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Z-score Calculator

Enter your raw score, the distribution mean, and the standard deviation to compute the Z-score. The Z-score shows how many standard deviations the raw score is from the mean.

Notes: This calculator assumes a normally distributed variable when converting Z-scores to probabilities/percentiles. Use a strictly positive standard deviation. For probabilities, choose the tail option that matches your hypothesis or interpretation.

Use this Z-score Calculator to standardize any raw score relative to a distribution’s mean and standard deviation, and optionally convert the result into probabilities or percentiles under the normal distribution.

What is a Z-score?

A Z-score (also called a standard score) tells you how many standard deviations a raw value is above or below the mean. It is computed with a simple formula: Z = (X ? ?) / ?, where X is the raw score, ? is the mean, and ? is the standard deviation. Positive Z-scores indicate values above the mean; negative Z-scores indicate values below the mean. Because Z-scores are unitless, they allow direct comparisons across different scales and measurements.

Why standardize with Z-scores?

  • Comparability: Convert different metrics onto the same scale for apples-to-apples comparisons.
  • Outlier detection: Large absolute Z-scores (e.g., |Z| ? 2 or 3) may indicate unusual or extreme observations.
  • Probability estimates: With the normal distribution assumption, Z-scores map directly to tail probabilities and percentiles.
  • Feature scaling: In analytics and machine learning, standardizing features can improve model performance and interpretability.

How to use the Z-score Calculator

  1. Enter the raw score (X) you want to standardize.
  2. Provide the distribution’s mean (?) and standard deviation (?). Ensure ? is strictly positive.
  3. Optional: Choose a probability output. Select left-tail, right-tail, or two-tailed probability if you want p-values or percentiles based on the standard normal distribution.
  4. Set your preferred number of decimal places for display.
  5. Click “Calculate Z-score” to view the standardized score and any requested probabilities.

Interpreting results

A Z-score of 0 means the value equals the mean. A Z-score of +1 means one standard deviation above the mean; ?1 means one standard deviation below. For probability outputs:

  • Left-tail probability P(Z ? z): The proportion of the normal distribution at or below your Z.
  • Right-tail probability P(Z ? z): The proportion at or above your Z (1 minus the left-tail).
  • Two-tailed probability 2 × min(P(Z ? z), P(Z ? z)): Useful for two-sided tests, measuring extremeness on both sides of the mean.

Example

Suppose a test has a mean of 70 and a standard deviation of 10. If a student scores 85, then Z = (85 ? 70) / 10 = 1.5. Interpreted under the normal model, a Z of 1.5 corresponds to a left-tail probability around 0.9332 (93.32nd percentile). This means the score is higher than roughly 93% of scores in a normal distribution with that mean and standard deviation.

Best practices and considerations

  • Check distribution shape: Probabilities are derived assuming normality. If the underlying data are skewed or heavy-tailed, Z-based probabilities may be approximate.
  • Use the right ?: If you only have a sample standard deviation, it’s still common to compute a Z-like score, but formal inference might require t-distribution methods.
  • Context matters: A Z of 2 might be notable in some settings and routine in others. Always interpret in the domain context.
  • Precision: Choose an appropriate number of decimal places. Too few can obscure differences; too many can imply false precision.

Common applications

Z-scores are widely used in quality control, standardized testing, anomaly detection, A/B testing, and as a preprocessing step for statistical modeling and machine learning. They make it straightforward to compare performance across different instruments or time periods, and to translate raw results into interpretable percentiles.

Next steps

Enter your values into the Z-score Calculator, pick a probability output if needed, and interpret your results in context. For more rigorous analysis, pair Z-scores with confidence intervals, distribution checks, or nonparametric methods when normality is doubtful.

FAQs

What does the Z-score Calculator compute?

It standardizes a raw score using the mean and standard deviation, and can return probabilities or percentiles under the normal model.

How do I interpret results from the Z-score Calculator?

A positive Z means above the mean, negative means below. Large absolute values indicate more unusual observations.

Can the Z-score Calculator show percentiles?

Yes. Choose left-tail probability to see the percentile corresponding to your Z-score under the normal distribution.

What inputs are required for the Z-score Calculator?

You need the raw score (X), the mean (?), and a strictly positive standard deviation (?).

Does the Z-score Calculator assume a normal distribution?

Only for converting Z-scores to probabilities or percentiles. The Z itself does not require normality.

Is the Z-score Calculator suitable for small samples?

You can compute a Z-score, but for inference with unknown ? and small samples, a t-distribution may be more appropriate.

Can the Z-score Calculator handle right-tail or two-tailed p-values?

Yes. Select right-tail or two-tailed probability to get the corresponding p-value from the standard normal distribution.