Z-Score / P-Value Calculator

A z-score (also called a standard score) is a statistical measure that describes how many standard deviations a data point is from the mean of a distribution. It standardizes any value from a normal distribution N(μ, σ) to the standard normal distribution N(0, 1), making it possible to compare values from different distributions on a common scale. The formula for calculating a z-score is:

Where x is the raw score, μ (mu) is the population mean, and σ (sigma) is the population standard deviation. A z-score of 0 means the value equals the mean, a z-score of +1 means one standard deviation above the mean, and a z-score of -2 means two standard deviations below the mean. Large absolute values of z (e.g., |z| > 3) indicate the value is far from the mean and may be considered unusual or an outlier.

What is a P-Value?

A p-value (probability value) is the probability, under the assumption that a specified statistical model (the null hypothesis) is true, of observing a test statistic at least as extreme as the value that was actually observed. In the context of z-scores and the normal distribution, the p-value represents the area under the standard normal curve in the tail(s) beyond the observed z-score.

P-values are fundamental to hypothesis testing and help us make decisions about whether observed data are consistent with a null hypothesis. A small p-value (typically < 0.05 or < 0.01) suggests that the observed result is unlikely under the null model, providing evidence against the null hypothesis and in favor of the alternative.

One-Tailed vs Two-Tailed P-Values

Left-Tailed (Lower-Tail) P-Value: This is the probability that a value is less than or equal to the observed z-score, calculated as P(Z ≤ z). It represents the area under the standard normal curve to the left of z. Use left-tailed tests when your alternative hypothesis states that the parameter is less than the null value (e.g., "the new process reduces defect rate").

Right-Tailed (Upper-Tail) P-Value: This is the probability that a value is greater than or equal to the observed z-score, calculated as P(Z ≥ z) = 1 - P(Z ≤ z). It represents the area under the curve to the right of z. Use right-tailed tests when your alternative hypothesis states that the parameter is greater than the null value (e.g., "the treatment increases average response time").

Two-Tailed P-Value: This represents the probability of observing a z-score at least as extreme in either direction from the mean. It's calculated as 2 × min(P(Z ≤ z), P(Z ≥ z)) or equivalently 2 × (1 - Φ(|z|)), where Φ is the standard normal CDF. Two-tailed tests are used when the alternative hypothesis is "not equal to" rather than directional (e.g., "the average differs from the claimed value" without specifying higher or lower).

What This Calculator Supports

Our Z-Score / P-Value Calculator offers multiple calculation modes to handle different statistical tasks:

• X → Z → p: Given a raw score x, population mean μ, and standard deviation σ, compute the z-score and corresponding p-value for the selected tail type.
• Z → p: Given a z-score directly, compute the p-value for left-tail, right-tail, or two-tailed tests. This is useful when working with standardized test statistics.
• p → Critical Z: Given a desired significance level (p-value), find the critical z-value that defines the rejection region. This is essential for setting up hypothesis test thresholds.
• Between-Bounds Probability: Enter lower and upper bounds (in z-scores or raw x values) to compute the probability that a value falls within that interval: P(lower ≤ Z ≤ upper) = Φ(upper) - Φ(lower).

The calculator works with both the standard normal distribution (μ = 0, σ = 1) for direct z-score calculations, and custom normal distributions with any specified μ and σ for converting raw scores to z-scores and probabilities.

1. Choose Your Calculation Mode: Select the type of calculation you need based on what information you have and what you're trying to find:
   • Z → p: If you already have a z-score (standardized value) and want to find the corresponding p-value (probability). This is common when working with test statistics from hypothesis tests.
   • p → Z: If you have a significance level or probability and need to find the critical z-value. For example, finding the z-value for α = 0.05 in a hypothesis test.
   • X → z → p: If you have a raw score x and the population parameters (μ and σ) and want to standardize it to a z-score and find the probability.
2. Select Distribution Type: Choose between:
   • Standard Normal (μ = 0, σ = 1): Use this when working directly with z-scores or when your data has already been standardized. This is the default for most statistical tables and is what you'll use for hypothesis testing with standardized test statistics.
   • Custom Normal (specify μ and σ): Use this when working with raw scores from a population with known mean and standard deviation. For example, if you're analyzing test scores with μ = 75 and σ = 10, or measurements with μ = 50.0 and σ = 0.3.
3. Pick the Tail Type: Select the appropriate tail based on your hypothesis or question:
   • Left-tailed: P(Z ≤ z) — Use when testing if a value is significantly lower than expected (e.g., H₁: μ < μ₀).
   • Right-tailed: P(Z ≥ z) — Use when testing if a value is significantly higher than expected (e.g., H₁: μ > μ₀).
   • Two-tailed: 2 × P(|Z| ≥ |z|) — Use when testing for any significant difference in either direction (e.g., H₁: μ ≠ μ₀). This is the most conservative choice and is commonly used in scientific research.
4. Enter Values: Input the required values based on your selected mode:
   • For Z → p: Enter the z-score value.
   • For p → Z: Enter the p-value (probability) as a decimal between 0 and 1 (e.g., 0.05 for 5%).
   • For X → z → p: Enter the raw score x, mean μ, and standard deviation σ.
   • For between-bounds: Enter both lower and upper values (as z-scores or x values depending on distribution type).
5. Click Calculate: Once you've entered all required values, click the Calculate button. The calculator will display:
   • An interactive bell curve chart with the relevant area shaded (left tail, right tail, both tails, or interval)
   • The z-score (if you entered x) or the x-value (if you entered z)
   • The p-value for the selected tail type
   • Critical values if you're finding z from p
   • For between-bounds, the probability that a value falls in that range
6. Interpret and Use Results: Review the visual representation on the chart to confirm the shaded region matches your intended question. Use the numeric results for hypothesis testing (compare p-value to α), confidence interval construction (use critical z-values), or probability calculations. If available, use the share or copy link feature to save your calculation with all inputs and results encoded in the URL for easy reference or sharing with colleagues.

• Match the Hypothesis to the Tail Type: Your alternative hypothesis determines which tail to use. One-sided tests (H₁: μ < μ₀ or H₁: μ > μ₀) use left or right tail respectively, while two-sided tests (H₁: μ ≠ μ₀) use two-tailed. Choosing the wrong tail can lead to incorrect conclusions. Always align your tail selection with the scientific or practical question you're asking.
• Memorize Common Critical Values: For hypothesis testing and confidence intervals, certain z-values appear repeatedly:
  • 90% confidence (two-tailed, α = 0.10): z = ±1.645
  • 95% confidence (two-tailed, α = 0.05): z = ±1.96
  • 99% confidence (two-tailed, α = 0.01): z = ±2.576
  • One-tailed α = 0.05: z = 1.645 (right) or -1.645 (left)
  • One-tailed α = 0.01: z = 2.326 (right) or -2.326 (left)
  Knowing these values by heart speeds up hypothesis testing and allows you to quickly estimate rejection regions without a calculator.
• Use Custom μ, σ for Raw Scores: When working with real-world measurements that haven't been standardized, use the custom distribution option. For example, if you're analyzing classroom exam scores with a known mean of 75 and standard deviation of 10, enter those values and your actual score to get the z-score and percentile (p-value). This converts your specific context to the universal standard normal scale.
• Between-Bounds is Fastest for Interval Probabilities: Instead of manually computing Φ(upper) - Φ(lower), use the between-bounds feature by entering lower and upper values directly. This is especially useful for questions like "What's the probability a randomly selected value falls between 80 and 90?" or "What fraction of parts will be within specification limits of 49.5 to 50.5 mm?" The calculator handles the subtraction and shades the exact interval.
• Small Samples or Unknown σ? Use t-Distribution: The z-distribution assumes you know the population standard deviation σ and works best with large samples (n ≥ 30). For small samples (n < 30) or when using the sample standard deviation s instead of σ, use the t-distribution calculator instead. The t-distribution has heavier tails and provides more conservative (wider) confidence intervals and higher p-values, accounting for the additional uncertainty from estimating σ.
• P-Values Are Not Error Probabilities: A common misconception is that a p-value of 0.05 means there's a 5% chance the null hypothesis is true. This is incorrect. The p-value is the probability of observing data as extreme as yours if the null hypothesis were true, not the probability that the null is true given your data. Keep this distinction in mind when communicating results: "The data are unlikely under the null model" rather than "The null is unlikely."
• Visualize Before Deciding: Always look at the shaded region on the bell curve to confirm it matches your intended question. If you selected left-tail but the shading is on the right, you may have made an input error. The visual feedback helps catch mistakes before you use the results in important decisions or reports.
• Context Matters for Significance Levels: While α = 0.05 is conventional in many fields, it's not universal. Medical trials often use α = 0.01 for higher confidence, physics uses 5σ (p < 0.0000003) for discovery claims, and exploratory research might use α = 0.10. Choose your significance level based on the consequences of Type I error (false positive) vs Type II error (false negative) in your specific context.

Z-Score Interpretation

The z-score tells you how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean. Positive z-scores indicate values above the mean, negative z-scores indicate values below. The magnitude tells you the distance: |z| = 1 is one standard deviation away, |z| = 2 is two standard deviations, and so on. By the empirical rule, about 68% of values fall within |z| < 1, 95% within |z| < 2, and 99.7% within |z| < 3. Values with |z| > 3 are rare and often considered outliers.

One-Tailed P-Values

Left-tail p-value: This is P(Z ≤ z), the area under the standard normal curve to the left of your z-score. It represents the probability of observing a value less than or equal to z under the standard normal model. For example, if z = -1.5 gives a left-tail p = 0.0668, there's about a 6.68% chance of getting a z-score of -1.5 or lower.

Right-tail p-value: This is P(Z ≥ z) = 1 - P(Z ≤ z), the area to the right of your z-score. It represents the probability of observing a value greater than or equal to z. For example, if z = 2.0 gives a right-tail p = 0.0228, there's about a 2.28% chance of getting a z-score of 2.0 or higher. This is commonly used in one-sided tests where you're looking for evidence of an increase.

Two-Tailed P-Value

The two-tailed p-value represents the probability of observing a z-score at least as extreme in either direction from the mean. It's calculated as 2 × (1 - Φ(|z|)), where Φ is the standard normal CDF. For example, if |z| = 1.96, the two-tailed p ≈ 0.05 (5%). This means there's a 5% combined chance of observing z ≤ -1.96 or z ≥ +1.96. Two-tailed p-values are used in non-directional hypothesis tests where you care about deviations in either direction, not just increases or decreases.

Critical Z from P-Value

When you input a p-value and select a tail type, the calculator solves the inverse problem: finding the z-value where the cumulative probability equals p (or 1-p for right-tail, or the symmetric value for two-tailed). For example, entering p = 0.05 with two-tailed gives z = ±1.96, the critical values that define the rejection region for a 95% confidence interval or α = 0.05 hypothesis test. These critical values are the boundaries: if your test statistic exceeds them, you reject the null hypothesis.

Between Two Values

The between-bounds probability is P(a ≤ Z ≤ b) = Φ(b) - Φ(a), where Φ is the standard normal CDF. This tells you the fraction of the distribution that falls within the interval [a, b]. For example, P(-1 ≤ Z ≤ 1) ≈ 0.68 (68% of values), P(-1.96 ≤ Z ≤ 1.96) ≈ 0.95 (95%), and P(-2.576 ≤ Z ≤ 2.576) ≈ 0.99 (99%). When working with raw x values instead of z-scores, the calculator first standardizes them using z = (x - μ) / σ, then computes the interval probability.

Shaded Chart Area

The interactive bell curve chart visually represents your calculation. The shaded area corresponds to the probability region:

• Left-tail: shaded from the left edge up to your z-value
• Right-tail: shaded from your z-value to the right edge
• Two-tailed: shaded in both tails symmetrically around the mean
• Between-bounds: shaded only in the interval between lower and upper values

The shaded area's size corresponds to the probability (p-value). Always verify the shading matches your intent before using the numeric results.

Numerical Precision and Limitations

Our calculator uses high-precision numerical approximations for the standard normal CDF (Φ) and its inverse, based on established statistical algorithms. The results are accurate to many decimal places and match published z-tables and statistical software like R, Python SciPy, and Excel. For typical statistical work (p-values down to 0.0001 or beyond), the accuracy is more than sufficient. Extremely far into the tails (probabilities below 10⁻¹⁰), tiny numerical errors may appear, but these are well beyond the range of practical statistical inference. The z-distribution assumes exact normality; if your data are not normally distributed, consider transformations, non-parametric methods, or bootstrapping.