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The Empirical Rule

Also known as The 68-95-99.7 RuleOriginal content by D.R.S.

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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

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What’s different from Chebyshev

Chebyshev’s Rule• 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• 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.

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The Empirical Rule says…

• 68% of the data lives within one standard deviation of the mean, between z = _____ and z = _____

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The Empirical Rule says…

• 95% of the data lives within two standard deviations of the mean, between z = _____ and z = _____

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The Empirical Rule says…

• 99.7% of the data lives within three standard deviations of the mean,between z = _____ and z = _____

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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

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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

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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

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Practice with Areas: 0 < z < 1

• _____% of the data lies between z = -1 and z = +1

• And this area is ½ of that, or _____%

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Practice with Areas: 0 < z < 2

• _____% of the data lies between z = -2 and z = +2

• And this area is ½ of that, or _____%

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Practice with Areas: 0 < z < 3

• _____% of the data lies between z = -3 and z = +3

• And this area is ½ of that, or _____%

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Practice with Areas: 0 < z <∞

• Since the bell-shaped curve is SYMMETRIC, _____ % lies to the right of z = 0.

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Practice with Areas: -∞ < z < 0

• Since the bell-shaped curve is SYMMETRIC, _____ % lies to the right of z = 0.

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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 _____ % = _____%

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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 _____ % = _____%

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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:

____% + ____%= _____%

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Practice with Areas: -∞ < z < 2

• ____ % to the left of z = 0, • Plus the area between z = 0 and z = 2– Total area between z = -2 and z = +2 is ______%– Half of that

is ____%• Summary:

____% + ____%= _____%

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Practice with Areas: -∞ < z < 3

• ____ % to the left of z = 0, • Plus the area between z = 0 and z = 3– Total area between z = -3 and z = +3 is ______%– Half of that

is ____%• Summary:

____% + ____%= _____%

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Practice with Areas: -∞ < z < ∞

• Should be a cinch, right? ____ %

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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: ____%- ____% = ____%

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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: ____%- ____% = ____%

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Practice with Areas: 1 < z < ∞

• ____ % in the right half, 0 to ∞ • Minus the ____% between 0 and 1• Equals _____%

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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

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Practice with Areas: 2 < z < ∞

• ____% in the right half• Minus _____% between 0 and 2• Equals ____% to the right of z = 2.

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Practice with Areas: -3 < z < 0

• _____% between z = -3 and z = +3• Half of that is _____%

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Practice with Areas: -2 < z < 0

• _____% between z = -2 and z = +2• Half of that is _____%

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Practice with Areas: -1 < z < 0

• _____% between z = -1 and z = +1• Half of that is _____%

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Practice with Areas: -3 < z < 1

• ____ % between z = -3 and z = 0• ____ % between z = 0 and z = 1• Add, giving _____%

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Practice with Areas: -3 < z < 2

• ____ % between z = -3 and z = 0• ____ % between z = 0 and z = 2• Add, giving _____%

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Practice with Areas: -2 < z < -1

• ____% between z = -2 and z = 0• ____% between z = -1 and z = 0• Subtract, giving ____%

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Practice with Areas: -2 < z < 1

• ____ % between z = -2 and z = 0• ____ % between z = 0 and z = 1• Add, giving _____%

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Practice with Areas: -2 < z < 3

• ____ % between z = -2 and z = 0• ____ % between z = 0 and z = 3• Add, giving _____%

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Practice with Areas: -1 < z < 3

• ____ % between z = -1 and z = 0• ____ % between z = 0 and z = 3• Add, giving _____%

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Practice with Areas: -1 < z < 2

• ____ % between z = -1 and z = 0• ____ % between z = 0 and z = 2• Add, giving _____%