P-Value for a Z-Score of 1.96: Two- and One-Tailed

P-value from a z, t, chi-square or F statistic, and the critical value behind it

Every classical test ends the same way: a statistic, a distribution and a tail area. This calculator takes the statistic your software, textbook or paper printed, together with its degrees of freedom, and returns the exact p-value for a two-sided, right-tailed or left-tailed test. Switch to critical values and it gives the threshold for any alpha, including your own, such as a Bonferroni-corrected 0.0083. Degrees of freedom may be fractional, as Welch's test produces, and may run to ten million.

1.960

z for a two-sided p of 0.05

3.841

the same threshold squared, chi-square with 1 df

4.32 bits

the surprise in p = 0.05, like four heads in a row

Five fields between a statistic and its p

What to calculate - the p-value of a statistic you have, or the critical value for a significance level.
Distribution - z (standard normal), t, chi-square or F. The table below says which one your test uses.
Statistic and degrees of freedom - for example t = 2.2 with 18 df, or F = 3.5 with 2 and 27 df. Use a dot for decimals; fractions are fine for df.
Tail - two-sided for z and t unless a direction was fixed in advance; right-tailed for chi-square and F in almost every test.
Significance level - 0.10, 0.05, 0.01, 0.001 or your own value. The result shows the p-value, the rejection region, the S-value, the equivalent |z| and the tail areas.

Which distribution your test statistic follows

Test	Distribution	Degrees of freedom	Usual tail
One-sample or paired t-test	t	n - 1	two-sided
Two-sample t-test, pooled	t	n_A + n_B - 2	two-sided
Welch's t-test	t	Welch-Satterthwaite, fractional	two-sided
Pearson correlation r	t = r √(n - 2) / √(1 - r²)	n - 2	two-sided
Large-sample test of a mean or a proportion	z	none	two-sided
Chi-square test of independence	chi-square	(rows - 1) × (columns - 1)	right
Chi-square goodness of fit	chi-square	categories - 1 - estimated parameters	right
One-way ANOVA	F	groups - 1 and N - groups	right
Overall F in regression	F	predictors and n - predictors - 1	right
Ratio of two variances	F	n₁ - 1 and n₂ - 1	two-sided

Five printed statistics, five p-values

Chi-square = 7.2 on 3 df, right tail. p = 0.0658. The 0.05 threshold is 7.815, so a 2 × 4 table with this statistic falls just short.

F = 3.5 on 2 and 27 df, right tail. p = 0.0445, just past the threshold of 3.354: three groups of ten differ in an ANOVA at the 0.05 level.

t = 2.2 on 18 df, two-sided. p = 0.0411, against a threshold of 2.101. The same p comes from |z| = 2.042, a reminder that small samples need a larger t.

t = -2.1 on 25 df, left tail. p = 0.0230. A one-sided test in the predicted direction halves the two-sided p; in the other direction it would be 0.9770.

z = 2.33, two-sided. p = 0.0198, as in a test of two proportions with large samples.

Thresholds for alpha 0.05, by degrees of freedom

Two-sided t, right-tailed chi-square, and right-tailed F with 1 numerator df. The last column is the t column squared, because F(1, v) is t² with v degrees of freedom.

df	t, two-sided	chi-square	F(1, df)
1	12.706	3.841	161.448
2	4.303	5.991	18.513
5	2.571	11.070	6.608
10	2.228	18.307	4.965
20	2.086	31.410	4.351
30	2.042	43.773	4.171
60	2.000	79.082	4.001
120	1.980	146.567	3.920

One p-value, four ways to read it

The S-value, -log₂ p, counts the surprise in bits: p = 0.05 is about as surprising as 4 heads in a row from a fair coin. The equivalent |z| puts any p on the familiar normal scale.

p, two-sided	Stars just below it	S-value (bits)	Equivalent \|z\|
0.10	.	3.32	1.645
0.05	*	4.32	1.960
0.01	**	6.64	2.576
0.005	**	7.64	2.807
0.001	***	9.97	3.291
5.73 × 10^-7	***	20.73	5.000

The last row is five sigma. Particle physicists quote it one-sided, which gives p = 2.87 × 10^-7.

How the four distributions connect

Chi-square with 1 df is z squared. A chi-square of 4 on 1 df and z = 2 two-sided both give p = 0.0455, which is why 3.841 is 1.960 squared.

F with 1 numerator df is t squared. An ANOVA on two groups gives the same p as the pooled t-test: 4.965 = 2.228² at 10 df.

The t distribution approaches z as df grow. The two-sided 0.05 threshold falls from 12.706 at 1 df to 1.980 at 120 df; with ten million df it matches 1.960.

Chi-square averages its df. That is why its thresholds grow with df, while t thresholds shrink.

P-value questions that come with a printout

SPSS shows p = .000. What is the real value?

SPSS rounds to three decimals, so .000 means below 0.0005. Type the statistic and df here to get the actual value, for example z = 5 gives 5.73 × 10^-7. Report it as p < 0.001 rather than p = 0.

Can I halve a two-sided p to get a one-sided one?

Only when the statistic points in the direction you predicted before seeing the data. In the other direction the one-sided p is 1 minus half the two-sided one, as with t = -2.1: 0.0230 left-tailed, 0.9770 right-tailed.

Why does a chi-square test use only the right tail?

Because the statistic adds up squared gaps between observed and expected counts. Any departure from the null hypothesis makes it larger, so large values are the evidence and the right tail holds the p-value. A very small chi-square points to a fit that is too good, which the left tail checks.

What about a 2 × 2 table with small counts?

The chi-square p is an approximation that fails when expected counts are small. Fisher's exact test computes the p-value from the table itself; it has its own calculator, linked below.

Is a p-value of 0.06 significant?

Not at alpha 0.05, and the threshold was set before the data for a reason: moving it afterwards defeats its purpose. A p of 0.06 is weak evidence, close to the chi-square example above (0.0658), and the calculator flags any p between 0.8 and 1.25 times alpha as fragile. Report the exact value and the confidence interval rather than a trend toward significance.

Why can the degrees of freedom be a fraction?

Welch's t-test estimates them from the two variances, so R and SPSS print values such as 17.776. Printed tables list whole numbers only, but the t distribution is defined for any positive df, and this calculator uses the exact value: t = -1.8608 on 17.776 df gives p = 0.0794, the figure R reports.

Which alpha should I use for several tests?

With a Bonferroni correction, divide alpha by the number of tests: 0.05 over six tests gives 0.0083. Choose your own alpha and type it in; the critical value and the table row for it appear in the result.

Related tools

Fisher Exact Test Calculator

The exact p-value for a 2 × 2 table, with no chi-square approximation - See calculator

One-Sample T-Test Calculator

Get t and its p straight from your measurements and a reference value - See calculator

Two-Sample T-Test Calculator

Welch, Student and paired tests from raw data or summaries - See calculator

Z-Score Calculator

Percentiles and tail areas of the normal distribution for a single value - See calculator

Pearson Correlation Calculator

r with its t statistic and p-value for paired data - See calculator

Wilcoxon Test Calculator

Rank-based p-values when the t-test assumptions look shaky - See calculator

What is the p-value for a z-score of 1.96? Two-tailed, alpha 0.05

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