## How To Use the Z-Table to Find Area and Z-Scores

In this article, I’ll walk you through how to use the z-table, or z-score table. The z-table is a chart of numbers that we use to identify the area under the normal curve to the left of a z-score.  Using some simple subtraction, you can also find the area to the right of a z-score, or the area between z-scores with the z-table.  And finally, I’ll guide you through finding area given z-scores.

Some teachers require students to use z-tables, but there are other ways to find area under the normal curve.  You may have a calculator with a normal cumulative distribution function (normalCDF) like the TI 84.  Even better, try the fantastic Normal CDF calculator here!  But, if you are using a z-table, read on.

## How to Use the Z-Table for Positive Z-Scores

Here is a full picture of the positive z-table. Notice that all the values for z in the first column are positive.  The z-table shows areas as 4 digit decimal values throughout the rows and columns.  The first area shown is .5000.  Note that the z-score is 0.0 in the first column, first row.  Note that the number in the column header for .5000 is .00.  Next, we put these two values together, and we have a z-score of 0.00.  Therefore, a z-score of 0.00 has an area to its left of .5000.  This should make sense, since a z-score of 0.00 is exactly in the center of the bell curve, with exactly 50% of the area to its left.

Suppose that a problem gives you a z-score of 1.53, and you need to find the area to its left.  The area under the normal curve to the left of z = 1.53 would be graphically represented like this:

The vertical line dividing the black shaded region from the white un-shaded region is z = 1.53.  Using the z-table, we will find the area to the left of z = 1.53.  Since this shaded area is more than 50% of the bell curve, the area we get as an answer will be more than 0.5.  The z-table has areas expressed as decimals rounded to 4 places, not percents.

To find the area to the left of z = 1.53, first, break up the number 1.53 into two parts, the first is 1.5, and the second is .03.

1.53 = 1.5 +.03

Then, go to the row with 1.5, and go to the column .03.  The area answer is .9370.

So, 93.70% of area is to the left of z = 1.53.

## How to Use the Z-Table for Negative Z-Scores

Here is a full picture of the negative z-table. Notice that all the values for z in the first column are negative.  They increase from -3.4 to -0.0.  A z-score of -0.00 is the same as a z-score of 0.00.  You’ll find the area for -0.00 in the last row of the first area column.  The value shown is .5000, that’s the same area we found for a z-score of 0.00.

Suppose that a problem gives you a z-score of -.76, and you need to find the area to its left.  The area under the normal curve to the left of z = -.76 would be graphically represented like this:

The vertical line dividing the black shaded region from the white un-shaded region is z =-.76.  Using the negative z-table, we will find the area to the left of z = -.76.  Since this shaded area is less than 50% of the bell curve, the area we get as an answer will be less than 0.5.

To find the area to the left of z = -.76, first, break up the number –.76 into two parts, the first is -0.7, and the second is .06.

-.76 = -0.7 – .06

Then, go to the row with 0.7, and go to the column .06.  The area answer is .2236.

So, 22.36% of area is to the left of z = -.76.

## Find the Area to the Right of a Z-Score

To find the area to the right of a z-score, first understand that the bell curve has 100% of area beneath it in total.  For any z-score, the area to its right plus the area to its left must equal 1, or 100%.  In this picture, 100% of the bell curve is shaded.

In our first example, the area found to the left of z = 1.53 was .9370.  Let’s change the problem and find the area to the right of 1.53.  That’s represented by the shaded area below.

The white area is the area to the left of 1.53.  To find the area to the right, we first find the area to the left of the z-score, then we subtract that area from 1.  By simple subtraction from 1, or 100%, we have

1 – .9370 = .0630

Therefore, the area to the right of z = 1.53 is .0630.

## Find the Area Between Two Z-Scores

We will use subtraction to find the area between two z-scores.  First, we’ll find the area to the left of each z-score.  Next, we’ll subtract the smaller area from the larger area.  For example, let’s say we need t find the area between z-scores of -.75 and 1.21.  Graphically, this area would be represented by the picture below.

First, we use the negative z-table to find the area to the left of -.75.  This is .2266.

Next, use the positive z-table to find the area to the left of 1.21.  This is .8869.

Finally, subtract the smaller area, .2266, from the larger area, .8869.  That gives us an area of:

.8869 – .2266 = .6603

Therefore, the area between z-scores -.75 and 1.21 is .6603.

## Find the Z-Score Given Area

The last thing we’ll look at is reversing the use of the z-table.  Instead of finding an area given a z-score, we’re going to work backwards and find a z-score given area.

Example: Find the z-score in which the area to its left is .40.

The first thing you should always do in any z-score / area problem is to draw a picture of the normal curve.  In this example, we want to draw a picture where the shaded area is 40%, and the non-shaded area is 60%.  If the area to the left is .40, as the problem state, then the z-score has to be negative.  This is clear when we draw a picture with the left-most 40% of area shaded.

You don’t have to draw your picture perfectly, but make sure that the vertical line separating the two regions is on the left side of z = 0, since 40% is less than 50% (area to the left of z = 0).

Since we know the z-score is negative, we are going to scan through the areas on the negative z-table of the number that is closest to .40. The closest value we see is .4013.  The next to the right is .3974.  While this is close to .40, .4013 is closer.

The z-score that corresponds to an area of .4013 is -.25.  Therefore the z-score in which the area to the left is 40% is -.25.

## Find the Z-Score Given Area to the Right

To find the z-score given area to the right, we need to first subtract that area from 1 to get the area to the left, then we use the method shown just above.

For example, suppose we need to find the z-score in which the area to its right is 80%.

80% is expressed as .80 when dealing with areas on the z-table, but we DON’T want to scan the table for an area of .80.  Instead, we first subtract the area from 1 to get area to the left of the z-score.

1 – .80 = .20

So, we are looking for a z-score in which the area to its left is .20.  Next, we look through the negative z-table for the area closest to .20.

The z-score that corresponds to an area of .2005 is -.84.  Therefore the z-score in which the area to the left is 20% is -.84.

## What’s Next?

Now that you know how to use the z-table for finding areas and finding z-scores, try using the Normal CDF calculator.  It’s a great way to check your work and verify that you are using the table correctly.  Are you interested in understanding where the values on the z-table actually come from? They are found by integrating the normal function using calculus.  You can find out more here: https://en.wikipedia.org/wiki/Gaussian_integral .