Probability and Statistics > Normal Distributions > Two Tailed Normal Curve

## Two Tailed Normal Curve: How to find the area

A two tailed normal curve is one where there’s an area in each of the two tails. In order to find the area for a two tailed normal curve, all you have to do is know how to read a z-table. Z-tables are just lists of percentages. The total area under a normal curve is 100%(1.) and the z-table lists areas as a fraction of that percentage. For example, you could look up a z-score for 60% of a normal curve (.6) or 6% (0.06).

If you are looking for other variations on finding areas under curves, see the area under a normal distribution curve index. The index lists several variations on area finding under a curve, like finding areas for right-tailed normal curves or left-tailed normal curves.

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## Two Tailed Normal Curve: How to find the area: Steps

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**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*. In this example, if you found .1772 as your z-value, then 0.500 – .1772 = .3228. Set this number aside for a moment.

**Step 3:** *Repeat steps 1 and 2 for the other tail. *For example, you might have symmetrical tails (that’s the most common spread for two-tailed problems). So if you repeat the steps you would get .3228 again.

**Step 4:** *Add both z-values together.*In this example, the two z-values are .3228 and .3228, so:

.3228 + .3228 = .6456

*That’s it!*

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## References

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

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