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The Empirical Rule. Standard Deviation and the Normally Distributed Data Set. Normal Distribution. Data Values 5 6 7 8 9 10 11. S -3 -2 -1 0 1 2 3. x. Data values 2 4 6 8 10 12 14.
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The Empirical Rule Standard Deviation and the Normally Distributed Data Set
Normal Distribution Data Values 5 6 7 8 9 10 11 S -3 -2 -1 0 1 2 3 x Data values 2 4 6 8 10 12 14
The Empirical Rule • When • The standard deviation is known • The mean is known • & the data is normally distributed (like a bell-shaped mound) • Then • The Empirical Rule can be used to generalize some properties about the distribution (data) • Also known as the sigma rule; 68-95-99.7 rule
Normal Distribution Percent of values under portions of the normal curve 34.13% 34.13% 13.59% 13.59% 2.14% 2.14% 0.13% 0.13% S -3 -2 -1 0 1 2 3 x
The Empirical Rule • When data is normally distributed: • Approx. 68% of the values will fall within +/- 1 S from the x 68% S -3 -2 -1 0 1 2 3
The Empirical Rule • When data is normally distributed: • Approx. 95% of the values will fall within +/- 2 S from the x 95% S -3 -2 -1 0 1 2 3
The Empirical Rule • When data is normally distributed: • Approx. 99% of the values will fall within +/- 3 S from the x 99.7% S -3 -2 -1 0 1 2 3
Approximate Empirical Calculations • Approximate Calculations • 68% = x +/- 1(s) • [ x – 1(s), x + 1(s)] • 95% = x +/- 2(S) • [ x – 2(s), x + 2(s)] • 99% = x +/- 3(S) • [ x – 3(s), x + 3(s)]
SAT Distribution m = 490 and s = 100