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Normal Distribution Calculator

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Given a normal distribution with μ = 73 and σ = 12., calculate the 68-95-99.7 rule, or three-sigma rule, or empirical rule ranges

Calculate Range 1:

Range 1, or the 68% range, states that 68% of the normal distribution values lie within 1 standard deviation of the mean
68% of values are within μ ± σ
μ ± σ = 73 ± 12.
73 - 12. <= 68% of values <= 73 + 12.
61 <= 68% of values <= 85

Calculate Range 2:

Range 2, or the 95% range, states that 95% of the normal distribution values lie within 2 standard deviations of the mean
95% of values are within μ ± 2σ
μ ± 2σ = 73 ± 2(12.)
73 - 2 x 12. <= 95% of values <= 73 + 2 x 12.
73 - 24 <= 95% of values <= 73 + 24
49 <= 95% of values <= 97

Calculate Range 3:

Range 3, or the 99.7% range, states that 99.7% (virtually ALL) of the normal distribution values lie within 3 standard deviations of the mean
99.7% of values are within μ ± 3σ
μ ± 3σ = 73 ± 3(12.)
73 - 3 x 12. <= 99.7% of values <= 73 + 3 x 12.
73 - 36 <= 99.7% of values <= 73 + 36
37 <= 99.7% of values <= 109





What is the Answer?

37 <= 99.7% of values <= 109

How does the Normal Distribution Calculator work?

Calculates the probability that a random variable is less than or greater than a value or between 2 values using the Normal Distribution z-score (z value) method (Central Limit Theorem).
Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability
This calculator has 4 inputs.

What 1 formula is used for the Normal Distribution Calculator?

  1. Z = (X - μ)/σ/√n

For more math formulas, check out our Formula Dossier

What 9 concepts are covered in the Normal Distribution Calculator?

distribution
value range for a variable
empirical rule
Provides estimate for the spread of data in a normal distribution. 68% of the data will fall within one standard deviation of the mean. 95% of the data will fall within two standard deviations of the mean. 99.7% of the data will fall within three standard deviations of the mean
event
a set of outcomes of an experiment to which a probability is assigned.
mean
A statistical measurement also known as the average
normal distribution
an arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme.
probability
the likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes
standard deviation
a measure of the amount of variation or dispersion of a set of values. The square root of variance
variance
How far a set of random numbers are spead out from the mean
z score
the number of standard deviations from the mean a data point is. Also known as a standard score

Example calculations for the Normal Distribution Calculator

  1. P(Z>1.5)
  2. P(Z<2.15)
  3. The amount of Jens monthly phone bill is normally distributed with a mean of $73 and a standard deviation of $12. What percentage of her phone bills are between $37 and $109?

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