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Standardizing - the standardized normal probability function

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

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 ,

is transformed.

Making the substitution standard deviation σ = '1', and mean μ = zero, the equation becomes:

The value of z is calculated from the 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.

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

*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.

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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:

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

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.

μ mean, σ² variance

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

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)

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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 .

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