How to find the area under the normal distribution curve between two z-scores on opposite sides of the mean

critical values and z-tables, Normal Distribution 3 Comments

There are a few ways to find the area under a normal distribution curve for two z-scores on opposite sides of the mean 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 on the same side, see this normal distribution curve index).

normal distribution opposite side of mean

normal distribution with z-scores on opposite side of mean

Step 1: Look in the z-table for the given z-scores (you should have two) by finding the intersections. For example, if you are asked to find the area from z= -0.46 to z= +0.16, look up both 0.46* and 0.16. The table below illustrates the result for 0.46  (0.4 in the left hand colum and 0.06 in the top row. the intersection is .6772).

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: Add both of the values you found in step 1 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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3 Responses to “How to find the area under the normal distribution curve between two z-scores on opposite sides of the mean”

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

    Sep 01, 09 at 6:11 am

    [...] How to Do Everything Statistics » How to find the area under the normal distribution curve bet… September 1st, 2009 at 6:07 am [...]

  2. Joe said:

    Aug 29, 11 at 6:29 pm

    What if you are using z-scores produced with class boundaries in a CHi Square test

  3. Stephanie said:

    Sep 27, 11 at 4:53 pm

    Hi, Joe,
    Can you post your question on the forum? Unfortunately, I don’t have the time to answer math questions here.
    Thanks,
    Stephanie

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