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.

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

**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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Sir please let me know how do i use ‘Area of the left or right (one tail)’ or ‘Area of the left and right (two tail) to find value of z.

It depends on what you are trying to test. If you are looking for > or < a certain value then that is a one-tailed test. If you are looking for both, that's a two-tailed test.