H Algebra 2/Trig · Unit 8
Sketch the mean and the standard deviations, then use 68%, 95%, and 99.7%.
Jump to section:
68, 95, 99.7
About 68% of the data is within 1 standard deviation of the mean.
About 95% is within 2. About 99.7% is within 3.
A sketch first
Mark the mean. Then mark 1, 2, and 3 standard deviations on each side.
The percent comes from how far the boundaries are from the mean.
Label the mean and the standard deviations
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 72, and the standard deviation is 4
60, 64, 68, 72, 76, 80, 84
Answer: 60, 64, 68, 72, 76, 80, 84
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 50, and the standard deviation is 5
35, 40, 45, 50, 55, 60, 65
Answer: 35, 40, 45, 50, 55, 60, 65
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 64, and the standard deviation is 6
46, 52, 58, 64, 70, 76, 82
Answer: 46, 52, 58, 64, 70, 76, 82
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 40, and the standard deviation is 3
31, 34, 37, 40, 43, 46, 49
Answer: 31, 34, 37, 40, 43, 46, 49
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 90, and the standard deviation is 7
69, 76, 83, 90, 97, 104, 111
Answer: 69, 76, 83, 90, 97, 104, 111
The middle label is the mean.
Step one standard deviation at a time, three steps each way.
mean = 28, and the standard deviation is 2
22, 24, 26, 28, 30, 32, 34
Answer: 22, 24, 26, 28, 30, 32, 34
A percent from the sketch
Find the distance from the mean to each end.
48 − 43 = 5
5 ÷ 5 = 1
53 − 48 = 5
Each end is 1 standard deviation from the mean.
About 68% of the data sits in that band.
Answer: about 68%
Find the distance from the mean to each end.
30 − 22 = 8
8 ÷ 4 = 2
38 − 30 = 8
Each end is 2 standard deviations from the mean.
About 95% of the data sits in that band.
Answer: about 95%
Find the distance from the mean to each end.
55 − 37 = 18
18 ÷ 6 = 3
73 − 55 = 18
Each end is 3 standard deviations from the mean.
About 99.7% of the data sits in that band.
Answer: about 99.7%
Count how many standard deviations the boundary is from the mean.
85 = 80 + 1(5)
The boundary is 1 standard deviation above the mean.
About 68% sits within 1 standard deviation, so about 32% sits outside that band.
Half of that outside amount is above the mean.
About 16%
Answer: about 16%
Count the standard deviations from the mean down to the boundary.
36 = 42 − 2(3)
The boundary is 2 standard deviations below the mean.
About 95% sits within 2 standard deviations, so about 5% sits farther away.
Half of that 5% is below the mean.
About 2.5%
Answer: about 2.5%
Within 2 standard deviations runs from the mean minus 2 standard deviations to the mean plus 2 standard deviations.
lower = 67 − 2(8)
lower = 51
upper = 67 + 2(8)
upper = 83
About 95% of the data sits in that band.
Answer: about 95%
The interval for the middle percent
The middle 95% sits within 2 standard deviations of the mean.
lower = 110 − 2(7)
lower = 96
upper = 110 + 2(7)
upper = 124
Answer: 96 to 124
The middle 95% sits within 2 standard deviations of the mean.
lower = 60 − 2(9)
lower = 42
upper = 60 + 2(9)
upper = 78
Answer: 42 to 78
The middle 99.7% sits within 3 standard deviations of the mean.
lower = 45 − 3(4)
lower = 33
upper = 45 + 3(4)
upper = 57
Answer: 33 to 57
The middle 95% sits within 2 standard deviations of the mean.
lower = 88 − 2(5)
lower = 78
upper = 88 + 2(5)
upper = 98
Answer: 78 to 98
The middle 99.7% sits within 3 standard deviations of the mean.
lower = 25 − 3(2)
lower = 19
upper = 25 + 3(2)
upper = 31
Answer: 19 to 31
The middle 95% sits within 2 standard deviations of the mean.
lower = 140 − 2(6)
lower = 128
upper = 140 + 2(6)
upper = 152
Answer: 128 to 152
Need help with the empirical rule?
Book a Session with Amy