Statistics and Probability Dictionary
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Statistics Dictionary
Absolute Value
Accuracy
Addition Rule
Alpha
Alternative Hypothesis
Back-to-Back Stemplots
Bar Chart
Bayes Rule
Bayes Theorem
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Biased Estimate
Bimodal Distribution
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Bivariate Data
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Critical Parameter Value
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Nominal Scale
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One-Sample t-Test
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Ratio Scale
Reduced Row Echelon Form
Region of Acceptance
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Regression
Relative Frequency
Relative Frequency Table
Replication
Representative
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Residual Plot
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Set
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Statistical Experiment
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Stemplot
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Subtraction Rule
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T Distribution
T Score
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Test Statistic
Transpose
Treatment
t-Test
Two-Sample t-Test
Two-stage Sampling
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Type I Error
Type II Error
Unbiased Estimate
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Uniform Distribution
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Univariate Data
Variable
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Vector Inner Product
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Vectors
Voluntary Response Bias
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Y Intercept
z Score

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 statistic, follow these steps.
Find the
degrees of freedom
(df). Often, df is equal to the sample
size minus one.
The critical t statistic (t*) is
the t statistic 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 statistic 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 statistic is preferred. Nevertheless, many
introductory texts and the Advanced Placement Statistics Exam use the
z score exclusively. On this website, 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 statistic. You can also use a
graphing calculator or
standard statistical tables (found in the appendix of
most introductory statistics texts).