This how-to covers solving normal distribution problems that contain the phrase “less than” (or a similar phrase such as “fewer than”). 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 number associated with “less than” (i.e. “under $5,000″ would have X as 5,000)

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 “less than” on the graph (take a guess at where the number would fall–it doesn’t have to be exact). For example, if your mean was $15, and you were asked for “dinners under $4″) your graph will look something like this:

Step 3:  Figure out the z value. Plug the values into the z value formula and solve. The “X” in our sample graph is 4, and the μ (or mean) is 15. 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: Take the number from step 3 and find the area using the z-table. (If you don’t remember how to do that you can find instructions for finding an area under a curve and reading a z-table  here).

Step 5: 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 6a

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!

Related posts:

1. Normal Distribution Word Problems: “Between”
2. Normal Distribution Word Problems: “Above”
3. Normal Distribution Word Problems: Finding upper and lower cut off points
4. Normal distribution word problems: finding a lower cut off point
5. Normal distribution word problems Index