Important properties of the Normal distribution

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 at. Hence the mean, median and mode of the normal distribution coincide.

3. It ranges between to

4. The value of the maximum ordinate is

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

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

7. The area under the normal curve distribution are:

a. covers 68.27% area

b. covers 95.45% area

c. 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	0.50
Z = ± 1.0	0.6826
Z = ± 1.96	0.95
Z = ± 2.00	0.9544
Z = ± 2.58	0.99
Z = ± 3.0	0.9973

9. All odd moments are equal to zero.

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

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Some Key Abbreviations used in Statistics

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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Indian and international number system–explained

It is common people confuse between lakhs and millions, and crores and billions. So its common question people ask .  how many lakhs makes a million? Well, I present you clear answer here. Indian number system use terms such as ones, tens, hundreds, thousands and then lakhs and crores. While International number system use terms such as ones, tens, hundreds, thousands, millions and so on.  For better understanding refer the table below.

Number	Indian system	Number	International system
1	Ones	1	Ones
10	Tens	10	Tens
100	Hundreds	100	Hundreds
1,000	Thousands	1,000	Thousands
10,000	Ten Thousands	10,000	Ten Thousands
1,00,000	One lakh	100,000	Hundred thousand
10,00,000	Ten lakhs	1,000,000	One million
1,00,00,000	One Crore	10,000,000	Ten million
10,00,00,000	Ten Crore	100,000,000	Hundred million
1,00,00,00,000	Arab	1,000,000,000	One billion
10,00,00,00,000	Ten Arab	10,000,000,000	Ten billion
10000,00,00,000	Kharab	100,000,000,000	Hundred billion
100000,00,00,000	Ten Kharab	1,000,000,000,000	One trillion
1000000,00,00,000	Neel	10,000,000,000,000	Ten trillion
10000000,00,00,000	Ten Neel	100,000,000,000,000	Hundred trillion

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Facts about infinity–What is Infinity?

What if I say Infinity is simple to understand ? Will you agree?. Well, Here I give you some facts about Infinity to better understand it. After this you can accept that infinity is really simple. don't forget to subscribe my blog.

• Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
• Infinity, is not big, it's not huge, it's not tremendously large, or it's neither extremely humongously enormous. It is Endless!
• If there is no reason something should stop, then it is infinite.
• Infinity is not a real number, it is an idea. An idea of something without an end.
• Infinity itself, symbolized by a figure that resembles a sideways 8, is not a number. You could write it in the format of an infinite number, such as a 1 followed by an infinite number of zeroes. This, however, is a concept, not a number.
• Here comes a beautiful fact, Negative infinity is less than any Real number,
  and Infinity is greater than any Real number.

These are just common facts about infinity for simple understanding. In advanced discussions, such as Theoretical applications of physical infinity , Cosmology and etc.., infinity is dealt in complex manner which is not a simple concept. Unless a care is exercised, especially when dealing infinity as a common number, paradoxes arise readily.

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List of Greek alphabets, name ,Pronunciation and its English equivalent

The Greek alphabets have been used to write the Greek language since the 8th century BC. Here is a list of alphabets, its name , How it is pronounced and What is its English equivalent that is used in Maths and statistics. Don't forget to subscribe the blog for more interesting updates.

Upper Case	Lower Case	Greek Letter Name	English Equivalent	Pronunciation
Α	α	Alpha	a	al-fa
Β	β	Beta	b	be-ta
Γ	γ	Gamma	g	ga-ma
Δ	δ	Delta	d	del-ta
Ε	ε	Epsilon	e	ep-si-lon
Ζ	ζ	Zeta	z	ze-ta
Η	η	Eta	h	eh-ta
Θ	θ	Theta	th	te-ta
Ι	ι	Iota	i	io-ta
Κ	κ	Kappa	k	ka-pa
Λ	λ	Lambda	l	lam-da
Μ	μ	Mu	m	m-yoo
Ν	ν	Nu	n	noo
Ξ	ξ	Xi	x	x-ee
Ο	ο	Omicron	o	o-mee-c-ron
Π	π	Pi	p	pa-yee
Ρ	ρ	Rho	r	row
Σ	σ	Sigma	s	sig-ma
Τ	τ	Tau	t	ta-oo
Υ	υ	Upsilon	u	oo-psi-lon
Φ	φ	Phi	ph	f-ee
Χ	χ	Chi	ch	kh-ee
Ψ	ψ	Psi	ps	p-see
Ω	ω	Omega	o	o-me-ga

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List of Numeral symbols

List of numeral symbols in European, Roman, Hindu Arabic and Hebrew

Name	European	Roman	Hindu Arabic	Hebrew
zero	0		٠
one	1	I	١	א
two	2	II	٢	ב
three	3	III	٣	ג
four	4	IV	٤	ד
five	5	V	٥	ה
six	6	VI	٦	ו
seven	7	VII	٧	ז
eight	8	VIII	٨	ח
nine	9	IX	٩	ט
ten	10	X	١٠	י
eleven	11	XI	١١	יא
twelve	12	XII	١٢	יב
thirteen	13	XIII	١٣	יג
fourteen	14	XIV	١٤	יד
fifteen	15	XV	١٥	טו
sixteen	16	XVI	١٦	טז
seventeen	17	XVII	١٧	יז
eighteen	18	XVIII	١٨	יח
nineteen	19	XIX	١٩	יט
twenty	20	XX	٢٠	כ
thirty	30	XXX	٣٠	ל
forty	40	XL	٤٠	מ
fifty	50	L	٥٠	נ
sixty	60	LX	٦٠	ס
seventy	70	LXX	٧٠	ע
eighty	80	LXXX	٨٠	פ
ninety	90	XC	٩٠	צ
one hundred	100	C	١٠٠	ק

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Basic symbols in Mathematics and Statistics

Image via Pixabay

Basic math symbols

Symbol	Symbol Name	Meaning / definition	Example
=	equals sign	equality	3 = 1+2 3 is equal to 1+2
≠	not equal sign	inequality	2 ≠ 3 2 is not equal to 2
≈	approximately equal	approximation	sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y
>	strict inequality	greater than	3 > 2 3 is greater than 2
<	strict inequality	less than	2 < 3 2 is less than 3
≥	inequality	greater than or equal to	5 ≥ 4, x ≥ y means x is greater than or equal to y
≤	inequality	less than or equal to	4 ≤ 5, x ≤ y means x is greater than or equal to y
( )	parentheses	calculate expression inside first	2 × (3+5) = 16
[ ]	brackets	calculate expression inside first	[(1+2)*(1+5)] = 18
+	plus sign	addition	1 + 1 = 2
−	minus sign	subtraction	2 − 1 = 1
±	plus - minus	both plus and minus operations	3 ± 5 = 8 and -2
∓	minus - plus	both minus and plus operations	3 ∓ 5 = -2 and 8
*	asterisk	multiplication	2 * 3 = 6
×	times sign	multiplication	2 × 3 = 6
∙	multiplication dot	multiplication	2 ∙ 3 = 6
÷	division sign / obelus	division	6 ÷ 2 = 3
/	division slash	division	6 / 2 = 3
–	horizontal line	division / fraction
mod	modulo	remainder calculation	7 mod 2 = 1
.	period	decimal point, decimal separator	2.56 = 2+56/100
ab	power	exponent	23 = 8
a^b	caret	exponent	2 ^ 3 = 8
√a	square root	√a · √a  = a	√9 = ±3
3√a	cube root	3√a · 3√a  · 3√a  = a	3√8 = 2
4√a	fourth root	4√a · 4√a  · 4√a  · 4√a  = a	4√16 = ±2
n√a	n-th root (radical)		for n=3, n√8 = 2
%	percent	1% = 1/100	10% × 30 = 3
‰	per-mille	1‰ = 1/1000 = 0.1%	10‰ × 30 = 0.3
ppm	per-million	1ppm = 1/1000000	10ppm × 30 = 0.0003
ppb	per-billion	1ppb = 1/1000000000	10ppb × 30 = 3×10-7
ppt	per-trillion	1ppt = 10-12	10ppt × 30 = 3×10-10

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