The Empirical Rule

Standard Deviation and the Normally Distributed Data Set

2

Normal Distribution

S -3 -2 -1 0 1 2 3

xData values 2 4 6 8 10 12 14

Data Values567891011

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

4

Normal Distribution

2.14% 2.14%

S -3 -2 -1 0 1 2 3

0.13% 0.13%

13.59%

34.13%34.13%

13.59%

x

Percent of values under portions of the normal curve

The Empirical Rule• When data is normally distributed:– Approx. 68% of the values will fall within +/- 1 S from

the x

S -3 -2 -1 0 1 2 3

68%

The Empirical Rule• When data is normally distributed:– Approx. 95% of the values will fall within +/- 2 S from

the x

S -3 -2 -1 0 1 2 3

95%

The Empirical Rule• When data is normally distributed:– Approx. 99% of the values will fall within +/- 3 S from

the x

S -3 -2 -1 0 1 2 3

99.7%

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