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Z-Score Calculator

Enter a value, mean, and standard deviation to calculate the z-score and the corresponding probability from the standard normal distribution.

Calculate:

Results

Z-Score:0.5
P(Z < 0.5):0.6915 (69.15%)
P(Z > 0.5):0.3085 (30.85%)
Step-by-step work:
Step 1 — Z-Score formula:
  z = (x - μ) / σ

Step 2 — Substitute values:
  z = (75 - 70) / 10
  z = 5 / 10
  z = 0.5

Step 3 — Interpretation:
  A z-score of 0.5 means the value 75
  is 0.5 standard deviations above the mean.
  This is within 1 standard deviation of the mean (common).

Step 4 — Probability (area under curve):
  P(Z < 0.5) = 0.6915 (69.15%)
  P(Z > 0.5) = 0.3085 (30.85%)

What Is a Z-Score?

A z-score, also called a standard score, expresses how many standard deviations a particular data point lies above or below the mean of its distribution. It converts any normally distributed variable into a common scale — the standard normal distribution with mean 0 and standard deviation 1.

The formula is simple:

z = (x − μ) / σ

Where x is the observed value, μ is the population mean, and σ is the population standard deviation. A positive z-score means the value is above the mean; a negative z-score means it is below.

The Standard Normal Distribution

When you convert raw values to z-scores, you are standardizing the data to fit the standard normal distribution — a bell-shaped curve centered at z = 0 with a standard deviation of 1. This distribution has well-studied mathematical properties that allow you to compute probabilities precisely.

The total area under the curve equals 1 (representing 100% probability). The area to the left of a z-score gives the cumulative probability — the fraction of the distribution that falls at or below that value.

The 68-95-99.7 Rule

One of the most useful properties of a normal distribution is how values concentrate around the mean:

Z-score range Probability within range Description
−1 to +1 68.27% Within 1 standard deviation
−2 to +2 95.45% Within 2 standard deviations
−3 to +3 99.73% Within 3 standard deviations
|z| > 3 0.27% Extreme outliers

These percentages apply to any normally distributed variable. Whether you are looking at heights, test scores, or manufacturing measurements, the same proportions hold.

Step-by-Step Example

Scenario: A class has a mean test score of 75 and a standard deviation of 10. A student scored 92. What is their z-score, and what percentage of students scored lower?

Step 1 — Calculate the z-score:
z = (92 − 75) / 10 = 17 / 10 = 1.70

Step 2 — Find the cumulative probability:
From a z-table, P(z < 1.70) ≈ 0.9554.
Approximately 95.54% of students scored lower than 92.

Step 3 — Interpret:
The student is in the top 4.46% (1 − 0.9554 = 0.0446) of the class.

Reading a Z-Table

A z-table (standard normal table) lists cumulative probabilities for z-scores from about −3.4 to +3.4. To read the table:

  1. Find the row corresponding to the ones and tenths digit of your z-score (e.g., 1.7 for z = 1.73).
  2. Find the column corresponding to the hundredths digit (e.g., 0.03 for z = 1.73).
  3. The cell gives P(Z < z) — the probability that a random value is below this z-score.
  4. To find the probability above z, subtract from 1: P(Z > z) = 1 − P(Z < z).
  5. For a two-tailed test between −z and +z: P(−z < Z < z) = P(Z < z) − P(Z < −z).

Practical Uses of Z-Scores

Z-scores appear across many fields where data follows (or approximates) a normal distribution:

Frequently Asked Questions

What is a z-score?

A z-score (also called a standard score) measures how many standard deviations a particular value is above or below the mean of its distribution. The formula is z = (x − μ) / σ, where x is the value, μ is the population mean, and σ is the population standard deviation. A z-score of 0 means the value equals the mean. A z-score of +1 means the value is one standard deviation above the mean. A z-score of −2 means the value is two standard deviations below the mean.

How do you calculate a z-score?

To calculate a z-score: (1) subtract the mean from the value (x − μ), giving the deviation from the mean; (2) divide by the standard deviation (σ). The result tells you how many standard deviations the value lies from the mean. Example: mean = 100, standard deviation = 15, value = 130. z = (130 − 100) / 15 = 30 / 15 = 2.0. The value of 130 is exactly 2 standard deviations above the mean.

What is the 68-95-99.7 rule?

For a normally distributed dataset, the 68-95-99.7 rule (also called the empirical rule) states: approximately 68% of values fall within 1 standard deviation of the mean (z between −1 and +1); approximately 95% fall within 2 standard deviations (z between −2 and +2); approximately 99.7% fall within 3 standard deviations (z between −3 and +3). Values with |z| > 3 are very rare — less than 0.3% of the distribution.

How do you find the probability associated with a z-score?

Probabilities from z-scores are read from a standard normal distribution table (z-table) or computed using statistical software. The table gives the cumulative probability — the probability that a random value from the distribution falls at or below a given z. For example, z = 1.96 corresponds to a cumulative probability of 0.975 (97.5%). This means 97.5% of values fall at or below z = 1.96, and 2.5% fall above. For a two-tailed probability, subtract both tails: P(−1.96 < z < 1.96) = 0.975 − 0.025 = 0.95 (95%).

What are z-scores used for in practice?

Z-scores are used across many fields: standardized testing (SAT, IQ scores are scaled so mean = 100/500, SD = 15/100), medical lab results (values flagged as abnormal if |z| > 2), finance (z-score is used in the Altman Z-score model for predicting corporate bankruptcy), quality control (Six Sigma defines defects as outside ±6σ), and research (calculating p-values for hypothesis tests). Z-scores allow comparison of values from different distributions by putting everything on a common scale.

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