# Find a Critical Value in Any Tail: Easy Steps

Probability and Statistics > Critical Values & Hypothesis Testing > Find a critical value

Critical values are used in hypothesis testing. The alpha level is the maximum probability where you reject the null hypothesis if the null hypothesis is true. The technique for looking up a critical value is very straightforward as long as you know if you have a left tailed test or right tailed test (or potentially, both).

## Find a critical value for a confidence level

Sample question:Find a critical value for a 90% confidence level.

Step 1: Subtract the confidence level from 100% to find the α level: 100% – 90% = 10%.

Step 2: Convert Step 1 to a decimal: 10% = 0.10.

Step 3: Divide Step 2 by 2 (this is called “α/2”).
0.10 = 0.05. This is the area in each tail.

Step 4: Subtract Step 3 from 1 (because we want the area in the middle, not the area in the tail):
1 – 0.05 = .95.

Step 5: Look up the area from Step in the z-table. The area is at z=1.645. This is your critical value for a confidence level of 90%.

## Common confidence levels and their critical values

You don’t have to perform the above calculations every time. This list of critical values and their associated two-tailed test confidence levels were calculated using the above steps:

 Confidence Level Critical Value (Z-score) 0.90 1.645 0.91 1.70 0.92 1.75 0.93 1.81 0.94 1.88 0.95 1.96 0.96 2.05 0.97 2.17 0.98 2.33 0.99 2.575

## Find a Critical Value: Two-Tailed Test

Watch the video or read the steps below:

Sample question: Find the critical value for alpha of .05.

Step 1: Subtract alpha from 1.

1 – .05 = .95

Step 2 :Divide Step 1 by 2 (because we are looking for a two-tailed test).

.95 / 2 = .475

Step 3: Look at your z-table and locate the answer from Step 2 in the middle section of the z-table. The fastest way to do this is to use the find function of your browser (usually CTRL+F). In this example we’re going to look for .475, so go ahead and press CTRL+F, then type in .475.

Step 4:. In this example, you should have found the number .4750. Look to the far left or the row, you’ll see the number 1.9 and look to the top of the column, you’ll see .06. Add them together to get 1.96. That’s the critical value!

Tip: The critical value appears twice in the z table because you’re looking for both a left hand and a right hand tail, so don’t forget to add the plus or minus sign! In our example you’d get ±1.96.

## Find a Critical Value: Right-Tailed Test

Sample question: Find a critical value in the z-table for an alpha level of 0.0079.

Step 1: Draw a diagram, like the one above. Shade in the area in the right tail.
This area represents alpha, α. A diagram helps you to visualize what area you are looking for (i.e. if you want an area to the right of the mean or the left of the mean).

Step 2: Subtract alpha (α) from 0.5.
0.5-0.0079 = 0.4921.

Step 3: Find the result from step 2 in the center part of the z-table.
The closest area to 0.4921 is 0.4922 at z=2.42.

The closest to .4921 is the value (.4922) at z = 2.42

That’s it!

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step solutions, just like this one!

## Find a Critical Value: Left-Tailed Test

Sample question: find the critical value in the z-table for α=.012 (left-tailed test).

Step 1: Draw a diagram, like the one above. Shade in the area in the left tail (because you’re looking for a critical value for a left-tailed test).
This area represents alpha, α.

Step 2:Subtract alpha (α) from 0.5.
0.5 – 0.012 = 0.488.

Step 3: Find the result from step 2 in the center part of the z-table.

The closest area to 0.488 is at z=2.26. If you can’t find the exact area, just find the closest number and read the z value for that number.

Step 4:Add a negative sign to Step 3 (left-tail critical values are always negative).
-2.26.

That’s it!

Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.

Find a Critical Value in Any Tail: Easy Steps was last modified: February 22nd, 2016 by

# 26 thoughts on “Find a Critical Value in Any Tail: Easy Steps”

1. What is wrong with my z chart

Uhmm….followed the instructions, looked at my Z chart and did not find step 4. My z chart gets me -0.07….what gives? where did I go wrong?

2. Kristie

I cannot figure this out for the life of me! The table isnt working for me. This is my question: Compute the critical value z a/2 that corresponds to a 94% level of confidence.

3. Andale

Hi, Kristie,

Thanks for dropping by. Can you ask your question on the forum? One of our mods will be happy to help you with any statistics question :)

statisticshowto.com/forums

4. Bas

Hi,

I am confused. Checking the Ztable for teh critcal value when the level of significance is 0.05 like stated in step 1 and two. I am seraching in the Z table and find the value +/- 1.6 instead of 1.9. What do I forget or dont understan looking up for the critical value in the z table?
Thanks, Bas

5. Andale

Bas, it’s hard to say without seeing exactly what you’re doing. One thought — are you sure you are using the correct z-table? That’s usually why students come up with a different number…

6. sumir

Best explanation I have come across for determining where the critical value of 1.96 comes from – thanks.

7. Victor

I came up with 0.0 on the left column and .06 on the top of the z-table I’m I missing something here. Please help.

8. Andale

Victor,
Are you looking at the right z-table? If you find .475 using the search function, you should see it. The 0.0 row has VERY tiny values, so it makes me think you might be looking at a different z-table in your textbook? (Some texts, especially those for biological sciences, will have z-tables for very tiny incremenets).
Stephanie

9. B

Im not sure whats happening here, but this isnt the correct way to find those values. You just have to divide .05 by 2 to get .025. Subject .025 from 1 (1-.025) to get .975. THEN you search the Z Table for .975 to come to the 1.96. Either I am missing something, which may certainly be the case, or your work on this is just way wrong (respectfully).

10. Andale

Hello, Bernard,

Thanks for your comment. The technique here assumes you’re using a standard z-table (right of curve). This table only goes up to .5 (which makes sense because the area under the curve is 1). So it wouldn’t be possible to find .975 in a standard (right of curve) table. You have to divide the alpha level by two when using the standard z-table, because of the two tails.

If you’re using the FULL z-table, THEN your method would work to come to the same answer — 1.96.

I guess it all depends on which method your instructor chooses to use. As with many cases in math, there’s always more than one way to do things :)

I hope that helps!

Stephanie

11. Learing

Wouldn’t you divide .05 by 2 to get .025 then subtact that from which will give you .975 or as shown on the z chart 1.96

12. Andale

As with many things in math, there’s more than one way of getting the “correct” answer “)

Best,
Stephanie

13. Kristof Decleer

Hi, I’m currently working on my master thesis and in one of my models I use a Generalized Error Distribution but I can’t seem to find the critical values in the appropriate table. could you help me with this?

14. Andale

It would depend on the shape parameter. For example, if you have shape parameter = 1/2, use the z-table.

15. maryam irfan

16. Andale

What z scores did you get? Does your z-table look the same as the one I am using? The most likely problem is that you’re using a different table (i.e. a left instead of a right). If you can, post a screen shot of your table at our statistics help forum.

17. Monica

Thanks guys this has been helpful, but my question is; do I have to use the z-table to get the critical value? Isn’t there a way I can calculate it to work it out?. Or will one be allowed to go into the examination hall with a z-table?

18. Andale

You have to use the z-table to get the area under a curve unless you’re using one of the common ones in the list (which you can possibly memorize). A z-table should be provided for you in any exam. Some exams (like AP statistics) allow you to take in a TI83, which has a built-in z table.

19. Steve

Sample question:Find a critical value for a 90% confidence level.
Step 1: Subtract the confidence level from 100% to find the α level: 100% – 90% = 10%.
Step 2: Convert Step 1 to a decimal: 10% = 0.10.
Step 3: Divide Step 2 by 2 (this is called “α/2”).
0.10 = 0.05. This is the area in each tail.
Step 4: Subtract Step 3 from 1 (because we want the area in the middle, not the area in the tail):
1 – 0.05 = .95.
Step 5: Look up the area from Step in the z-table. The area is at z=1.645. This is your critical value for a confidence level of 90%.

Wouldn’t the critical value be twice the value of 0.5 minus whatever the probability is for z=1.645?
The probability for that z-score will give you the probability of a score *within* the confidence interval. Am I wrong?