Z-Score Calculator with P-Values and Critical Values

This z-score calculator does three related jobs. It standardises a raw score, turning "85 when the average is 70 and the standard deviation is 10" into "1.5 standard deviations above the mean". It turns a z-score into probabilities: the area to the left, the area to the right, and the two-tailed p-value.

And it runs the other way, giving the critical z for a confidence level such as 95% or 99%.

A z-score on its own is just a distance measured in standard deviations, and it is meaningful for any data. The probabilities are a further step: they assume the values follow a normal distribution, the familiar bell curve. When that assumption holds, a z-score tells you how unusual a value is, and a z statistic from a test tells you its p-value.

The probabilities are computed from the error function to 50 significant digits, not looked up in a printed table, and they agree with the standard normal table in the NIST/SEMATECH e-Handbook of Statistical Methods at every entry checked.

1. Choose what you have: a raw score with the mean and standard deviation, a z-score, or a confidence level.
2. Fill in the boxes that mode uses. The others are hidden and keep their values if you switch back.
3. Read the z-score (or the critical z in confidence mode) at the top. The cards give the left and right areas, the two-tailed p-value, the area between -|z| and |z|, and the percentile.

The presets load z = 1.96, a raw score one standard deviation below its mean, and the critical values for 99% and 90% confidence.

1. Standardise the score. In raw score mode the mean is subtracted from the score and the difference is divided by the standard deviation. A standard deviation of zero or less is rejected, since it would make the division meaningless. In z-score mode, z is taken as typed.

1. Area to the left. The standard normal cumulative probability Φ(z) is computed from the complementary error function, erfc. Below an argument of 4, erf comes from its all-positive power series (DLMF 7.6.2); above 4, erfc comes from its continued fraction (DLMF 7.9), which stays accurate deep into the tail where 1 minus a number close to 1 would lose every digit.

1. Area to the right. Computed directly as half of erfc(z/√2), not as 1 minus the left area, so a tiny right tail keeps its precision.

1. Two-tailed p-value and central area. The two-tailed p-value is twice the area beyond |z|. The area between -|z| and |z| is 1 minus that.

1. Critical z for a confidence level. In confidence mode, α is 1 minus the confidence level. The two-sided critical value is the z with Φ(z) = 1 - α/2, and the one-sided value has Φ(z) = 1 - α. Each is found by starting from a rational approximation accurate to about nine digits and taking Newton steps on the 50-digit Φ until the change is far below the digits shown.

1. Show the results. z-scores are rounded half up to 12 significant figures and probabilities to 10, with trailing zeros dropped.

The default inputs are a score of 85, a mean of 70 and a standard deviation of 10.

1. z = (85 - 70) ÷ 10 = 1.5.
2. Area to the left: Φ(1.5) = 0.9331927987, so the score sits at the 93.31927987% percentile.
3. Area to the right: 0.06680720127. That is the right-tailed p-value.
4. Two-tailed p-value: twice the area beyond 1.5, 0.1336144025.

The Score of 62 (mean 70, SD 8) preset gives z = -1. The area to its left is 0.1586552539 and the area between -1 and 1 is 0.6826894921, the "about 68%" that people quote for one standard deviation.

Pick the tail that matches your test. For a right-tailed test, the p-value is the area to the right of z. For a left-tailed test, it is the area to the left. For a two-tailed test, it is twice the area beyond |z|.

For z = 1.96, the z = 1.96 preset gives a right-tail area of 0.02499789515 and a two-tailed p-value of 0.0499957903, just under 0.05. That is why 1.96 is the famous cut-off for a two-sided test at the 5% level. The NIST/SEMATECH table lists the area from 0 to 1.96 as 0.47500, which matches the calculator's left area of 0.9750021049 minus one half.

The two-sided critical z for 95% confidence is 1.95996398454, usually rounded to 1.96. For 99% it is 2.57582930355 and for 90% it is 1.64485362695. The one-sided values are smaller because the whole α sits in one tail: 1.64485362695 for 95% and 2.32634787404 for 99%. The second reference table lists both for common levels.

The first reference table gives the left area, right area and two-tailed p-value for z-scores people use most, from -3 to 4. Each row is a run of this calculator, so the figures match what the tool shows for the same z.

• Non-normal data. The z-score itself is valid for any data, but the probabilities assume a normal distribution. For skewed data or data with heavy tails, they can be far off.
• Small samples with an estimated standard deviation. When σ is estimated from a small sample, a test statistic follows Student's t distribution, not the normal. Use the t-test calculator in that case.
• Sample means. To standardise a sample mean rather than a single value, divide by the standard error, σ divided by the square root of n, not by σ. Enter that standard error as the standard deviation.
• Discrete data. Counts such as the number of heads in 20 tosses are only approximately normal. No continuity correction is applied.

• Using the wrong tail. A two-tailed p-value is twice a one-tailed one. Decide which test you are doing before looking at the numbers.
• Mixing up the area to the left with the p-value. For a positive z the left area is large, near 1. The right area is the small p-value.
• Rounding z early. Rounding 1.645 to 1.6 changes a right-tail area of 0.04998490554 into something noticeably larger. Keep the full z.
• Dividing by the variance instead of the standard deviation. The denominator of a z-score is σ, not σ².
• Calling the percentile a probability of being correct. The percentile is the share of the distribution below z, nothing more.

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.1, Cumulative Distribution Function of the Standard Normal Distribution: the table the calculator is checked against. itl.nist.gov/div898/handbook/eda/section3/eda3671.htm
• NIST Digital Library of Mathematical Functions, section 7.6: power series for the error function. dlmf.nist.gov/7.6
• NIST Digital Library of Mathematical Functions, section 7.9: continued fraction for the complementary error function. dlmf.nist.gov/7.9
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1, Normal Distribution. itl.nist.gov/div898/handbook/eda/section3/eda3661.htm