Standardizing - the standardized normal probability function
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 ,
is transformed. The standard deviation σ becomes '1', while the mean μ becomes equal to zero.
The value of z is calculated from the formula:
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%.
0.0 - 0.5 standard deviation
0.5 - 1.0 standard deviations
1.0 - 1.5 standard deviations
*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.
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).
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)
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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|goodness of fit|
|distrib. sample mean|
|mean,median comprd 1|
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|type I & II errors|
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