = \sqrt{\frac{\sum^n_{i=1}(x_i-\bar{x})^2}{n-1}}

Definition

Standard deviation is a measure of how data is spread out in relation to the mean.

Empirical Rule

The Empirical Rule, also known as the 68–95–99.7 rule, applies when the population follows a Normal Distribution. According to this rule:

About 68% of the data falls within one standard deviation (σ) of the mean;
About 95% falls within two standard deviations (2σ);
About 99.7% falls within three standard deviations (3σ).

When dealing with samples, the percentages might not be precisely accurate, but you can expect them to be quite close to the values in the rule, especially with larger sample sizes.

Example

To illustrate this, examine a sample of kitten weights measured in grams:

In this scenario, the following data is being used:

Mean value ( $\mu$ ) is 100 grams;
Standard deviation ( $\sigma$ ) is 20 grams.

As mentioned earlier, one standard deviation above and below the mean encompasses 68% of the values. In this instance, those values range:

\textbf{from:}\ \mu - \sigma = 100 - 20 = 80;\\ \textbf{to:}\ \mu + \sigma = 100 + 20 = 120.

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