Step by step
The question
Question: the numbers are normal, with mean 40 and SD 3: what percent is between 40 and 46?
Normal means the numbers make a bell shape, the same on both sides of the mean.
1 Draw the bell
The mean, 40, is in the middle.
Each mark is one SD, 3, further: 40 plus 3 is 43, then 46, then 49.
And going down: 37, 34, 31.
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.
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.
We write each piece on the curve.
3 Shade the part
40 is the mean.
46 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.
4 Add the pieces
We add the shaded pieces: 34 plus 13.5 is 47.5%.
5 Check
Let's check: the pieces not shaded add up to 52.5 percent.
47.5 plus 52.5 is 100 percent: the whole bell.
So it is right!
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.