Normal Distribution Calculator

Calculate z-scores, probabilities, and percentiles for normal distributions. Interactive bell curve visualization with step-by-step solutions.

Formula:Z = (X - mu) / sigma

Results

Z-Score

1.0000

84.13th percentile

P(X < value)84.1345%
P(X > value)15.8655%
Percentile84.13th

Distribution Parameters

Center of the distribution

Spread of the distribution

Calculation Mode

Find probability for this value

Bell Curve Visualization

Shaded area represents the probability. Total area under the curve = 1.

Results

Z-Score

1.0000

P(X < value)

84.1345%

P(X > value)

15.8655%

Percentile

84.13th

Step-by-Step Calculation

Step 1: Calculate Z-Score

Z = (X - mu) / sigma

Z = (1 - 0) / 1

Z = 1.0000 / 1

Z = 1.0000

Step 2: Find Cumulative Probability

P(Z < 1.0000) using standard normal CDF

P(X < 1) = 84.1345%

The 68-95-99.7 Rule (Empirical Rule)

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

Common Z-Scores Reference

Z-ScorePercentileP(Z < z)Description
-30.13%0.00130.13th percentile
-22.28%0.02282.28th percentile
-115.87%0.158715.87th percentile
050.00%0.500050th percentile (mean)
184.13%0.841384.13th percentile
297.72%0.977297.72th percentile
399.87%0.998799.87th percentile

Results

Z-Score

1.0000

P(X < value)84.1345%
Percentile84.13th

?How Do You Calculate Normal Distribution Probability?

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.

What is Normal Distribution?

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.

Key Facts About Normal Distribution

  • Z-score formula: Z = (X - mu) / sigma
  • 68% of data falls within 1 standard deviation of mean
  • 95% of data falls within 2 standard deviations of mean
  • 99.7% of data falls within 3 standard deviations of mean
  • Standard normal distribution has mu = 0 and sigma = 1
  • The bell curve is symmetric around the mean
  • Total area under the curve equals 1 (100%)
  • Mean = Median = Mode for normal distribution

Quick Answer

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.

Frequently Asked Questions

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