This how-to covers solving normal distribution problems that contain the phrase “between” and includes an upper and lower limit (i.e. “find the number of houses priced between $50K and 200K”. Note that this is different from finding the “middle percentage” of something; in that case you will want to follow this: How To Find a Middle Percentage on a Normal Distribution Word Problem. For other problems, see the Normal Distribution problem Index.

Step 1: Identify the parts of the word problem. The word problem will identify:

1. the mean (average or μ)
2. standard deviation (σ)
3. number selected (i.e. “choose one at random” or “select ten at random”)
4. X: the numbers associated with “between” (i.e. “between $5,000 and $10,000″ would have X as 5,000 and as $10,000)

In addition, you will be given EITHER:

1. sample size (i.e. 400 houses, 33 people, 99 factories, 378 plumbers etc.). OR
2. you might be asked for a probability (in which case your sample size will probably be everyone, i.e. “Journeyman plumbers” or “First year pilots.”

Step 2: Draw a graph. Put the mean you identified in Step 1 in the center. Put the number associated with “between” on the graph (take a guess at where the numbers would fall–it doesn’t have to be exact). For example, if your mean was $100, and you were asked for “hourly wages between $75 and $125″) your graph will look something like this:

Step 3:Figure out the z values. Plug the first X value (in my graph above, it’s 75) into the z value formula and solve. The μ (the mean), is 100 from the sample graph. You can obtain these figures (including σ, the standard deviation) from your answers in step 1 :

*Note: if the formula confuses you, all this formula is asking you to do is:
1. subtract the mean from X
2. divide by the standard deviation.

Step 4: Repeat step 3 for the second X.

Step 5: Take the numbers from step 3 and 4 and use them to find the area in the z-table. (If you don’t remember how to find an area you can find instructions for finding an area under a normal distribution curve here).

If you were asked to find a probability in your question, go to step 6a. If you were asked to find a number from a specific given sample size, go to step 6b.

Step 6a:

Convert the answer from step 5 into percentage.

For example, 0.1293 is 12.93%.

That’s it–skip step 6b!

Step 6b

Multiply the sample size (found in step 1) by the z-value you found in step 4. For example, 0.300 * 100 = 30.

That’s it!