Z-Score Calculator

Q: What is a good Z-Score?

Whether a z-score is 'good' depends on context. For test scores where higher is better, a positive z-score (above average) is desirable. For data quality, z-scores between -2 and +2 indicate typical values, while values beyond ±3 may indicate errors or outliers. In research, z-scores help identify significant deviations from expected values.

Calculate z-scores (standard scores) instantly with step-by-step formulas, interactive normal distribution visualization, probability lookup, percentile ranking, and statistical interpretation.

Z-Score Calculator

Calculation Mode

Quick Examples

Test Score x=85, μ=75, σ=10 IQ Score x=100, μ=100, σ=15 Height (in) x=72, μ=68, σ=3 95% CI (z→x) z=1.96, μ=0, σ=1

Data Value (x)

The raw score or observed value

Z-Score (z)

The standardized score (number of standard deviations)

Mean (μ)

Population or sample mean

Standard Deviation (σ)

Must be greater than 0

Decimal Precision

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About Z-Score Calculator

Welcome to the Z-Score Calculator, a comprehensive statistical tool that calculates z-scores (standard scores) with step-by-step explanations, interactive normal distribution visualization, probability calculations, and percentile ranking. Whether you are analyzing test scores, conducting statistical research, performing quality control analysis, or studying normal distributions, this calculator provides professional-grade analysis with intuitive visual feedback.

What is a Z-Score?

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. It transforms raw data into a standardized scale, making it possible to compare values from different distributions or identify unusual values.

Z-Score Formula

Standard Z-Score Formula

$$z = \frac{x - \mu}{\sigma}$$

Where:

z = Z-score (standard score)
x = Data value (raw score)
$\mu$ = Population mean (average)
$\sigma$ = Population standard deviation

Inverse Z-Score Formula

To find a data value from a known z-score:

Value from Z-Score

$$x = z \cdot \sigma + \mu$$

How to Interpret Z-Scores

Z-scores indicate the relative position of a value within a distribution:

z = 0: The value equals the mean (50th percentile)
z = 1: One standard deviation above the mean (approximately 84th percentile)
z = -1: One standard deviation below the mean (approximately 16th percentile)
z = 2: Two standard deviations above the mean (approximately 98th percentile)
z = -2: Two standard deviations below the mean (approximately 2nd percentile)

The Empirical Rule (68-95-99.7 Rule)

In a normal distribution:

68% of values fall within z = ±1 (within 1 standard deviation of the mean)
95% of values fall within z = ±2 (within 2 standard deviations)
99.7% of values fall within z = ±3 (within 3 standard deviations)

Common Z-Score Reference Table

Z-Score	Percentile	P(Z ≤ z)	Interpretation
-3.0	0.13%	0.0013	Extremely below average
-2.0	2.28%	0.0228	Well below average
-1.0	15.87%	0.1587	Below average
0	50.00%	0.5000	Average (mean)
1.0	84.13%	0.8413	Above average
1.96	97.50%	0.9750	Significantly above (95% CI)
2.0	97.72%	0.9772	Well above average
3.0	99.87%	0.9987	Extremely above average

Applications of Z-Scores

Standardized Testing

Z-scores are foundational to standardized test interpretation. Tests like the SAT, GRE, and IQ tests convert raw scores to standardized scores. This allows fair comparison of performance across different test versions or years.

Quality Control

In manufacturing and Six Sigma methodology, z-scores identify products or processes that deviate significantly from specifications. Values beyond ±3 sigma typically indicate defects or special cause variation requiring investigation.

Financial Analysis

Z-scores help assess the relative performance of investments, identify unusual market movements, and evaluate risk. The Altman Z-score is a famous formula using weighted financial ratios to predict bankruptcy risk.

Medical and Research Applications

Healthcare uses z-scores for growth charts (BMI-for-age, height-for-age), bone density measurements (T-scores and Z-scores), and identifying abnormal lab values. Research uses z-scores for meta-analysis and combining results from different studies.

Outlier Detection

Data points with z-scores beyond ±2 or ±3 are often considered outliers. This threshold helps identify data entry errors, unusual observations, or special cases requiring further investigation.

Z-Score vs Percentile

While related, z-scores and percentiles measure different things:

Z-score: Measures distance from the mean in standard deviation units (can be negative, zero, or positive)
Percentile: Indicates the percentage of values that fall below a given value (ranges from 0 to 100)

You can convert between them using the standard normal distribution. For example, z = 1.0 corresponds to approximately the 84th percentile.

Frequently Asked Questions

What is a Z-Score?

A z-score (also called standard score) measures how many standard deviations a data point is from the mean of a distribution. The formula is z = (x - μ) / σ, where x is the data value, μ is the mean, and σ is the standard deviation. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

How do you interpret a Z-Score?

Z-scores indicate relative position: z = 0 means the value equals the mean; z = 1 means 1 standard deviation above the mean; z = -1 means 1 standard deviation below. In a normal distribution, about 68% of values fall within z = ±1, about 95% within z = ±2, and about 99.7% within z = ±3. Values beyond ±3 are often considered outliers.

What is the difference between Z-Score and Percentile?

A z-score measures distance from the mean in standard deviation units, while a percentile indicates the percentage of values that fall below a given value. They are related: z = 0 corresponds to the 50th percentile; z = 1 is approximately the 84th percentile; z = 2 is approximately the 98th percentile.

When should I use Z-Scores?

Z-scores are useful for: comparing values from different distributions (like test scores from different exams), identifying outliers in data, standardizing data for statistical analysis, calculating probabilities in normal distributions, and creating standardized test scores. They are essential in statistics, quality control, psychology, and many scientific fields.

Can a Z-Score be negative?

Yes, a z-score can be negative, positive, or zero. A negative z-score means the data value is below the mean; a positive z-score means it is above the mean; and a z-score of zero means the value equals the mean.

What is a good Z-Score?

Whether a z-score is "good" depends on context. For test scores where higher is better, a positive z-score (above average) is desirable. For data quality, z-scores between -2 and +2 indicate typical values, while values beyond ±3 may indicate errors or outliers.

Additional Resources

Reference this content, page, or tool as:

"Z-Score Calculator" at https://MiniWebtool.com/zscore-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 30, 2026

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About Z-Score Calculator

What is a Z-Score?

Z-Score Formula

Inverse Z-Score Formula

How to Interpret Z-Scores

The Empirical Rule (68-95-99.7 Rule)

Common Z-Score Reference Table

Applications of Z-Scores

Standardized Testing

Quality Control

Financial Analysis

Medical and Research Applications

Outlier Detection

Z-Score vs Percentile

Frequently Asked Questions

What is a Z-Score?

How do you interpret a Z-Score?

What is the difference between Z-Score and Percentile?

When should I use Z-Scores?

Can a Z-Score be negative?

What is a good Z-Score?

Additional Resources

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