Training and consultancy for testing laboratories.

Posts tagged ‘Normal distribution’

Descriptive statistics of Excel Data Analysis Tools

It is convenient for us to use Excel to analyze our data. Indeed, Excel comes equipped with a Descriptive Statistics tool in the Data Analysis add-in package, termed Analysis ToolPak or ATP. With this tool, we get as many as 16 different descriptive statistical parameters without having to enter a single function on the worksheet….

Descriptive statistics of Excel Data Analysis Tools


R computations with normal distribution

Std normal distribution density A

R computations with normal distributions

There are various R functions which are useful for computation with normal distributions, such as pnorm( ), qnorm( ), and dnorm( ).

The pnorm( ) function gives the cumulative distribution function, and the alphabet ‘p’ stands for probability.  The qnorm( ) is for quantiles whilst the dnorm( ) function, the density.

Let’s use the statistical notation for normal distribution: X ~ N(µ,sigma2).  We shall illustrate the usage of these R functions.

R function pnorm( )

For example, let X ~ N(8,4), then

(a)  the probability P(X < 2) can be computed via pnorm( ) in several different ways:

> pnorm(2,mean=8,sd=2)  #P(X<=2) in N(8,4)

[1] 0.001349898

> pnorm(2,8,2)  #P(X<=2) in N(8,4) simplified

[1] 0.001349898

(b)  the probability P(X < 1.96) for x ~ N(8,4) by R language is:

> pnorm(1.96,8,2)  #P(X<=1.96) in N(8,4)

[1] 0.001263873

Remember that for f(1.96) = 0.975 and f(1.645) = 0.950, respectively from the statistics table, the R gives us the same answers:

> pnorm(1.96,0,1)  #P(X<=1.96) in N(0,1)

[1] 0.9750021

> pnorm(1.645,0,1)  #P(X<=1.645) in N(0,1)

[1] 0.9500151

> pnorm(1.645)  #P(X<=1.645) in N(0,1) simplified

[1] 0.9500151


And, when P(X < -1.645), the R result indicates the area on the left hand side of the normal distribution curve:

> pnorm(-1.645)  #P(X<=-1.645) in N(0,1) simplified

[1] 0.04998491


 R function qnorm( )

In layman’s language, a quantile is where a series of sample data is sub-divided into equal proportions. In statistics, we divide a probability distribution into areas of equal probability. The simplest division that can be envisioned is into two equal halves, i.e., 50%.

The R function: qnorm( ) is used to compute the quantiles for the standard normal distribution using its density function f.

For example,

> qnorm(0.95)  #95.0% quantile of N(0,1)

[1] 1.644854

> qnorm(0.975)  #97.5% quantile of N(0,1)

[1] 1.959964


 R function dnorm( )

The density of a Gaussian formulae for normal distribution can be shown to be close to 0.4 when x = 0.

The R function dnorm(0) indeed gives the same result as below:

> dnorm(0)  # Density of N(0,1) evaluated at x= 0

[1] 0.3989423


Further remarks

Like pnorm( ), the functions qnorm( ) and dnorm( ) can also be used for normal distributions with non-zero mean and non-zero standard deviation or variance, simply by supplying the mean and standard deviation as extra arguments.

For example, for the N(8,4) distribution,  the results are self-explanatory:

> qnorm(0.975,8,2)  # 97.5% quantile of N(8,4)

[1] 11.91993

> dnorm(1,8,2)  #Density of N(8,4) at x=1

[1] 0.0004363413

> dnorm(4,8,2)  #Density of N(8,4) at x=4

[1] 0.02699548



Review of normal probability distribution – Part II

Histogram AA

Review of normal probability distribution Part II

Review of normal probability distribution – Part I

Histogram A

Review of normal probability distribution Part I


How to use Excel on AD statistic test for data normality?

A-D Calculation on 25 data Chloride

How to use R to generate random numbers?