Z-score Calculator
Calculate the z-score and p-values for a normal distribution. The z-score tells you how many standard deviations a value is from the mean.
About Z-scores
A z-score (also called a standard score) indicates how many standard deviations a data point is from the mean of a dataset. It allows you to compare values from different datasets by standardizing them to a common scale.
Z-score Formula
Z-score:
z = (x - μ) / σ
Where:
x = raw score (data point)
μ = population mean
σ = population standard deviation
Interpreting Z-scores
- z = 0: The raw score equals the mean
- z > 0: The raw score is above the mean
- z < 0: The raw score is below the mean
- |z| = 1: The raw score is 1 standard deviation away from the mean
- |z| = 2: The raw score is 2 standard deviations away from the mean
- |z| = 3: The raw score is 3 standard deviations away from the mean
Z-scores and the Normal Distribution
In a normal distribution:
- Approximately 68% of values have z-scores between -1 and 1
- Approximately 95% of values have z-scores between -2 and 2
- Approximately 99.7% of values have z-scores between -3 and 3
Example Calculation
Let's calculate the z-score for a student who scored 85 on a test where the mean score was 75 with a standard deviation of 5:
Raw score (x) = 85
Mean (μ) = 75
Standard deviation (σ) = 5
z = (x - μ) / σ = (85 - 75) / 5 = 10 / 5 = 2
The z-score is 2, meaning the student's score is 2 standard deviations above the mean.
Applications of Z-scores
| Field | Application |
|---|---|
| Education | Standardizing test scores, grading on a curve |
| Finance | Risk assessment, portfolio analysis |
| Quality Control | Identifying outliers in manufacturing processes |
| Medicine | Comparing patient data to population norms |
| Psychology | Standardizing psychological test results |
P-values and Z-scores
The p-value associated with a z-score represents the probability of obtaining a value at least as extreme as the observed value, assuming the null hypothesis is true. In other words:
- P(x ≤ X): Probability of obtaining a value less than or equal to X
- P(x > X): Probability of obtaining a value greater than X
- P(a ≤ x ≤ b): Probability of obtaining a value between a and b