How to Find the Critical Value

The critical value is a factor used in the calculation of the margin of error (ME). It depends on:

The confidence level. Researchers often choose confidence levels of 90%, 95%, or 99%.
The shape of the sampling distribution. If the sampling distribution is shaped like a t distribution, the critical value is expressed as a t-score. If it is shaped like a normal distribution, the critical value is expressed as a z-score.

t-Score vs. z-Score

Should you express the critical value as a t-score or as a z-score? The answer depends several factors, including the type of statistic (e.g., proportion vs. mean), sample size, and knowledge of population standard deviation. Here are some "rules of thumb" to help you choose:

For a sample mean, use z-scores when the population standard deviation is known and the sample size is large. Otherwise, use t-scores.
For a sample proportion, use z-scores when n * p ≥ 10 and n * (1-p) ≥ 10. (This condition will be satisfied when the sample includes at least 10 "successes" and at least 10 "failures".)

Bottom line: Use a z-score when the sampling distribution of the statistic more closely approximates a normal distribution. Use a t-score when it more closely approximates a t distribution. Elsewhere on this site, we provide more detailed guidance on how to choose between the normal and t distributions.

Warning: If the sample size is small and the population distribution is distinctly not normal (e.g., badly skewed, with outliers), neither the t-score nor the z-score should be used to compute critical values.

How to Express Critical Value as t-Score

To express the critical value as a t-score, follow these steps.

Compute alpha (α): α = 1 - (confidence level / 100)
- When the confidence level is 99%, α is 1 - 99/100 or 0.01.
- When the confidence level is 95%, α is 1 - 95/100 or 0.05.
- When the confidence level is 90%, α is 1 - 90/100 or 0.1.
Find the critical probability (p*): p* = 1 - α/2
Find the degrees of freedom (df): df = n - 1 (for a mean score from a single sample)
When estimating a mean score from a single sample, degrees of freedom are equal to the sample size (n) minus one. For other applications, the degrees of freedom may be calculated differently. (Elsewhere on this site, we explain how to calculate degrees of freedom for different applications.)
Find the t-score having degrees of freedom equal to df and a cumulative probability equal to the critical probability (p*).

To find the critical t-score, use an online calculator (e.g.,Stat Trek's t Distribution Calculator), a graphing calculator, or a t-distribution statistical table (found in the appendix of most introductory statistics texts). For an example that uses an online calculator, see Problem 1.

How to Express Critical Value as z-Score

The critical value depends on the confidence level. The table below shows z-score critical values for three common confidence levels - 90% confidence, 95% confidence, and 99% confidence.

Confidence level	Critical value
90%	1.645
95%	1.96
99%	2.576

To express the critical value as a z-score when the confidence level is not 90%, 95%, or 99%, follow these steps.

Compute alpha (α): α = 1 - (confidence level / 100)
Find the critical probability (p*): p* = 1 - α/2
Find the z-score having a cumulative probability equal to the critical probability (p*).

To find the critical z-score, use an online calculator (e.g, Stat Trek's Normal Distribution Calculator), a graphing calculator, or a normal distribution statistical table (found in the appendix of most introductory statistics texts).

Test Your Understanding

Problem 1: Find t-score critical value

A battery manufacturer drew a random sample of 25 D-cell batteries from its production line. Tested for continuous use in a flashlight, the batteries lasted an average of 77 hours with a standard deviation of 13 hours. What is the critical value for battery life? Assume a 95% confidence level.

(A) 1.645
(B) 1.96
(C) 2.064
(D) 2.576
(E) None of the above.

Solution

The correct answer is C. Because the sample size is small (n=25) and the population standard deviation is unknown, the sampling distribution for mean battery life should follow a t distribution. Therefore, the critical value will be expressed as a t-score. To find the t-score critical value, we take the following steps:

Compute alpha (α):
α = 1 - (confidence level / 100)

α = 1 - 0.95 = 0.05
Find the critical probability (p*):
p* = 1 - α/2

p* = 1 - 0.05/2 = 0.975
Find the degrees of freedom (df):
df = n - 1 = 25 -1 = 24
Find the critical value. The t-score critical value will be the t-score having 24 degrees of freedom and a cumulative probability equal to 0.975. Using the t Distribution Calculator, we find that the critical value is 2.064.

Problem 2: Find z-score critical value

Nine hundred (900) high school freshmen were randomly selected for an IQ test. For the test, survey participants demonstrated an average IQ of 103. The population standard deviation for the IQ test was 10. What is the critical value, assuming a 99% confidence level?

(A) 1.645
(B) 1.96
(C) 2.576
(D) 2.96
(E) None of the above.

Solution

The correct answer is C. In this study, statistic of interest is a sample mean, the sample size is large (n=100); and population standard deviation is known. Therefore, the sampling distribution will be normally distributed; and we can express the critical value as a z-score. A 99% confidence interval is one of the common confidence intervals for which the z-score critical value is known to be 2.576, as shown in the table below.

When you need a z-score critical value for one of the common confidence intervals (90%, 95%, or 99%), you can read it right off the table.

Confidence level	Critical value
90%	1.645
95%	1.96
99%	2.576

When you need a z-score critical value for one of the common confidence intervals (90%, 95%, or 99%), you can read it right off the table.