Z-Score Calculator

How to Use the Z-Score Calculator

Choose an operation type: calculate z-score, find original value, compute probability, or compare values. Enter the required values and click Calculate.

Calculator Modes

Calculate Z-Score

Find how many standard deviations a value is from the mean:

• Formula: z = (x - μ) / σ
• x: The value you want to standardize
• μ: The population mean
• σ: The population standard deviation

Find Original Value

Reverse calculation - find the original value given a z-score:

• Formula: x = z × σ + μ
• Useful for finding values at specific percentiles

Probability

Calculate the area under the standard normal curve:

• Single z-score: Find P(Z ≤ z) and P(Z > z)
• Range: Find P(a ≤ Z ≤ b) between two z-scores

Compare Values

Compare relative positions of values from different distributions. The higher z-score indicates a relatively higher position in its distribution.

What is a Z-Score?

A z-score (or standard score) indicates how many standard deviations an element is from the mean. A z-score of 0 means the value equals the mean; positive values are above the mean, negative values are below.

Common Z-Score Values

• z = 0: At the mean (50th percentile)
• z = 1: ~84.1th percentile (1 std dev above)
• z = -1: ~15.9th percentile (1 std dev below)
• z = 1.645: ~95th percentile
• z = 1.96: ~97.5th percentile
• z = 2: ~97.7th percentile
• z = 2.576: ~99.5th percentile

Common Applications

• Comparing test scores across different exams
• Quality control in manufacturing
• Financial analysis and risk assessment
• Medical research and clinical trials
• Sports statistics and performance analysis
• Academic grading on a curve

Properties of Z-Scores

• Z-scores have a mean of 0 and standard deviation of 1
• Most values (99.7%) fall between z = -3 and z = 3
• Z-scores are dimensionless (no units)
• Adding a constant to all values doesn't change z-scores