Normal Distribution Simulator
Adjust mean, standard deviation, and shaded probability range
Normal Distribution Formulas
| Concept | Formula | Meaning |
|---|
| z-score | z = (x - mean) / sd | How many standard deviations x is from the mean. |
| Density | f(x) = exp(-z^2 / 2) / (sd sqrt(2pi)) | Height of the bell curve at x. |
| Probability range | P(a <= X <= b) = CDF(b) - CDF(a) | Area under the curve between two values. |
Practical Rules
- About 68% of values are within 1 standard deviation of the mean.
- About 95% of values are within 2 standard deviations of the mean.
- About 99.7% of values are within 3 standard deviations of the mean.
- A larger standard deviation makes the curve wider and lower.
Example: for mean = 0 and sd = 1, the range from -1 to 1 contains about 68.27% of the distribution.
Step-by-Step Probability Example
Suppose scores are normally distributed with mean = 100 and sd = 15. To estimate the probability between 85 and 115, convert each value to a z-score: z1 = (85 - 100) / 15 = -1 and z2 = (115 - 100) / 15 = 1. The area between z = -1 and z = 1 is about 68.27%, so roughly 68.27% of scores fall in that range.
Common Z-Score Areas
| Range | Approximate area | Use |
|---|
| mean +/- 1 sd | 68.27% | Typical central range. |
| mean +/- 2 sd | 95.45% | Common range for unusual-value checks. |
| mean +/- 3 sd | 99.73% | Very broad normal range. |
Common Mistakes
- Using a negative or zero standard deviation. Standard deviation must be positive.
- Reading curve height as probability. Probability is area under the curve, not the height of one point.
- Forgetting to standardize x-values before comparing them with a standard normal table.
Frequently Asked Questions
What does this normal distribution simulator show?▼
It shows the bell curve for a chosen mean and standard deviation and shades the probability between two x-values.
What is a z-score?▼
A z-score is the number of standard deviations a value is from the mean: z = (x - mean) / standard deviation.
What does standard deviation change?▼
Larger standard deviation makes the bell curve wider and lower. Smaller standard deviation makes it narrower and taller.
Is the probability exact?▼
The simulator uses a standard normal CDF approximation, which is accurate enough for learning and quick estimation.
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