Worked examples
The question
Question: the numbers are normal, with mean 50 and SD 5: what percent is between 45 and 55?
mean 50, SD 5: percent between 45 and 55?
Normal means the numbers make a bell shape, the same on both sides of the mean.
normal: a bell shape, the same on both sides
1 Draw the bell
The mean, 50, is in the middle.
Each mark is one SD, 5, further: 50 plus 5 is 55, then 60, then 65.
And going down: 45, 40, 35.
2 The 68-95-99.7 rule
The rule: 68 percent are within 1 SD of the mean.
95 percent within 2 SDs.
99.7 percent within 3 SDs.
within 1 SD: 68%
within 2 SD: 95%
within 3 SD: 99.7%
Split into pieces, half on each side.
68 divided by 2 is 34.
95 minus 68 is 27, half is 13.5.
99.7 minus 95 is 4.7, half is 2.35.
And 0.15 at each end.
68 ÷ 2 = 34
(95 − 68) ÷ 2 = 13.5
(99.7 − 95) ÷ 2 = 2.35
(100 − 99.7) ÷ 2 = 0.15
We write each piece on the curve.
3 Shade the part
45 is 1 SD below the mean.
55 is 1 SD above the mean.
We shade between them.
Careful!
The rule already counts both sides of the mean together.
Careful: the rule counts both sides of the mean
4 Add the pieces
We add the shaded pieces: 34 plus 34 is 68%.
= between 45 and 55: 34 + 34 = 68%
5 Check
Let's check: the pieces not shaded add up to 32 percent.
rest: 0.15 + 2.35 + 13.5 + 13.5 + 2.35 + 0.15 = 32
68 plus 32 is 100 percent: the whole bell.
So it is right!
68% + 32% = 100%
Let's look back.
First: draw the bell.
Then: the 68-95-99.7 rule.
Then: shade the part.
Then: add the pieces.
Last: check.
Answer = between 45 and 55: 34 + 34 = 68%
▶ Watch it solved
The question
Question: the numbers are normal, with mean 50 and SD 5: what percent is above 60?
mean 50, SD 5: percent above 60?
Normal means the numbers make a bell shape, the same on both sides of the mean.
normal: a bell shape, the same on both sides
1 Draw the bell
The mean, 50, is in the middle.
Each mark is one SD, 5, further: 50 plus 5 is 55, then 60, then 65.
And going down: 45, 40, 35.
2 The 68-95-99.7 rule
The rule: 68 percent are within 1 SD of the mean.
95 percent within 2 SDs.
99.7 percent within 3 SDs.
within 1 SD: 68%
within 2 SD: 95%
within 3 SD: 99.7%
Split into pieces, half on each side.
68 divided by 2 is 34.
95 minus 68 is 27, half is 13.5.
99.7 minus 95 is 4.7, half is 2.35.
And 0.15 at each end.
68 ÷ 2 = 34
(95 − 68) ÷ 2 = 13.5
(99.7 − 95) ÷ 2 = 2.35
(100 − 99.7) ÷ 2 = 0.15
We write each piece on the curve.
3 Shade the part
60 is 2 SD above the mean.
Above it: everything to the right.
Careful!
Outside 2 SD is 5 percent, but that is both ends together.
One end is half of it.
Careful: not 5% ✗
4 Add the pieces
We add the shaded pieces: 2.35 plus 0.15 is 2.5%.
= above 60: 2.35 + 0.15 = 2.5%
5 Check
Let's check: the pieces not shaded add up to 97.5 percent.
rest: 0.15 + 2.35 + 13.5 + 34 + 34 + 13.5 = 97.5
2.5 plus 97.5 is 100 percent: the whole bell.
So it is right!
2.5% + 97.5% = 100%
Let's look back.
First: draw the bell.
Then: the 68-95-99.7 rule.
Then: shade the part.
Then: add the pieces.
Last: check.
Answer = above 60: 2.35 + 0.15 = 2.5%
▶ Watch it solved
The question
Question: the numbers are normal, with mean 100 and SD 15: what percent is between 85 and 130?
mean 100, SD 15: percent between 85 and 130?
Normal means the numbers make a bell shape, the same on both sides of the mean.
normal: a bell shape, the same on both sides
1 Draw the bell
The mean, 100, is in the middle.
Each mark is one SD, 15, further: 100 plus 15 is 115, then 130, then 145.
And going down: 85, 70, 55.
2 The 68-95-99.7 rule
The rule: 68 percent are within 1 SD of the mean.
95 percent within 2 SDs.
99.7 percent within 3 SDs.
within 1 SD: 68%
within 2 SD: 95%
within 3 SD: 99.7%
Split into pieces, half on each side.
68 divided by 2 is 34.
95 minus 68 is 27, half is 13.5.
99.7 minus 95 is 4.7, half is 2.35.
And 0.15 at each end.
68 ÷ 2 = 34
(95 − 68) ÷ 2 = 13.5
(99.7 − 95) ÷ 2 = 2.35
(100 − 99.7) ÷ 2 = 0.15
We write each piece on the curve.
3 Shade the part
85 is 1 SD below the mean.
130 is 2 SD above the mean.
We shade between them.
Careful!
Each rule counts both sides of the mean together.
One side is only half.
Careful: 68% is both sides: one side is 34%
4 Add the pieces
We add the shaded pieces: 34 plus 34 plus 13.5 is 81.5%.
= between 85 and 130: 34 + 34 + 13.5 = 81.5%
5 Check
Let's check: the pieces not shaded add up to 18.5 percent.
rest: 0.15 + 2.35 + 13.5 + 2.35 + 0.15 = 18.5
81.5 plus 18.5 is 100 percent: the whole bell.
So it is right!
81.5% + 18.5% = 100%
Let's look back.
First: draw the bell.
Then: the 68-95-99.7 rule.
Then: shade the part.
Then: add the pieces.
Last: check.
Answer = between 85 and 130: 34 + 34 + 13.5 = 81.5%
▶ Watch it solved
