The normal distribution, also known as the Gaussian distribution, is one of the most important concepts in statistics and engineering. It describes how continuous variables are distributed when influenced by many small random factors, such as manufacturing tolerances, measurement errors, or natural variations.
Unlike discrete distributions, continuous probability distributions allow variables to take any value within a range. Instead of assigning probability to a single value, probabilities are represented by the area under a curve. The total area under the curve equals 1, representing all possible outcomes.
This page explains the fundamentals of the normal distribution, including mean and variance, probability density functions, standardisation using z-values, and how to use normal distribution tables for probability calculations.
Probability distributions for continuous random variables differs from discrete distributions variables in the following important aspects:
1) A variable can take on any, of an infinite number of values within the range
2) The probability of any specific value is zero.e.g The probability 50mm ± 0 is zero.
3) Probabilities are expressed in terms of an area under a curve of the continuous probability distribution.
In engineering as in most other disciplines it is useful to determine patterns
which indicate the probability of variables and events e.g in the mass production of 50mm diameter
shaft on a lathe... what is the probability of the shafts being within the acceptable tolerance limits . Probability curves show
the variable on the x axis and the probability f(x) of the variable on the y axis.
The total area under the curve represents the probability of all possible events and equals 1
There are a number of continuous probability distributions which are used for different
probability conditions. The most widely used is the normal probability distribution also called the Gaussian probability distribution. This represents the
probability conditions most representative of events that occur in the natural world subject
to countless variables.
The normal probability distribution has the following characteristics
1) The graphed distribution curve is bell shaped and is symmetrical about the mean value
2) The mean, median, and mode are the same.
3) The distribution is defined by the mean span>μ and the standard deviation σ
4) The distribution extends from negative infinity to positive infinity
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f(x) = probability function. (values between 0 and 1) F(x) = probability distribution function. Φ (z) = Probability distribution function.(Standardised probability ) μ = mean σ 2 = variance σ = Standard deviation z = (x - μ ) / σ Equation to standardise probability distribution function/ X = random variable . (value identified by x ) x = value of random variable |
The mean value of a distribution is denoted by μ and is defined by
The variance of a continuous distribution σ2 and is defined by
f(x) is the probability function of the random variable X under consideration.
The defintion of f(x) for a normal distribution is provided below.
The probability that is a single trial the variable will assume a value ≤ x is called the distribution function
A typical normal probability curve drawn with a mean value μ= 0 and a variance σ 2 = 1 is shown above.
For this case the probability that is a single trial the variable will assume a value ≤ x becomes
(This function is plotted on the webpage Standardised Normal Distribution Table:
The values for this table have been obtained using the standard functions available on Microsoft Excel.)
This corresponds to a probability function
Note:
If v = (t - μ ) / σ then dv / dt = 1 / σ . Therefore dt = dv σ
Now considering the equation above for F(x) and letting v = (t - μ ) / σ
This is clearly = Φ(z) where z = (x - μ ) / σ
that is
It is very easy to obtain the probability that a variable x meets the criteria ( a < x < b ) using the tabled values by using the relationship.
Standardised Normal Distribution Table
The webpage linked provides values for Φ(z) for the range z = 0 to 4,19 .
It can be used for values -4,19 to 4,19 by using the relationship Φ(-z) = 1 - Φ(z)
a) Determining the probability of value is between a = μ - σand setting b = μ + σ using the tables result in simply looking up the difference in values Φ (1) - Φ (-1) = 0.841345 - (1- 0.841345) = 0.68269
b) Determining the probability of value is between a = μ - 2σand setting b = μ + 2σ using the tables result in simply looking up the difference in values Φ (2) - Φ (-2) = 0.97725 - (1- 0.97725) = 0.9545
c) Determining the probability of value is between a = μ - 3σand setting b = μ + 3σ using the tables result in simply looking up the difference in values Φ (2) - Φ (-2) = 0.0.99865 - (1- 0.0.99865) = 0.9973
The above simple calculations show that if a population conforms to a normal probability distribution
About 2/3 of the population values lie within the range μ - σ and μ + σ
About 95% of the population values lie within the range μ - 2σ and μ + 2σ
About 99% of the population values lie within the range μ - 3σ and μ + 3σ
The relationship applies to all normal distribution.
Example using the tables.Determine the probability P( x ≤ 10,5 ) for a x is normal with a mean (μ) = 10 and a standard deviation (σ ) = 2.
There is a 60% probability of x being less than or equal to 10,5
The values of p in the binomial distribution and μ and σ are the called the parameters of the distributions and it is frequently necessary to obtain values for these parameters from a given sample of a population. It is often sufficient to obtain an estimate of μ ( xm is approximately equal to μ ) using the equation ref.Sample Variables
and it is possible to obtain an estimate of the variance ( as sx is approximately = to σ using the equation ref.Sample Variables
In the case of the binomial distribution the p is easily estimated as being approximately
p = approximately xm / n
For normal probability distributions good evaluations of the parameters
can be obtained using normal probability paper. This is graph paper with
the lines arranged such that a normal probability distribution is plotted as a straight line.
Normal probability paper can be downloaded free from
Weibull Probability Plotting Papers
A normal distribution is a continuous probability distribution that is symmetric about its mean, where most values cluster around the mean and probabilities decrease as values move away.
A z-score is a standardised value that shows how many standard deviations a data point is from the mean. It is calculated as z = (x − μ) / σ.
It models real-world phenomena such as measurement errors, manufacturing tolerances, and natural variations, making it essential in engineering and statistics.
The area under the normal distribution curve represents probability. The total area equals 1, meaning 100% of all possible outcomes.
Approximately 68% of values lie within ±1 standard deviation, 95% within ±2, and 99.7% within ±3.