Normal Curve (Z-Score) Calculator
Calculate probabilities and percentiles for a normal distribution.
Understanding the Normal Distribution Curve
The normal distribution, often called the "Bell Curve" or Gaussian distribution, is a fundamental concept in statistics. It describes a symmetrical distribution where most of the observations cluster around the central peak (the mean), and the probabilities for values taper off equally in both directions.
How to Use This Calculator
To find the probability associated with a specific data point, you need three values:
- Mean (μ): The average of your entire data set.
- Standard Deviation (σ): A measure of the spread or dispersion of the data.
- Test Value (x): The specific number you want to analyze within the distribution.
What is a Z-Score?
The Z-score represents how many standard deviations a specific value is away from the mean. If a Z-score is 0, it is exactly at the mean. If it is 1, it is one standard deviation above the mean. The formula used is:
Z = (x – μ) / σ
Real-World Example: IQ Scores
Standardized IQ tests are designed to follow a normal distribution with a mean of 100 and a standard deviation of 15. If a person has an IQ of 130:
- Mean: 100
- Standard Deviation: 15
- Test Value: 130
- Z-Score: (130 – 100) / 15 = 2.0
A Z-score of 2.0 indicates the person is in approximately the 97.7th percentile, meaning they scored higher than 97.7% of the population.
Why the Normal Curve Matters
In data science and quality control, the normal curve allows us to predict the likelihood of certain outcomes. According to the Empirical Rule (68-95-99.7):
- 68% of data falls within 1 standard deviation.
- 95% of data falls within 2 standard deviations.
- 99.7% of data falls within 3 standard deviations.