12 December 2017

Important properties of the Normal distribution

December 12, 2017


The normal distribution was first described by Abraham Demoivre (1667-1754) as the limiting form of binomial model in 1733. Normal distribution was rediscovered by Gauss in 1809 and by Laplace in 1812. Both Gauss and Laplace were led to the distribution by their work on the theory of errors of observations arising in physical measuring processes particularly in astronomy. Here I will show you some important properties of Normal distribution

1. The normal curve is “bell shaped” and symmetrical in nature. The distribution of the frequencies on either side of the maximum ordinate of the curve is similar with each other.

2. The maximum ordinate of the normal curve is atclip_image002. Hence the mean, median and mode of the normal distribution coincide.

3. It ranges between clip_image004 to clip_image006

4. The value of the maximum ordinate is clip_image008

5. The points where the curve change from convex to concave or vice versa is at clip_image010

6. The first and third quartiles are equidistant from median.

7. The area under the normal curve distribution are:

a. clip_image012 covers 68.27% area

b. clip_image014 covers 95.45% area

c. clip_image016 covers 99.73% area


8. When μ = 0 and σ = 1, then the normal distribution will be a standard normal curve. The probability function of standard normal curve is


The following table gives the area under the normal probability curve for some important value of Z.

Distance from the mean ordinate in

Terms of ± σ

Area under the curve

Z = ± 0.6745


Z = ± 1.0


Z = ± 1.96


Z = ± 2.00


Z = ± 2.58


Z = ± 3.0


9. All odd moments are equal to zero.

10. Skewness = 0 and Kurtosis = 3 in normal distribution.

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06 December 2017

Some Key Abbreviations used in Statistics

December 06, 2017


FNR - False Negative Ratio
FPR - False Positive Ratio
iff  - if an only if
i.i.d. - independent and identically distributed
IRQ - inter-quartile range
pdf - probability density function
LSE - Least Square Error
ML - Maximum Likelihood
MSE - Mean Square Error
PDF – probability distribution function
RMS - Root Mean Square Error
r.v. - Random variable
ROC - Receiver Operating Characteristic
SSB - Between-group Sum of Squares
SSE - Error Sum of Squares
SSLF - Lack of Fit Sum of Squares
SSPE - Pure Error Sum of Squares
SSR - Regression Sum of Squares
xxiv - Symbols and Abbreviations
SST - Total Sum of Squares
SSW - Within-group Sum of Squares
TNR - True Negative Ratio
TPR - True Positive Ratio
VIF - Variance Inflation Factor

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27 November 2017

Multivariate normality Tests with R - Mardia's Test, Henze-Zirkler, Royston

November 27, 2017
Most multivariate techniques, such as Linear Discriminant Analysis (LDA), Factor Analysis, MANOVA and Multivariate Regression are based on an assumption of multivariate normality. So, In this post, I am going to show you how you can assess the multivariate normality for the variables in your sample. The above test multivariate techniques can be used in a sample only when the variables follow a Multivariate normal distribution.

For this, you need to install a package called MVN

Type install.packages("MVN")and then load the package by library(“MVN”)

There are 3 multivariate normality tests available in this package
  1. Mardia's Multivariate Normality Test
  2. Henze-Zirkler's Multivariate Normality Test
  3. Royston's Multivariate Normality Test

Let's discuss one by one,

I am using inbuilt trees data here data(“trees”). This data consists of 3 variables i.e Girth, Height and volumeFirst, we use Mardia's test to verify the normality for the above data

Type mardiaTest(trees)

This will return the results of normality test with 3 variables in it. Data is not multivariate normal when the p-value is less than 0.05

If you want to check Multivariate normality of selected variables. Create a subset. Here I am creating a subset under name trees1 that includes 1st and 3rd variables


Now let's check normality of trees1 using Henze-Zirkler's Test

Type hzTest(trees1)To use Royston's Multivariate Normality Test

Type roystonTest(trees1)

So, That is how you can test the multivariate normality of  variables using R. Give your queries and suggestions in comment section below. Subscribe my blog and YouTube channel for more posts and videos.

12 November 2017

How to Search and Select UGC listed Academic journals?

November 12, 2017

Visit the UGC Journal list to search and select the journals. I strongly recommend you to publish fro one of these journals so that you can stay safe from Predatory/Fake/Unethical Academic journals.

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19 October 2017

Generating Even / Odd numbers using R

October 19, 2017
In the below tutorial I have explained how you can generate Even or Odd numbers using R. You can generate using any one of the following methods.
Method 1:
Generating even numbers that lie between 1 to 100
          even <- seq(1,100,2)

Method 2:
Generating 100 odd numbers starting from 1
         odd <- seq(1,by=2, len=100)

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13 October 2017

Verifying and Selecting a Scopus indexed journals - Video

October 13, 2017
In the following video, I will show you how you can verify if a journal is Scopus indexed as is claim and How to select journal based on your discipline.
Visit the Scopus website for verifying.

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