Over 9,508,000 live tutoring sessions served!

Sales Toll Free No: 1-866-617-6020

Home How it works About Us Algebra Booster !

Math ▶ Statistics ▶ Probability Distribution ▶ Normal Distribution

Normal Distribution 0

I need help with "Normal Distributio Get a Tutor

Probability is an essential and a very special branch of mathematics. In probability, we 

study about the probable chances of happening of a certain event. In the language of 

probability, an event is an outcome of an experiment or a process. Probability usually 

deals with the events which have two possible results or outcomes, such as - either 

pass or fail, either success or failure, either true or false, either one or zero etc. 

Probability of an event must lie between 0 to 1. A definite or sure event has a probability 

of 1; while an impossible event has a probability zero.

Probability distribution is an important term that assigns the probabilities for the all 

possible outcomes for a finite set of events. There are different types of probability 

distributions. Normal distribution is most common of all. The normal distribution refers to 

type of the family of a continuous probability distributions defined in terms of the normal 

equation. 

Normal distribution is denoted by the equation with the random variable Y is -

Y = 

Where, is the mean, 

X is a random normal variable, 

 referred to the standard deviation, 

the value of is approximately 3.14, 

also e is approximately 2.71828. 

The random variable X defined in above normal equation is known as the normal 

random variable. 

exp(( ) 1

σ√2π

−1

2

x−μ

σ )

2

μ

σ

π

Normal Distribution 

Definition

Standard Normal 

Distribution

Normal Probability 

Distribution

Normal Distribution Curve

Properties of Normal 

Distribution

Inverse Normal Distribution

Log Normal Distribution

Bivariate Normal 

Distribution

Multivariate Normal 

Distribution

Cumulative Normal 

Distribution

Normal Distribution 

Standard Deviation

Truncated Normal 

Distribution

Half Normal Distribution

Skew Normal Distribution

Normal Distribution 

Percentile

Normal Distribution 

Example

Related Concepts

Confidence Interval Normal 

Distribution

Normal Approximation Binomial 

Distribution

Distribution Probability

Beta Probability Distribution

Binomial Distribution Probability

Binomial Distribution Probability 

Examples

Binomial Distribution Probability 

Formula

Conditional Probability Distribution

Related Formulas

Formula Normal Distribution

A Probability Distribution is an 

Equation that

Normal Distribution

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
Normal Distribution Definition Back to Top

Standard Normal Distribution Back to Top

Normal Probability Distribution Back to Top

The normal distribution is a probability distribution that associates the normal random 

variable X with a cumulative probability. The term bell curve is used to describe the 

mathematical concept called normal distribution. A bell curve graph depends on two 

factors, the mean and the standard deviation. The mean identifies the position of the 

center and the standard deviation determines the the height and width of the bell. All 

normal distributions look like a symmetric, bell-shaped curve.

The normal distribution, also called as Gaussian distribution, is a continuous distribution 

of great importance. A standard normal distribution is a normal distribution with mean 0 

and standard deviation 1. The Normal distribution is often called as the "bell curve", 

when the random variable takes the values in the set of real numbers. Standard normal 

distribution is a special case of the normal distribution, also called Z distribution. The pdf

of the standard normal distribution is same as normal distribution, except that it has = 

0 and = 1 

Normal distributions can be transformed to standard normal distributions by the 

formula:

z = 

where, X is a score from the original normal distribution, is the mean of the 

original normal distribution, and is the standard deviation of original normal 

distribution.

The normal distribution is a continuous probability distribution. This has several 

implications for probability.

1. The total area under the normal curve is equal to 1.

2. The probability that a normal random variable X equals any particular value is 0.

Free Math Tutor 

Online

Get Answers to all Math 

Problems with Expert Online 

Tutors, Try Now

μ

σ

X−μ

σ

μ

σ

Binomial Distribution Formula 

Example

Chi Square Distribution Formula

Related Calculators

Cumulative Normal Distribution 

Calculator

Normality Calculations

Area under Normal Curve 

Calculator

Related Worksheets

Normal Distribution Word Problems

Normal Distribution Worksheet

Standard Normal Distribution 

Worksheet

Binomial Distribution Worksheet

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
Normal Distribution Curve Back to Top

Properties of Normal Distribution Back to Top

Inverse Normal Distribution Back to Top

3. The probability that X is greater than a equals the area under the normal curve 

bounded by a and plus infinity (as indicated by the non-shaded area in the figure 

below).

4. The probability that X is less than a equals the area under the normal curve 

bounded by a and minus infinity (as indicated by the shaded area in the figure 

below).

Additionally, every normal curve (regardless of its mean or standard deviation) 

conforms to the following rules:

1. About 68% of the area under the curve falls within 1 standard deviation of the 

mean.

2. About 95% of the area under the curve falls within 2 standard deviations of the 

mean.

3. About 99.7% of the area under the curve falls within 3 standard deviations of the 

mean.

The normal distribution graph depends on two factors - the mean and the standard 

deviation. The mean of the distribution determines the location of the centre of the 

graph, and the standard deviation determines the height and width of the graph. When 

the standard deviation is large, the curve is short and wide. When the standard 

deviation is small, the curve is tall and narrow.

Normal Distribution Bell Curve

All normal distributions looks like a symmetric, bell-shaped curve, as shown below. The 

curve on the left is shorter and wider than the curve on the right, because the curve on 

the left has a bigger standard deviation.

Some properties of the normal distribution:

• The normal curve is a symmetric bell-shaped curve.

• 50% of the scores lie above and 50% below the midpoint of the distribution. 

• The total area under the curve is equal to 1.

• Curve is asymptotic to the x axis

• Mean, median, and mode are equal and located at the midpoint of the x axis.

The inverse normal probability distribution function will find the precise value at a given 

percent based upon the mean and standard deviation. the inverse Gaussian distribution 

is a two-parameter family of continuous probability distributions with support on (0, ). 

Its probability density function is given by 

 =

for and .

∞

f(x, μ, λ) [ exp λ

2πx3 ]

1/2 −λ(x−μ) )

2

2μ2x

x > 0, μ > 0 λ > 0

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
Log Normal Distribution Back to Top

Bivariate Normal Distribution Back to Top

Multivariate Normal Distribution Back to Top

Cumulative Normal Distribution Back to Top

Normal Distribution Standard Deviation Back to Top

where, is the mean and is the shape parameter.

A continuous distribution in which the logarithm of a variable has a normal distribution. 

Log-normal distributions can model a random variable X where log(X) is normally 

distributed. The probability density functions for the log normal distribution is

 = 

The bivariate normal distribution is the statistical distribution with probability density 

function. 

The bivariate normal distribution has two variables, and , possibly dependent. 

 =

where is correlation of and .

The multivariate normal distribution model extends the univariate normal distribution 

model to fit vector observations. A p-dimensional vector of random variables, 

 , is said to have a multivariate normal distribution if its 

density function is of the form 

 =

The cumulative distribution function describes the probability that a real-valued random 

variable X with a given probability distribution will be found at a value less than or equal 

to x. A cumulative distribution function gives the cumulative area with a pdf. The shape 

of the various distribution cdfs are distinctive, and the cdf is merely another way of 

illustrating the distribution. The cumulative distribution function is probability of a value 

less than x.

For the normal distribution, this gives in terms of the complimentary error function 

by

 =

μ λ

P(x) exp 1

s x √2π

−(lnx−M)

2

2s2

x1 x2

P(, ) x1 x2 exp[ ] 1

2πσ1σ2 1−ρ √ 2

−z

2(1− ) ρ2

ρ x1 x2

X = , ,..., X1 X2 Xp

−∞ < < ∞ Xi i = 1, . . . , p

f(X)

f(x) ( |∑ exp( (X − m) (X − m)) 1

2π )

ρ

2 |

1/2 −1

2 ∑−1

cdf = P(< x) = ∫ P(x)dx x

−∞ erfc( ( )) 1

2

−1

√2

x−μ

σ

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
Truncated Normal Distribution Back to Top

Half Normal Distribution Back to Top

Skew Normal Distribution Back to Top

The Normal distribution is often called as the "bell curve", when the random variable 

takes the values in the set of real numbers. 

Normal Distribution Formula

The normal distribution is defined by the equation with the random variable is 

 = 

where, is a normal random variable, is the mean, is the standard deviation. The 

distribution with and is called the standard normal distribution or the unit 

normal distribution. 

The truncated normal distribution assumed for x generalizes the one parameter half 

normal distribution, by allowing the normal distribution, which is truncated below at zero, 

to have a nonzero mode. The truncated normal distribution contains an additional 

parameter to be estimated and so provides a more flexible representation of the 

pattern of the efficiency in the data. 

The truncated normal density function for is given by

 = 

The half-normal distribution is a normal distribution with mean 0 and parameter limited 

to the domain, lies between to .

It has probability and distribution functions given by

 = , .

The half-normal distribution is a special case of the truncated normal distribution, with 

the estimated being mode 0. The probability density function for the half-normal 

distribution with variance parameter , in the domain as defined as

 = 

The skew normal distribution is a continuous probability distribution that generalized the 

normal distribution to allow for non-zero skewness. 

The skew-normal distribution has density as follows:

f(x, μ, ) σ2 exp(( ) 1

σ√2π

−1

2

x−μ

σ )

2

x μσ

μ =0 =1 σ2

μ

f(x) x ≥ 0

f(x) exp(− ) 1

√2πσφ( ) μ

σ

(x−μ)

2

2σ2

x 0 ∞

f(x, σ) exp( ) √2

σ π √

−y2

2σ2 y > 0

θ 0 ≤ x ≤ ∞

f(x) exp( ) 2θ

π

−θ2x2

π

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
Normal Distribution Percentile Back to Top

Normal Distribution Example Back to Top

 = for 

where is the standard normal cdf, and is constant.

• For = 0, the skew-normal distribution is the normal 

distribution, and if | | is relatively small, the distribution is close 

to the normal.

• For large | |, the distribution is more skewed.

The percentile of a normal distribution with mean and standard deviation is easily 

related to the percentile of the standard normal distribution. An important aspect of the 

normal distribution is that we can easily find the percentiles of the distribution.

Percentile for normal( ) = + (Percentile for standard normal) 

If z is the desired percentile for the standard normal distribution, then the desired 

percentile for the normal distribution is z standard deviation from .

Listed below are some of the normal distribution examples.

Solved Examples

Question 1: An average light bulb manufactured by the Acme Corporation 

lasts 300 days with a standard deviation of 50 days. Assuming that bulb life 

is normally distributed, what is the probability that an Acme light bulb will 

last at most 365 days? 

Solution: 

Given a mean score of 300 days and a standard deviation of 50 

days, we want to find the cumulative probability that bulb life is less 

than or equal to 365 days. Thus, we know the following:

1) The value of the normal random variable is 365 days.

2) The mean is equal to 300 days.

3)The standard deviation is equal to 50 days.

We enter these values into the Normal Distribution Calculator and 

compute the cumulative probability. The answer is: P( X < 365) = 

0.90. Hence, there is a 90% chance that a light bulb will burn out 

within 365 days.

Question 2: Suppose scores on an IQ test are normally distributed. If the 

test has a mean of 100 and a standard deviation of 10, what is the 

probability that a person who takes the test will score between 90 and 110?

Solution: 

Here, we want to know the probability that the test score falls 

between 90 and 110. The "trick" to solving this problem is to realize 

f(x)

exp( )φ(λx) 2

√2π

−x2

2

−∞ ≤ x ≤ ∞

φ λ

λ

λ

λ

μ σ

μ, σμ σ

μ

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014
◀◀ Binomial Distribution Bernoulli Distribution ▶▶

the following:

P( 90 < X < 110 ) = P( X < 110) - P( X < 90)

We use the Normal Distribution Calculator to compute both 

probabilities on the right side of the above equation.

1) To compute P( X < 110), we enter the following inputs into the 

calculator: The value of the normal random variable is 110, the 

mean is 100, and the standard deviation is 10. We find that P( X < 

110) is 0.84.

2) To compute P( X < 90), we enter the following inputs into the 

calculator: The value of the normal random variable is 90, the mean 

is 100, and the standard deviation is 10.

We find that P( X < 90) is 0.16. We use these findings to compute 

our final answer as follows:

P( 90 < X < 110 ) = P( X < 110) - P( X < 90)

P( 90 < X < 110 ) = 0.84 - 0.16

P( 90 < X < 110 ) = 0.68

Thus, about 68% of the test scores will fall between 90 and 110.

More topics in Normal Distribution 

Z Score Table

Related Topics 

Math Help Online Online Math Tutor

*AP and SAT are registered trademarks of the College Board.

About Us | Careers | Contact Us | Blog | Homework Help | Teaching Jobs | Search Lessons | Answers | Calculators | Worksheets | Formulas 

Copyright © 2014 - NCS Pearson, All rights reserved.

http://math.tutorvista.com/statistics/normal-distribution.html

10/29/2014