Before you can solve for the area under a normal curve, you must be able to imagine what the area looks like. The best (albeit optional) way do this is to make a sketch. For example, let’s say you were give a z-score and were asked to find the area between that score and z = 0 (the mean). Your sketch might look like this:

There are **seven **ways your sketch could look, depending on what z-values you were given. **Once you have drawn your sketch, look at the pictures below**. Click on the image that looks most like your sketch. The link will take you to a step-by-step guide on how to find the area under a normal curve for that shape. Many of these also have short videos showing the steps.

This video shows you how to find the area under a normal curve for a tail (either a left or right tail):

Can’t see the video? Click here to watch it on YouTube

## Choose One

**Tip:** Drawing sketches in probability and statistics isn’t just limited to normal distribution curves. If you get used to making a sketch, you’ll also have an easier time with creating complicated graphs (like Contingency Table: What is it used for?.

## Find an area under a normal curve from z=0 to z=?

## How to find the area under a curve (between 0 and any z-score)

You can look up numbers in the z-table, like 0.92 or 1.32. The values you get from the table give you How to Calculate Percentages: Simple Steps for the area under a curve in decimal form. For example, a table value of .6700 is are area of 67%.

**Note on using the table**: In order to look up a z-score in the table, you have to split up your z-value at the tenths place. For example, to look up 1.32 you would look up 1.3 and then look at .02. See the example below for a visual on what finding the intersection looks like. If you need more help, watch the video on this z-table page.

**Step 1:** *Look in the z-table** for the given z-score by finding the intersection*. For example, if you are asked to find the area between 0 and 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 |

That’s it!

***Note**. Because the graphs are symmetrical, you can ignore the negative z-scores and just look up their positive counterparts. For example, if you are asked for the area of 0 to -0.46, just look up 0.46.

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## References

Beyer, W. H. CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 2002.

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.