Section 1 : The Normal Distribution(2)
 

[standardizing][z tables][N(μ,σ2) , N(0,1) ]

[cumulative distribution function P(Z < z)]

 

 

 

Standardizing - the standardized normal probability function

 

normal and standardized distributions compared

 

Standardizing is the conversion of a normal distribution into a more useful form where :

the curve is symmetrical about the line z = 0 (the mean μ = 0)

the area below the curve and the z-axis is '1'

the units of z are 'standard deviations'    σ *

*z is also called 'the standard score' , 'sigma' , z-score

The f(x) function is transformed into the Φ(z) function.

Φ(z) is called the standardized normal probability function. This has a particular value for any value of z (this is what we look up in z-tables).

The normal probability density function f(x) equation ,

normal distribution equation expanded

is transformed. The standard deviation σ becomes '1', while the mean μ becomes equal to zero.

standardized normal probability function

 

The value of z is calculated from the formula:

the z-score formula

Example

A student gains a score of 57% in a test.

i) If the mean result is 47% and the standard deviation 20% , calculate the z-score for the student.

z-score problem #1

ii) Using the table, estimate what % of students scored lower than 47%.

values within*:
probability
0.0 - 0.5 standard deviation
0.191 (19.1%)
0.5 - 1.0 standard deviations
0.150 (15%)
1.0 - 1.5 standard deviations
0.092 (9.2%)

*on one side of the mean

Between the score 57% and the mean 47% represents 0.5 of a standard deviation(calculated in (i ).

According to the table this represents 19.1% of the scores.

Between the score 0% and the mean 47% represents 3 standard deviations.

This is half the total area under the curve (i.e. 50% of the scores).

So adding together these results: 19.1% + 50%.

The total % of students with scores less than 57% is 69.1%.

iii) Sketch a normal distribution curve illustrating the problem.

standard normal distribution problem #1

 

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z-tables

z-tables give the area under the f(z) graph between minus infinity and a particular value of z. This area is called the cumulative probability function or 'phi of z' , written Φ(z).
Mathematically this is expressed as:

derivation of CDF
This is part of the table used by the Edexcel Exam board, UK.

how to use z-tables

Readings of z are incremental by 0.01, from 0.00 to 4.00 .

The cumulative probability function Φ(z) ranges from 0.5000 to 1.0000 .

Only positive values of z are given. Using the symmetry of the curve, negative values can easily be inferred (see below).

 

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Nomenclature for Normal Distributions - N(μ,σ2) , N(0,1)

This is simply a short-hand way of describing a normal distribution.

N(μ,σ2)

μ mean, σ2 variance

So a standardized normal distribution with mean (μ) = 0 and variation (σ2) = 1 is written:

N(0,1)

Other distributions give flatter or sharper 'bell curves' depending on their value for σ2 .

N(0, 0.5) is a sharp curve(less range)

N(0, 2.0) is a shallow curve(wide range)

 

standard deviations compared

 

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Cumulative Distribution Function(CDF)   P(Z < z)

This form of the CDF is the area under the bell curve to one side of a typical value of z . The area gives a value for the probability of z in the stated range .

P(Z<z)

P(Z>z)

P(Z>-z)

P(Z,-z)

 

 

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