Main > Normal Distributions > How to find area left of a z score

## How to find area left of a z score

You can find area to the left of a z score (where z is greater than the mean) by using a **z-table**. The trick to using a z-table is knowing that the numbers in a z-table represent percentages. For example, a value in the table of 0.5000 represents 50% of the area under the curve and a value of .9999 represents 99% of the area under a curve. Once you know how to read the table, finding the area only takes a step or two!

If you are looking for other variations on area, see the index article, area under a normal distribution curve. You’ll find several articles for all different possibilities of areas. For example, finding the area for a value between 0 and any z-score, between two z-scores, or an area to the right of a z-score.

## How to find area left of a z score: Steps

**Step 1:**Split your given decimal into two by decimal places. For example, if you’re given 0.46, split that into 0.4 + 0.06.

**Step 2:** *Look up your decimals from Step 1 in the z-table*. The z-table below gives the result from looking up 0.4 in the left column and 0.06 in the top row. The intersection (i.e. the area under the curve) 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*.

Note: You’re adding .500 because that’s the 50% of the graph between the mean at zero and the far left of the graph. The above steps only gave you the sliver between 0 and the z-score.

That’s it!

*note on How to find area left of a z score with negative values. The bell curve is symmetrical, so if you are given negative values you can just look up their absolute values. For example, if you are asked for the area of a tail on the left to -0.96, look up the absolute value of -0.96 (0.96).

Am I missing something? The green highlighted intersection is 0.1772 but step 1 says it is .6772.

Mark,

You definitely have to add if your problem looks like the one in the picture, because you are finding the area left of the mean and then adding .5 for the area above the mean. I’m not sure what problem you are working on–if you could let me know what question you are working on, perhaps I could sort out the confusion.

Stephanie

I think this information is incorrect.

The area value for z=0.46 on Standard Normal Distribution Table (z-table as it’s called here) is .6772. The vaulue for z=-0.46 on a NDT is .3228. So no, you shouldn’t ignore negative z values. A NDT always gives you the area to the left of z. Therefore an area the left of -1 and 1 are going to be different. Just look at the graph, z=1 is going to encompass some of the area of the bell shape while z=-1 wont excompass as much area because it doesn’t have that extra area from the bell.

No where on a NDT is there a value of .1772. So this adding business doesn’t really jive. If your question looks like the one above just go to your NDT and find 0.46 and you’ll get .6772, no need to add anything to it, that’s your answer.

I think the writer of this is confused. The area a NDT gives is always the area to the LEFT of the z value. If you’re looking for the area to the RIGHT of a z value you would need to take 1-(z value).

Hi, Heather,

There are different kinds of NDT. This site (and the texts I use to teach), have areas to the left and right of z. Therefore, you need to add or subtract .5 if you are using these tables.

You are correct that the area value for z=0.46 on some types of Standard Normal Distribution Table (z-table) — for example the one that’s probably in YOUR text — is .6772. You can get the same answer by following the steps listed. In this sample problem, you would add .5 to .1772 to get .6772.

Thanks for stopping by,

Stephanie