Statistics How To

Find a Critical Value in Any Tail: Easy Steps

Main > 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 critical values is very straightforward as long as you know if you have a left tailed test or right tailed test (or potentially, both).

Click to skip to the steps for the type of test:
Find a Critical Value: Two-Tailed Test
Find a Critical Value: Right-Tailed Test
Find a Critical Value: Left-Tailed Test


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.

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Find a Critical Value: Right-Tailed Test

find a critical value

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 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!

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Find a Critical Value: Left-Tailed Test

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

Area under a normal distribution curve (one tail)

Area under a normal distribution curve (alpha is shaded yellow)

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.
critical value z table 2

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

That’s it!

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18 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