The Empirical Rule

The Empirical Rule Also known as The 68-95-99.7 Rule Original content by D.R.S.

What The Empircal Rule Says • If the data is distributed in a bell-shaped distribution, then • Approximately 68% of the data falls within one standard deviation, plus or minus, from 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

What’s different from Chebyshev Chebyshev’s Rule The Empirical Rule Applies only to data sets which have a bell-shaped distribution pattern It makes stronger claims, because we know about the bell shape It talks in firm values, a definite approximate % of the data. • Applies to any old distribution, any data set, no restrictions • It makes timid claims because we have no guarantees about the data distribution pattern. • It talks in terms of “ at least ____% of the data,” could be more, could be lots more

The Empirical Rule says… • 68% of the data lives within one standard deviation of the mean, between z = _____ and z = _____

The Empirical Rule says… • 95% of the data lives within two standard deviations of the mean, between z = _____ and z = _____

The Empirical Rule says… • 99.7% of the data lives within three standard deviations of the mean,between z = _____ and z = _____

Outside of the middle • 68% of the data lives within 1 stdev of mean • So _____% lives outside, >1 stdev of the mean • _____% in the left tail, _____% in the right tail

Outside of the middle • 95% of the data lives within 2 stdevs of mean • So _____% lives outside,>2 stdevs of the mean • _____% in the left tail, _____% in the right tail

Outside of the middle • 99.7% of the data within 3 stdevs of mean • So _____% lives outside,>3 stdevs of the mean • _____% in the left tail, _____% in the right tail

Practice with Areas: 0 < z < 1 • _____% of the data lies between z = -1 and z = +1 • And this area is ½ of that, or _____%

Practice with Areas: 0 < z <∞ • Since the bell-shaped curve is SYMMETRIC, _____ % lies to the right of z = 0.

Practice with Areas: -∞ < z < 0 • Since the bell-shaped curve is SYMMETRIC, _____ % lies to the right of z = 0.

Practice with Areas: -∞ < z < -1 • _____ % of the area lies to the left of z = 0 • But we take away the area between z = -1 to 0 • ½ of the area between z = -1 and z = +1, ½ of ___% • Summary: _____ % minus _____ % = _____%

Practice with Areas: -∞ < z < -2 • _____ % of the area lies to the left of z = 0 • But we take away the area between z = -2 to 0 • ½ of the area between z = -2 and z = +2, ½ of ___% • Summary: _____ % minus _____ % = _____%

Practice with Areas: -∞ < z < 1 • ____ % to the left of z = 0, • Plus the area between z = 0 and z = 1 • Total area between z = -1 and z = +1 is ______% • Half of that is ____% • Summary:____% + ____%= _____%

Practice with Areas: -∞ < z < ∞ • Should be a cinch, right? ____ %

Practice with Areas: 1 < z < 2 • Area between z = 0 and z = 2 is ½ of ____% which is ____% • Area between z = 0 and z = 1 is ½ of ____% which is ____% • Subtract: ____%- ____% = ____%

Practice with Areas: 1 < z < 3 • Area between z = 0 and z = 3 is ½ of ____% which is ____% • Area between z = 0 and z = 1 is ½ of ____% which is ____% • Subtract: ____%- ____% = ____%

Practice with Areas: 1 < z < ∞ • ____ % in the right half, 0 to ∞ • Minus the ____% between 0 and 1 • Equals _____%

Practice with Areas: 2 < z < 3 • Area between z = 0 and z = 3 = _____% • Area between z = 0 and z = 2 = _____ % • Subtract, giving ____% between 2 and 3

Practice with Areas: 2 < z < ∞ • ____% in the right half • Minus _____% between 0 and 2 • Equals ____% to the right of z = 2.

Practice with Areas: -3 < z < 0 • _____% between z = -3 and z = +3 • Half of that is _____%

Practice with Areas: -3 < z < 1 • ____ % between z = -3 and z = 0 • ____ % between z = 0 and z = 1 • Add, giving _____%

Practice with Areas: -2 < z < -1 • ____% between z = -2 and z = 0 • ____% between z = -1 and z = 0 • Subtract, giving ____%

The Empirical Rule