Z-Score Calculator

Convert between raw scores and standard scores, calculate probabilities, and visualize normal distribution. Essential tool for statistics and data analysis.

Z-Score Calculator

Calculate the standard score (z-score) for a raw value in a normal distribution.

Z-Score: 1.00
Percentile Rank: 84.13%
Interpretation: Above average
Formula:

z = (X - μ) / σ

Where:

  • z is the z-score
  • X is the raw score
  • μ is the population mean
  • σ is the standard deviation
Probability Calculator

Calculate the probability (p-value) for a given z-score in a standard normal distribution.

Z-Score: 1.96
Probability: 2.50%
P(Z ≥ 1.96) = 2.50%
Standard Normal Distribution Table (Z-Table):

The standard normal distribution has:

  • Mean (μ) = 0
  • Standard deviation (σ) = 1
  • Total area under curve = 1 (100%)
Between Two Scores Calculator

Calculate the probability between two z-scores or raw scores in a normal distribution.

Probability Between: 68.27%
P(-1 ≤ Z ≤ 1) = 68.27%
Empirical Rule (68-95-99.7 Rule):

For a normal distribution:

  • ≈68% of data falls within ±1σ of the mean
  • ≈95% within ±2σ
  • ≈99.7% within ±3σ
Understanding Z-Scores and Normal Distribution

What is a Z-Score?

A z-score (or standard score) represents how many standard deviations an element is from the mean. It's a statistical measurement that describes a value's relationship to the mean of a group of values.

Key Characteristics

  • Positive z-score: Value is above the mean
  • Negative z-score: Value is below the mean
  • Zero z-score: Value equals the mean
  • Magnitude: Indicates how unusual the value is

Normal Distribution

  • Bell-shaped, symmetric curve
  • Mean, median, and mode are equal
  • Defined by mean (μ) and standard deviation (σ)
  • Total area under curve = 1 (100% probability)

Why Use Z-Scores?

Z-scores are incredibly useful for:

Standardization

Compare scores from different normal distributions by converting them to a common scale.

Probability Estimation

Determine how unusual a value is by finding its probability in the standard normal distribution.

Hypothesis Testing

Essential for statistical tests like z-tests, t-tests, and confidence intervals.

Real-World Applications

Z-scores are used across many fields:

  • Education: Standardized test scoring (SAT, GRE)
  • Finance: Assessing investment performance
  • Quality Control: Identifying defects in manufacturing
  • Medicine: Growth charts, lab results interpretation
  • Psychology: IQ scores, personality assessments

How Our Z-Score Calculator Helps

Our advanced calculator simplifies complex statistical calculations by:

  • Converting between raw scores and z-scores instantly
  • Calculating exact probabilities for any z-score
  • Visualizing the normal distribution with highlighted areas
  • Providing clear interpretations of results
  • Supporting both one-tailed and two-tailed probability calculations

Whether you're a student learning statistics, a researcher analyzing data, or a professional making data-driven decisions, this tool makes z-score calculations accessible and intuitive.