Z-Score Calculator
P(Z < 0.49) =
| Left Tail Area | 0.687933 |
| Percentile / Probability | 68.79% |
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The Z-Score Formula
A z-score measures how many standard deviations a data point is from the mean. x is the raw value, μ (mu) is the population mean, and σ (sigma) is the population standard deviation. Once you know the z-score, you can find the associated probability using the standard normal distribution.
z = (x − μ) / σ
How It Works
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. A positive z-score indicates the value is above the mean, while a negative z-score means it is below the mean. Z-scores are fundamental in statistics for comparing data across different scales, performing hypothesis testing, and determining statistical significance. For example, a z-score of 1.96 means the data point is 1.96 standard deviations above the mean. In a standard normal distribution, approximately 95% of values fall between z = −1.96 and z = 1.96.
Example Problem
A class has a mean test score of 80 with a standard deviation of 5. A student scored 90. What percentage of students scored below this student?
- Calculate the z-score: z = (90 − 80) / 5 = 2.0
- Look up z = 2.0: the left-tail probability P(Z ≤ 2.0) = 0.9772
- Approximately 97.72% of students scored below 90 (97.72nd percentile)
Key Concepts
The standard normal distribution has a mean of 0 and standard deviation of 1. The cumulative distribution function (CDF) gives the probability that Z is less than or equal to a given z-score. Left tail gives P(Z ≤ z), right tail gives P(Z ≥ z), and two-tailed tests measure probability in both tails combined. The empirical rule states that 68.27% of data falls within 1 standard deviation, 95.45% within 2, and 99.73% within 3.
Frequently Asked Questions
What does a z-score of 0 mean?
A z-score of 0 means the data point is exactly at the mean. The probability of being at or below the mean (left-tail area) is 0.5, or 50%.
What z-score corresponds to a 95% confidence level?
For a 95% confidence interval (two-tailed), the critical z-score is 1.96. For a one-tailed test at 95% confidence, the critical z-score is 1.645.
Can z-scores be negative?
Yes. A negative z-score indicates the value is below the mean. For example, z = −2.0 means the value is 2 standard deviations below the mean.
How accurate is this calculator?
This calculator uses the Abramowitz and Stegun numerical approximation for the cumulative normal distribution function, providing accuracy to better than 7.5 × 10⁻⁸.
What is the empirical rule (68-95-99.7 rule)?
In a normal distribution, approximately 68.27% of data falls within 1 standard deviation of the mean, 95.45% within 2, and 99.73% within 3 standard deviations.
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