Statistics and Probability Dictionary

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Critical Value

The critical value is a factor used to compute the margin of error, as shown in the equations below.

Margin of error = Critical value x Standard deviation of the statistic
Margin of error = Critical value x Standard error of the statistic

When the sampling distribution of the statistic is normal or nearly normal, the critical value can be expressed as a t score or as a z score. To find the critical value, follow these steps.

  • Compute alpha (α): α = 1 - (confidence level / 100)
  • Find the critical probability (p*): p* = 1 - α/2
  • To express the critical value as a z score, find the z score having a cumulative probability equal to the critical probability (p*).
  • To express the critical value as a t score, follow these steps.
    • Find the degrees of freedom (df). Often, df is equal to the sample size minus one.
    • The critical t score (t*) is the t score having degrees of freedom equal to df and a cumulative probability equal to the critical probability (p*).

Should you express the critical value as a t score or as a z score? There are several ways to answer this question. As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the t score is preferred. Nevertheless, many introductory texts and the Advanced Placement Statistics Exam use the z score exclusively. On this web site, we provide sample problems that illustrate both approaches.

You can use the Normal Distribution Calculator to find the critical z score, and the t Distribution Calculator to find the critical t score. You can also use a graphing calculator or standard statistical tables (found in the appendix of most introductory statistics texts).

See also:  Margin of Error