How to find the area under a normal distribution curve (two tails)

Critical Values and Z-Tables, Normal Distribution 4 Comments

There are a few ways to find the area under a normal distribution curve for both tails using a z-table. Once you know how to read the table, finding the area only takes seconds!

If you are looking for other variations (finding the area for a value between 0 and any z-score, or between two z-scores, see this normal distribution curve index).

area under a normal distribution curve--two tails

area under a normal distribution curve--two tails

Step 1: Look in the z-table for one of  the given z-values by finding the intersection. For example, if you are asked to find the area in the tail to the left of z= -0.46, look up 0.46.* The table below illustrates the result for 0.46 (0.4 in the left hand column and 0.06 in the top row. the intersection is .1772).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

Step 2: Subtract the z-value you just found in step 1 from 0.500. Set this number aside for a moment.

Step 3: Repeat steps 1 and 2 for the other tail.

Step 4: Add both z-values together.

That’s it!

*note. Because the graphs are symmetrical, you can ignore the negative z-values and just look up their positive counterparts. For example, if you are asked for the area of a tail on the left to -0.46, just look up 0.46.

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4 Responses to “How to find the area under a normal distribution curve (two tails)”

  1. How to Do Everything Statistics » Area Under a Normal Distribution Curve Index said:

    Sep 01, 09 at 6:44 am

    [...] How to Do Everything Statistics » How to find the area under a normal distribution curve (two … September 1st, 2009 at 6:42 am [...]

  2. Sarah Joyner said:

    Oct 10, 09 at 1:11 pm

    These problems were a bit intimidating at first. However going over it a couple times and understanding the table finally, it became really easy to solve these types of problems.

  3. Vanessa DuBarry said:

    Oct 22, 09 at 7:29 pm

    I am still having problems reading these tables.where do I know it intersects? because in the above problem it says its.6772, where did that com from?. and where would I find if the number was 0.72? I dont see it in the table.. please help me asap

  4. angie widdows said:

    Nov 11, 09 at 6:08 pm

    It took me forever to understand this table. I think I was reading WAY too much into it.

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