How to find the area under a normal distribution curve (left of a z-score)

There are a few ways to find the area under a normal distribution curve for any area to the left of a z-score where z is greater than the mean using a z-table. Once you know how to read the table, finding the area only takes seconds!

If you are looking for other variations (finding the area for a value between 0 and any z-score, or between two z-scores, see this normal distribution curve index).

area shaded to the left of a z-score (z is greater than the mean)

Step 1: Look in the z-table for the given z-value by finding the intersection. 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: Add 0.500 to the  z-value you just found in step 1.

That’s it!

*note. Because the graphs are symmetrical, you can ignore the negative z-values and just look up their positive counterparts. For example, if you are asked for the area of a tail on the left to -0.46, just look up 0.46.

5 thoughts on “How to find the area under a normal distribution curve (left of a z-score)”

1. Stephanie

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

2. Heather

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).

3. Andale

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