Math & Engineering
Z-Score Calculator
Calculate how many standard deviations a data point is from the mean
Enter values to calculate the z-score
Related to Z-Score Calculator
The Z-Score Calculator determines how many standard deviations away from the mean a data point is. It converts raw scores into a standardized form that allows for comparison across different datasets. The formula used is:
Z-Score Formula
Z = (X - μ) / σ
Where:
- X is the raw score (value)
- μ (mu) is the population mean
- σ (sigma) is the population standard deviation
The calculator also converts the z-score into a percentile using the cumulative standard normal distribution function. This tells you what percentage of the data falls below your value in a normal distribution.
Z-scores help you understand where a value stands in relation to the mean of a dataset. The interpretation depends on the z-score value:
Common Z-Score Values
- Z = 0: The value equals the mean
- Z = 1: The value is one standard deviation above the mean
- Z = -1: The value is one standard deviation below the mean
- Z = 2: The value is two standard deviations above the mean
- Z = -2: The value is two standard deviations below the mean
The percentile tells you the proportion of values that fall below your data point. For example, a percentile of 84.13% means your value is higher than 84.13% of the data in a normal distribution.
1. What is a z-score used for?
Z-scores are used to standardize data and make comparisons across different datasets. They're commonly used in statistics, academic grading, quality control, and any field where you need to understand how far a value deviates from the mean.
2. What is considered a high z-score?
In a normal distribution, approximately 68% of the data falls within ±1 standard deviation, 95% within ±2, and 99.7% within ±3. Z-scores beyond ±3 are often considered unusual or significant, depending on the context.
3. Can z-scores be negative?
Yes, z-scores can be negative. A negative z-score indicates that the value is below the mean, while a positive z-score indicates it's above the mean. The magnitude tells you how many standard deviations away from the mean the value is.
4. How is the percentile calculated?
The percentile is calculated using the cumulative distribution function of the standard normal distribution. It represents the area under the normal curve up to the z-score value, multiplied by 100 to convert to a percentage.
5. What is the scientific source for this calculator?
This calculator implements the standardized normal score (z-score) formula, which is a fundamental concept in statistical theory. The calculations are based on established mathematical principles from statistical theory, particularly the work of Karl Pearson and Ronald Fisher in developing standardized scores. The percentile calculations use the error function approximation for the cumulative normal distribution, which is derived from statistical mathematics and probability theory as documented in standard statistical textbooks and mathematical literature.