Regarding relation between value of Chi-Square and Z-square

[Sundeep Kumar]

Respected members, 

My doubt is - I have read in a que that when degree of freedom is 1, value of Chi-square and Z-Square will be identical..... Can someone through light on this?

with regards,

Sundeep Kumar 

Research Scholar & Air Warrior 

LPU Univ 

[Neeraj Kaushik]

Dear Sundeep

Plz note that, unlike many statistical tests, Z-value does not have a table containing the degree of freedom.

Plz refer to https://byjus.com/maths/z-score-table/

Z-values represent the area under the standard normal curve.

On the other hand, the Chi-square test follows the Chi-square distribution and its value depends on DOF and LOS both. May refer https://www.statology.org/how-to-read-chi-square-distribution-table/

Now, please explain what exactly your question is.

Best wishes

Neeraj

 

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[Sundeep Kumar]

Good Morning Dr. Neeraj Sir,

Thanks for reaching out.  As you asked the exact question was :- 

with regards,

Sundeep Kumar

Research Scholar from LPU & 

Air Warrior of Indian Air Force 

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--

sandy

[Rohtash bhall]

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[Neeraj Kaushik]

Dear Sundeep

As Rohtash Bhall replied (in the subject of the email itself rather than the body of the email)

the chi-square distribution is identical to the square of the standard normal distribution. In other words, it is equivalent to the square of a Z-score.

Let me explain this.

When dof=1, we get n=2

Now if we have just the 2 values then z=(x-mean)/sd and Chi-sq = Summation of (O-E)^2 / E

In the formula of the z-test, there is a difference between the x & mean while in Chi-sq, other than the difference, a square is also there.

Now z=1.96 (at 5% alpha) and Chi-sq at 5% dof=1 is 3.84

Now you can see why the z2 (1.96*1.96)=Chi-square value (3.84) at df=1

Best wishes

Neeraj 

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[Narain]

Dear Sundeep,

I am attaching below the extract of the appendix of the book Basic Econometrics by Gujarati & Porter (page 819)

As evident from the definition of the Chi-Sq distribution, if degree of freedom is one (k=1 in above) the Chi-Sq distribution is the same as the square of Z (which is Normal distribution). In other words,

Let Z is standardized normal distributed then, Z^2 shall be Chi-Square distributed with d.o.f. equal to one.

Also, the same appendix explains the similar relations between distributions F as well as t-distribution to each other. For example. square of t (with d.o.f.=k) is the same as F (with Num d.o.f.=1 and Den d.o.f.=k).

I hope this clarifies your doubt.

With good wishes ...