Z-score Calculator Calculate standard scores to compare data points across different distributions — with probability int...

Z-score Calculator

Calculate standard scores to compare data points across different distributions — with probability interpretation and normal distribution visualization.

Single Z-score

Z to Probability

Probability to Z

Dataset Z-scores

Raw Score (X)

Population Mean (μ)

Standard Deviation (σ)

Z-score

Probability Type Left of Z (P(Z < z)) Right of Z (P(Z > z)) Between -|z| and |z| Outside -|z| and |z|

Probability

Tail Type Left Tail Right Tail Two-Tailed

Dataset Values 72, 78, 85, 90, 65, 88, 76, 82, 79, 84

Calculation Type Population Parameters Sample Parameters

0.00

Z-score

0.00%

Probability

0.00%

Percentile

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Interpretation

Statistical Interpretation

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Standard Normal Distribution

What is a Z-score?

A Z-score (standard score) indicates how many standard deviations a data point is from the mean of a distribution.

Z-score Formula**:

$$ Z = \frac{X - \mu}{\sigma} $$

Where:

• X = Raw score
• μ = Population mean
• σ = Population standard deviation

Interpretation**:

• Z = 0: Exactly at the mean
• Z > 0: Above the mean
• Z < 0: Below the mean
• |Z| > 2: Unusual value (outside 95% range)
• |Z| > 3: Extreme outlier

Z-score Common Mistakes

⚠️ Avoid these critical errors:

• Using sample SD for population Z-score**: Use population parameters when available
• Non-normal distributions**: Z-scores assume normal distribution for probability interpretation
• Small sample sizes**: Z-scores are less reliable with small samples (n < 30)
• Outliers**: Extreme values can distort mean and SD calculations
• Confusing Z with T**: Use T-scores for small samples with unknown population SD

✅ Best Practices**:

• Always verify your data follows a normal distribution before interpreting probabilities
• Use population parameters when calculating Z-scores for standardized testing
• For small samples, consider using T-distribution instead
• Report both Z-score and raw score for complete context

Real-World Applications

Z-scores are essential in:

• Education**: Standardized test scores (SAT, ACT, IQ tests)
• Healthcare**: Growth charts, bone density scores, lab test results
• Finance**: Credit scoring, risk assessment, anomaly detection
• Quality Control**: Process monitoring, defect detection
• Research**: Statistical analysis, hypothesis testing

📊 Example Use Cases**:

• Student test score**: Score=85, Mean=75, SD=10 → Z=1.0 (84th percentile)
• Bone density**: T-score=-2.5 → Z=-2.5 (Osteoporosis diagnosis)
• Manufacturing**: Part measurement=10.2mm, Mean=10.0mm, SD=0.1mm → Z=2.0 (Investigate process)

How to Use This Calculator

➡️ Single Z-score

"Score=85, mean=75, SD=10" → Z=1.0, 84.13% percentile

➡️ Z to Probability

"Z=1.5, left probability" → 93.32%

➡️ Probability to Z

"95% confidence, two-tailed" → Z=±1.96

➡️ Dataset Z-scores

"Calculate Z-scores for all values in dataset" → Individual Z-scores with statistics

Note: Uses precise normal distribution calculations. Probabilities are accurate to 4 decimal places. For two-tailed probabilities, the calculator returns the critical Z-value for the specified confidence level.