Home >>STATISTICS, Section 1, normal distribution 3

More on z-tables

From z-tables the area under the curve of f(z) can be determined.

Z is read from the extreme left(- ∞) up to any positive value of z.

You can download a Z-Table here. (.pdf)

This area Φ(z), is called the cumulative distribution function.

Note that the total area under the curve is 1.

Hence when z = 0 (in the middle) the area is 0.5 . Remember, z is read from the extreme left. So the referred area is to the left of centre.

If we want to measure the particular area(and hence cumulative probability) between discrete values we use a different form of the function: P(Z<z)

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The case of P(Z<z)

So to evaluate P(Z<z) all we have to do is read off the value of Φ(z) for z from the tables.

Since in this case,

Φ(z) = P(Z<z)

Example

i) For a Standardized Normal Distribution N(0,1), evaluate the Cumulative Distribution Function(CDF) for the condition where z<1.9

ii) Sketch a curve to illustrate your answer.

i)

P(Z<1.9) = Φ(1.9) = 0.9713

ii)

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The case of P(Z>z)

(area under the curve to the right of any value z) =

(area under whole curve) - (area under curve up to value z)

1  -  P(Z < z)  =   P(Z > z)

or

P(Z > z)>  =   1  -   P(Z < z)

Example

i) For a Standardized Normal Distribution N(0,1), evaluate the Cumulative Distribution Function(CDF) for the condition where z>1.9 .

ii) Sketch a curve to illustrate your answer.

i)

P(Z >1.9)  =   1  -   P(Z <1.9)

           =   1 -  0.9713

     =   0.0287

ii)

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The case of P(Z>-z)

By symmetry,

(area under the curve to the left of a positive value of z) =

(area under the curve to the right of a negative value of z)

P(Z<z)  =  P(Z>-z)   

Example

i) For a Standardized Normal Distribution N(0,1), evaluate the Cumulative Distribution Function(CDF) for the condition where z>-1.9

ii) Sketch a curve to illustrate your answer.

i)

     P(Z>-1.9) = P(Z<1.9)           

P(Z<1.9) = Φ(1.9) = 0.9713

ii)

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The case of P(Z<-z)

By symmetry,

(area under the curve to the left of a negative value of z) =

(area under the curve to the right of a positive value of z)

P(Z < -z)  =   P(Z > z)

from above,      

P(Z > z)   =   1  -  P(Z < z)

therefore,

P(Z < -z)  =   1  -  P(Z < z)   

Example

i) For a Standardized Normal Distribution N(0,1), evaluate the Cumulative Distribution Function(CDF) for the condition where z<-1.9

ii) Sketch a curve to illustrate your answer.

i)

   P(Z<-1.9)  =   1  -  P(Z <1.9)

 =   1 - Φ(1.9) =    1 -  0.9713   = 0.0287            

P(Z<-1.9) =   0.0287      

ii)

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