Normal distribution curve index >

## How to find the area between two z values on opposite sides of the mean

If you want to find the area between two z-scores, the technique will differ slightly depending on if you have two z-scores on one side of the mean or on opposite sides of the mean. This article will show you how to find the area between two z-scores on opposite sides of the mean. If you have z-scores on the same side, see: Area Between Two Z Values on One Side of the Mean.

**Step 1:** *Look in the z-table** for the z-scores by finding the intersections of both scores individually*. For example, you might be asked to find the area between two z values of -0.46 and +1.16, so look up both numbers: 0.46 (see note below about absolute values) and 0.16. The table below shows the z-table value for 0.46 (0.4 in the left hand column and 0.06 in the top row. The intersection = .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 the values you found in step 1 together*.

*That’s it!*

*note. Bell curves are symmetrical, so ignore negative z-values and just look up their absolute values. For example, if you want to find the area between two z values of -3 and -4, look up 3 and 4.

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## References

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.

Patel, J. K. and Read, C. B. Handbook of the Normal Distribution. New York: Dekker, 1982.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 285-290, 1999.