Empirical Rule Calculator

What is the Empirical Rule?

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical principle that describes the distribution of data in a normal distribution. It states that:

• Approximately 68% of the data falls within one standard deviation of the mean
• Approximately 95% of the data falls within two standard deviations of the mean
• Approximately 99.7% of the data falls within three standard deviations of the mean

Formulas and Their Meanings

1. Mean (\(\bar{x}\)): \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\] Where \(x_i\) are individual values and \(n\) is the number of values.

2. Variance (\(s^2\)): \[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}\] This measures the average squared deviation from the mean.

3. Standard Deviation (\(s\)): \[s = \sqrt{s^2}\] The square root of the variance, giving a measure of spread in the same units as the original data.

4. Skewness (\(g_1\)): \[g_1 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3}{s^3}\] A measure of the asymmetry of the probability distribution.

Calculation Steps

1. Calculate the mean by summing all values and dividing by the count.
2. Subtract the mean from each value and square the result for variance.
3. Sum these squared differences and divide by n to get the variance.
4. Take the square root of the variance to get the standard deviation.
5. For skewness, cube the differences from the mean, sum, divide by n and by the cube of the standard deviation.

Example Calculation

Let's calculate for the dataset: 2, 4, 4, 4, 5, 5, 7, 9

1. Mean: \(\bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5\)
2. Squared differences: \((2-5)^2, (4-5)^2, ..., (9-5)^2\)
3. Variance: \(s^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = 4\)
4. Standard Deviation: \(s = \sqrt{4} = 2\)
5. Skewness: \(g_1 = \frac{\frac{1}{8}(-27 + -1 + -1 + -1 + 0 + 0 + 8 + 64)}{2^3} = 0.37\)

Visual Representation

This bell curve represents a normal distribution. The red dashed line indicates the mean, and the green dashed lines show one standard deviation on either side of the mean.