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)*((x_{1}-x_{m})^{2}+(x_{2}-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.

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