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Calculate z-scores, percentiles, and probabilities with interactive bell curve visualization and Python code.
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Quick Reference:
A negative z-score means the value is below the mean. For example, z = -1.5 means the value is 1.5 standard deviations below the mean. About 6.68% of values fall below z = -1.5 in a normal distribution.
Use z-scores when you know the population standard deviation (σ). Use t-scores when estimating from a sample (s). For large samples (n > 30), z and t values converge.
Yes, but they are very rare in normal distributions. Only 0.27% of data falls beyond ±3σ. Z-scores beyond ±4 are extremely unlikely (0.006%) and often indicate outliers or non-normal data.
The z-score maps directly to a percentile via the standard normal CDF. For example: z = 0 → 50th percentile, z = 1 → 84.13th, z = 1.96 → 97.5th, z = -1 → 15.87th percentile.
scipy.stats.zscore(data) computes the z-score for each element in an array: z = (x - mean) / std. It standardizes the entire dataset in one call. Use ddof=1 for sample standard deviation.