Area under a normal distribution curve (right of a z-score)
There are a few ways to find the area under a normal distribution curve for any area to the right of a z-score where z is less 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 to the right of a z-score (z is less than the mean)
Step 1: Look in the z-tablefor the given z-value by finding the intersection. The table below illustrates the result for 0.46 (0.4 in the left hand colum and 0.06 in the top row. the intersection is .6772).
| 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.
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How to Do Everything Statistics » Area Under a Normal Distribution Curve Index said:
Sep 01, 09 at 6:36 am[...] How to Do Everything Statistics » Area under a normal distribution curve (right of a z-score) September 1st, 2009 at 6:32 am [...]
How to Do Everything Statistics » Central Limit Theorem: “Greater Than” Probability said:
Sep 24, 09 at 7:45 am[...] Step 4: Look up the z-value you calculated in step 3 in the z-table. If you don’t remember how to look up z-scores, you can find an explanation in step 1 of this article on finding the area under a normal distribution curve. [...]
How to Do Everything Statistics » Problems of Central Limit Theorem – Less Than said:
Sep 26, 09 at 8:30 am[...] Step 4: Look up the z-value you calculated in step 4 in the z-table. If you don’t remember how to look up z-scores, you can find an explanation in step 1 of this article on finding the area under a normal distribution curve. [...]
Rebecca Gamble said:
Oct 01, 09 at 1:19 pmI’m a little confused how do you get “.6772″ from the .4 and .06? Do we just replace the “.1772” in the chart with the 6 category?
Rebecca Gamble said:
Oct 01, 09 at 1:24 pmThe link for the ” look in the z-table” doesn’t link to anything?
Mark McCrea said:
Oct 02, 09 at 8:26 amNow these prolbem are become earier to do. Except in step two tell to add the 0.500 to the number that we have found. I end up having to substact 0.500 to get the correct answer. So is step two type O or is that the correct way to find that answer.
Stephanie said:
Oct 04, 09 at 8:11 amMark,
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
Stephanie said:
Oct 04, 09 at 8:22 amBroken link, now fixed. Thanks for spotting it,
Stephanie
Stephanie said:
Oct 04, 09 at 8:23 amYou add .5 to the .1772 in the chart (see step 2),
Stephanie
Ashanti Jones said:
Oct 04, 09 at 6:15 pmI found it easier to use the tab that says tables on the top of the screen.