Empirical Rule Example

This worksheet may help you to know about the Empirical Rule. The data set along with Bell Shape curve will be implied to Empirical Rule. We can apply the Empirical rule if we know the value of Standard Deviation and mean. The Empirical rule for Normal Distribution is defined as 68% of values fall in 1 Standard Deviation of the mean, 95% of values are fall in 2 standard deviation of the mean and 99.73% of values are fall in 4 standard deviation of the mean.

Formula for Empirical Rule
68% of values fall => mean ± sd
95% of value fall => mean ± 2 sd
99.73% of value fall => mean ± 3 sd
Where,
sd - standard deviation for the given value
Mean - mean for the given values

Example for Empirical Rule:
Find empirical rule for the values {12,32,45,53,21,43}

Step 1: Calculate Mean
mean = (12+32+45+53+21+43)/6
mean = 206/6
mean = 34.33

Step 2: Find Standard Deviation
You can use this Standard Deviation Calculator to find Standard deviation.
SD(σ)=&radical;(1/(N-1)*((x₁-x_m)²+(x₂-x_m)²+..+(x_n-x_m)²))

SD(σ)=√(1/(6-1)((12-34.33)²+(32-34.33)²+(45-34.33)²+(53-34.33)²+(21-34.33)²+(43-34.33)²))
SD(σ)=√(1/5((-22.3333)²+(-2.33)²+(10.6667)²+(18.6667)²+(-13.3333)²+(8.6667)²))
SD(σ)=√(1/5((498.7763)+(5.4443)+(113.7785)+(348.4457)+(177.7769)+(75.1117)))
SD(σ)=√(243.8667)
SD(σ)=15.6162

Step 3: Apply Empirical Rule
68% of values are between 18.7171 to 49.9496
95% of Values are between 3.1009 to 65.5658
97.73% of values are between -12.5154 to 81.182

The above worksheet is an walk through to understand the concept of understanding Empirical rule. When you try such calculations on your own, use this empirical rule calculator to verify your answers.