Normal distribution probability calculator with sample size

Note: This creates the graph based on the shape of the normal curve, which is a reasonable approximation to the t-distribution for a large sample size. The probabilities are accurate, but the graphs are not accurate if you are doing a t-distribution with small sample size (less than 30).

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 Frequently-Asked 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.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

Why is the normal distribution so important?

The normal distribution is important because it describes the statistical behavior of many real-world 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 bell-shaped 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 z-score. 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 z-score?

A z-score (aka, a standard score) is the normal random variable of a standard normal distribution.

To transform a normal random variable (x) into an equivalent z-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 first-graders. 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 scores in a set of data vary. It is a measure of the average distance of individual observations from the group mean.

Sample Problem

1. 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 raw score, 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 0.97725. Thus, there is a 97.7% probability that an Acme Light Bulb will burn out within 1200 hours.

1. Bill claims that he can do more push-ups than 90% of the boys in his school. Last year, the average boy did 50 push-ups, with a standard deviation of 10 pushups. Assume push-up 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 raw score 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.

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