Normal Distribution Calculator: Online Statistical Table
The Normal Distribution Calculator makes it easy to compute cumulative
probability, given a normal random variable; and vice versa. For help in using
the calculator, read the FrequentlyAsked Questions
or review the Sample Problems.
To learn more about the normal distribution, go to Stat Trek's
tutorial on the normal distribution.
Note: The normal distribution table, found in the appendix of
most statistics texts, is based on the
standard normal distribution, which has a
mean of 0 and a standard deviation of 1. To produce outputs
from a standard normal distribution with this calculator,
set the mean equal to 0 and the standard deviation equal to 1.
Instructions: To find the answer to a frequentlyasked
question, simply click on the question. If you don't see the answer you need,
try the Statistics Glossary or
check out Stat Trek's tutorial on the normal
distribution.
Why is the normal distribution so
important?
The normal distribution is important because it describes the
statistical behavior of many realworld events. The shape of the normal
distribution is completely described by the mean and the standard deviation.
Thus, given the mean and standard deviation, you can use the
properties of the normal distribution to quickly compute the cumulative
probability for any value. This process is illustrated in the
Sample Problems below.
What is a standard normal distribution?
There are an infinite number of normal distributions. Although
every normal distribution has a bellshaped curve, some normal distributions
have a curve that is tall and narrow; while others have a curve that is short
and wide.
The exact shape of a normal distribution is determined by its
mean and its standard deviation. The standard normal distribution is the
normal distribution that has a mean of zero and a standard deviation of one.
The normal
random variable of a standard normal distribution is called a standard
score or a zscore. The normal random variable X
from any normal distribution can be transformed into a z score from a
standard normal distribution via the following equation:
z = (X  μ) / σ
where X is a normal random variable, μ is the
mean, and σ is the standard deviation.
Because any normal random variable can be "transformed" into a z
score, the standard normal distribution provides a useful frame of reference.
In fact, it is the normal distribution that generally appears in the appendix
of statistics textbooks.
What is a normal random variable?
The normal distribution is defined by the following equation:
Normal equation. The value of the
random variable
Y is:
Y = { 1/[ σ * sqrt(2π) ] } *
e^{(x  μ)2/2σ2}
where
X is a normal random variable, μ is the
mean, σ is the standard deviation, π
is approximately 3.14159, and
e is approximately 2.71828.
In this equation, the
random variable X is called a normal random variable. A unique
cumulative probability can be associated with every normal random variable.
Given the normal random variable, the standard deviation of the normal
distribution, and the mean of the normal distribution, we can compute the
cumulative probability (i.e., the probability that a random selection from the
normal distribution will be less than or equal to the normal random variable.)
What is a standard score?
A standard score (aka, a zscore) is the
normal random variable
of a
standard normal distribution.
To transform a normal random variable (x) into an equivalent
standard score (z), use the following formula:
z = (x  μ) / σ
where μ is the mean, and σ is the standard deviation.
What is a probability?
A probability is a number expressing the chances that a specific
event will occur. This number can take on any value from 0 to 1. A probability
of 0 means that there is zero chance that the event will occur; a probability
of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify
the uncertainty associated with the event. For example, the probability of a
coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent
of the time, the coin flip would result in Heads; and fifty percent of the
time, it would result in Tails.
What is a cumulative probability?
A cumulative probability is a sum of probabilities. In connection
with the normal distribution, a cumulative probability refers to the
probability that a randomly selected score will be less than or equal to a
specified value, referred to as the normal random variable.
Suppose, for example, that we have a school with 100
firstgraders. If we ask about the probability that a randomly selected first
grader weighs exactly 70 pounds, we are asking about a simple probability  not
a cumulative probability.
But if we ask about the probability that a randomly selected
first grader is less than or equal to 70 pounds, we are really asking
about a sum of probabilities (i.e., the probability that the student is exactly
70 pounds plus the probability that he/she is 69 pounds plus the probability
that he/she is 68 pounds, etc.). Thus, we are asking about a cumulative
probability.
What is a mean score?
A mean score is an average score. It is the sum of individual
scores divided by the number of individuals.
What is a standard deviation?
The standard deviation is a numerical value used to indicate how
widely individuals in a group vary. It is a measure of the average distance of
individual observations from the group mean.

The Acme Light Bulb Company has found that an average light bulb lasts 1000
hours with a standard deviation of 100 hours. Assume that bulb life is normally
distributed. What is the probability that a randomly selected light bulb will
burn out in 1200 hours or less?
Solution:
We know the following:

The mean score is 1000.

The standard deviation is 100.

The normal random variable, for which we want to find a cumulative probability,
is 1200.
Therefore, we plug those numbers into the Normal
Distribution Calculator
and hit the Calculate button. The calculator reports that the cumulative
probability is 0.977. Thus, there is a 97.7% probability that an Acme Light
Bulb will burn out within 1200 hours.

Bill claims that he can do more pushups than 90% of the boys in his school.
Last year, the average boy did 50 pushups, with a standard deviation of 10
pushups. Assume pushup performance is normally distributed. How many pushups
would Bill have to do to beat 90% of the other boys?
Solution:
We know the following:

The mean score is 50.

The standard deviation is 10.

The cumulative probability is 0.90, since Bill has to outperform 90% of the
boys. (If he had claimed to outperform only 80% of the boys, the cumulative
probability would be 0.80.)
Therefore, we plug those numbers into the Normal
Distribution Calculator and hit the Calculate button. The calculator
reports that the normal random variable is 62.8. Therefore, Bill will need to
do at least 63 pushups to support his claim that he can do more pushups than
90% of the boys in his school.