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Calculate the z-score (standard score) of a value and its corresponding percentile using the standard normal distribution.
Z-Score Formula:
z = (x - μ) / σ
where x = value, μ = mean, σ = standard deviation
Percentile: area under standard normal curve to the left of z
A z-score (standard score) measures how many standard deviations a value is from the mean. z = 0 means exactly at the mean. z = 1 means one standard deviation above the mean. z = -2 means two standard deviations below.
The percentile tells you what percentage of values in a normal distribution fall below your value. A z-score of 0 is the 50th percentile (exactly half of values are below the mean).
A normal (Gaussian) distribution is a symmetric, bell-shaped probability distribution. About 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 (the empirical rule).
Z-scores standardize values to a common scale, making different datasets comparable. For example, a student who scored 85 on a test with mean 80 and std dev 5 has z = 1.0, which is better than a score of 90 on a test with mean 88 and std dev 1 (z = 2.0 is actually higher).
In a normal distribution: approximately 68% of values fall within 1 standard deviation (z between -1 and +1), 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is also called the 68-95-99.7 rule.