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Z-Score Formula:
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The z-score measures how many standard deviations a data point (x) is from the mean (μ) of a normal distribution. It standardizes values, allowing comparison across different normal distributions.
The calculator uses the z-score formula:
Where:
Explanation: The formula calculates how many standard deviations a value is above or below the mean. Positive z-scores indicate values above the mean, negative z-scores indicate values below the mean.
Details: Z-scores are crucial in statistics for probability calculations, hypothesis testing, and identifying outliers. They allow comparison of values from different normal distributions and help determine proportions and percentiles.
Tips: Enter the data point value (x), the mean of the distribution (μ), and the standard deviation (σ). Standard deviation must be greater than zero. The calculator will compute the corresponding z-score.
Q1: What does a z-score of 0 mean?
A: A z-score of 0 means the data point is exactly at the mean of the distribution.
Q2: How do I interpret a z-score of 1.5?
A: A z-score of 1.5 means the data point is 1.5 standard deviations above the mean.
Q3: What is the relationship between z-scores and percentiles?
A: Z-scores can be converted to percentiles using standard normal distribution tables. For example, a z-score of 1.0 corresponds to approximately the 84th percentile.
Q4: Can z-scores be used with non-normal distributions?
A: While z-scores can be calculated for any distribution, their interpretation in terms of probabilities is most meaningful for normal distributions.
Q5: What is considered an unusual z-score?
A: Typically, z-scores beyond ±2 are considered unusual, and beyond ±3 are considered outliers in a normal distribution.