Probability and Statistics > Normal Distributions > Area to the Right of a z score

## How to find the Area to the Right of a z score

There are a few ways to find the area to the right of a z-score *where z is less than the mean*. With any word problem like this, you’ll need to consult the z-table. The z-table gives you the area between points (i.e. the area to the right of a z score). Once you know how to read a z-table, finding the area only takes seconds!

If you are looking for **other variations of word problems like this** (for example, finding the area for a value between 0 and any z-score, or between two z-scores, see the normal distribution curve index). That index also includes pictures of all the different types of areas under the curve to help you with choosing the right article.

## How to find the Area to the Right of a z score: Steps

**Step 1: Split your z-score into two, after the tenths decimal place. **For example, 0.46 becomes 0.4 + 0.06.

**Step 2:** **Look in the z-table** *for the given z-value. In order to look up a value in the z-table, find the intersection*. The table below shows 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 3:** Add 0.500 to the * z-value you just found in step 2*.

That’s it!

**Note**: Because the graphs are symmetrical, you can ignore the negative z-values and just look up their positive counterparts (in other words, their absolute value). For example, if you are finding an area to the right of a z score and the area of a tail on the left is -0.46, just look up 0.46.

## Why am I Adding 50%?

Unfortunately, there are many variations on z-tables in textbooks. However, all of them work on the same assumption that the area under the graph of a normal distribution equals 100%. You might see a whole z-table from -3 to 3, or you might see two halves (like the ones I use on this site). You’ll know if you’re using a half table if the z-values range from zero to 3+.

You’re adding the .500 here because that’s the whole area of the right side of the graph (i.e. half of 100% is 50%). You need to add that to your calculation in order to cover the entire area (the yellow area in the top image). The golden rules:

- Always sketch a graph,
- Always check which z-table you’re using.

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