Calculate z-scores, probabilities, and percentiles for normal distributions. Interactive bell curve visualization with step-by-step solutions.
Z-Score
1.0000
84.13th percentile
Center of the distribution
Spread of the distribution
Find probability for this value
Shaded area represents the probability. Total area under the curve = 1.
Z-Score
1.0000
P(X < value)
84.1345%
P(X > value)
15.8655%
Percentile
84.13th
Z = (X - mu) / sigma
Z = (1 - 0) / 1
Z = 1.0000 / 1
Z = 1.0000
P(Z < 1.0000) using standard normal CDF
P(X < 1) = 84.1345%
68%
within 1 SD
mu - sigma to mu + sigma
-1.00 to 1.00
95%
within 2 SD
mu - 2sigma to mu + 2sigma
-2.00 to 2.00
99.7%
within 3 SD
mu - 3sigma to mu + 3sigma
-3.00 to 3.00
| Z-Score | Percentile | P(Z < z) | Description |
|---|---|---|---|
| -3 | 0.13% | 0.0013 | 0.13th percentile |
| -2 | 2.28% | 0.0228 | 2.28th percentile |
| -1 | 15.87% | 0.1587 | 15.87th percentile |
| 0 | 50.00% | 0.5000 | 50th percentile (mean) |
| 1 | 84.13% | 0.8413 | 84.13th percentile |
| 2 | 97.72% | 0.9772 | 97.72th percentile |
| 3 | 99.87% | 0.9987 | 99.87th percentile |
Z-Score
1.0000
The normal distribution (bell curve) is characterized by mean (mu) and standard deviation (sigma). To find probability for value X: 1) Calculate Z-score: Z = (X - mu) / sigma. 2) Look up cumulative probability in standard normal table. P(X < value) gives left-tail area. The 68-95-99.7 rule states that 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.
The normal distribution (Gaussian distribution) is a continuous probability distribution that is symmetric around its mean, with a bell-shaped curve. It is defined by two parameters: the mean (mu) which determines the center, and the standard deviation (sigma) which determines the spread. Many natural phenomena follow a normal distribution, including heights, test scores, measurement errors, and IQ scores. The standard normal distribution (Z-distribution) has mean 0 and standard deviation 1.
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The normal distribution (bell curve) is characterized by mean (mu) and standard deviation (sigma). To find probability for value X: 1) Calculate Z-score: Z = (X - mu) / sigma. 2) Look up cumulative probability in standard normal table. P(X < value) gives left-tail area. The 68-95-99.7 rule states that 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.
A z-score measures how many standard deviations a value is from the mean. Z = (X - mean) / standard deviation. A z-score of 0 means the value equals the mean, positive scores are above the mean, negative below.
The empirical rule states that for normal distributions: 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
Use the cumulative distribution function (CDF) or a z-table. P(Z < z) gives the area to the left of z under the standard normal curve. P(Z > z) = 1 - P(Z < z) for the right tail.
The standard normal distribution has mean = 0 and standard deviation = 1. Any normal distribution can be standardized by converting values to z-scores. This allows using a single reference table.
To find X given a probability: 1) Find the z-score for that probability using inverse CDF. 2) Convert back: X = mean + (z-score x standard deviation).
Last updated: 2025-01-15
Z-Score
1.0000
84.13th percentile