Z Test Statistic Calculator: Z-Score and P-Value

Reference: z and cumulative probability
z	P(Z ≤ z)	Interpretation
-3	0.0013499	unusual (\|z\| ≥ 2)
-2	0.02275	unusual (\|z\| ≥ 2)
-1	0.15866	typical
0	0.5	typical
1	0.84134	typical
2	0.97725	unusual (\|z\| ≥ 2)
3	0.99865	unusual (\|z\| ≥ 2)

IdeaDistance from H₀ in SE units

DistributionStandard normal

|z| ≥ 1.96Significant at 5% (two-tailed)

CoversMean, proportion, 2 proportions

What Is Z Test Statistic?

A z test statistic measures how many standard errors an observed value lies from its expected value under the null hypothesis. The general pattern is z = (observed − expected) ÷ standard error, and only the standard error changes between the four cases covered here.

The calculator gives all three p-values so you can choose the one that matches your alternative hypothesis, plus the percentile of the z-score and whether |z| ≥ 1.96, the usual two-tailed cut-off at the 5 % level. For a full test with a decision and confidence interval, use the z-test calculator.

Below you will find the formulas, a variables table, a worked example and answers to common questions such as how to compute z for a proportion and how to choose the tail.

How to Use the Z Test Statistic Calculator

Choose what you are standardising.
Fill in the matching inputs.
Read z and the p-value that matches your alternative hypothesis (two-tailed, right or left).
Compare |z| with the critical value, for example 1.96 at α = 0.05 (two-tailed).

Z Test Statistic Formula

z-score of one value

z=\frac{x-\mu}{\sigma}

z for a sample mean

z=\frac{\bar x-\mu_0}{\sigma/\sqrt n}

z for a proportion

z=\frac{\hat p-p_0}{\sqrt{p_0(1-p_0)/n}}

z for two proportions

z=\frac{\hat p_1-\hat p_2}{\sqrt{\hat p(1-\hat p)(1/n_1+1/n_2)}}

Z Test Statistic Variables Explained

Symbol	Meaning	Notes
z	Test statistic	(observed − expected) ÷ SE
x̄, μ₀, σ, n	Inputs for a mean	z = (x̄ − μ₀) ÷ (σ/√n)
p̂, p₀	Sample and hypothesised proportion	For a proportion
SE	Standard error	Depends on the case

Z Test Statistic Worked Example

With the example values loaded in the calculator above, the steps are:

z = (x̄ − μ₀) ÷ (σ/√n) = (52.3 − 50) ÷ 1.33333 = 1.725.
Left-tail area Φ(z) = 0.957736; right-tail = 0.0422637.
Two-tailed p-value = 2 × min(left, right) = 0.0845275.

Result: z for a sample mean = 1.725; Two-tailed p-value = 0.084527.

How to Read a z Test Statistic

A z statistic counts how many standard errors your result is from the value under the null hypothesis. A large absolute value means the result is unusual if the null is true. The calculator gives the two-tailed, right-tail and left-tail p-values, and tells you whether the result is significant at α = 0.05 for a two-tailed test.

At the 5% level the two-tailed critical z is ±1.96. Use z when the population standard deviation is known or when you test proportions in large samples. When the SD is estimated from a small sample, a t test is the safer choice.

Reading the z value

What different z values mean
\|z\|	Two-tailed p-value	Reading
0 to 1	above 0.32	Very ordinary
1.645	0.10	Significant at 10%
1.96	0.05	Significant at 5%
2.576	0.01	Significant at 1%
3.291	0.001	Significant at 0.1%
Above 4	below 0.0001	Extremely unusual under H₀

z or t?

Use a z statistic when the population standard deviation is known, or when you test a proportion (where the variance is determined by p₀). If σ is unknown and you estimate it from the sample, use a t statistic, which is wider in the tails for small samples. With n above 30 the two give almost identical answers.

Conditions

Random sample with independent observations.
For means: the population is roughly normal or n is large (about 30 or more).
For proportions: expected successes and failures at least 10 under H₀.
For two proportions: both groups have at least 10 successes and failures.

Where Z Test Statistic Is Used

Means with known σ

Test a sample mean against a hypothesised mean.

Proportions

Test a proportion, or compare two proportions.

Percentiles

See the percentile of the z value.

Reference table

Look up common z values and their probabilities.

Common Z Test Statistic Mistakes to Avoid

Use z only when σ is known, or for proportions with large samples.
Pick the tail from the research question before you compute anything.
The critical value of 1.96 is for a two-tailed test at 5%.
A z statistic is negative when the sample result is below the hypothesised value.

Z Test Statistic Terms and Definitions

z-score: The number of standard deviations a value lies from the mean.
Standard error: The standard deviation of a statistic such as the sample mean.

Z Test Statistic FAQs

What is a z test statistic?

A standardised value that tells you how many standard errors a sample estimate is from the hypothesised value.

How do I calculate the z test statistic for a mean?

Subtract the hypothesised mean from the sample mean and divide by σ/√n.

What is the critical z for 95% confidence?

1.96 for a two-tailed test. For a one-tailed 5% test it is 1.645.

How do I get a p-value from z?

Use the standard normal distribution: left tail Φ(z), right tail 1 − Φ(z), two-tailed twice the smaller tail.

What is the difference between z-score and z-statistic?

A z-score standardises a single observation. A z-statistic standardises a sample statistic such as a mean, using its standard error.

Can z be negative?

Yes. A negative z means the observed value is below the hypothesised value.

Z Test Statistic Calculator

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