Calculating Standard Normal Distribution in MS SQL

I am trying to convert an Excel function to an MS SQL function. One of the methods used in the VBA code is the Standard Normal Distribution function NormSDist(arg1 as double) as double. Does anyone know how to calculate this using T-SQL?

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Patrick MatthewsConnect With a Mentor Commented:
Hello garethmanuel,

The formula for the cumulative normal distribution is:

result = (1 / sqrt(2 * pi)) * (e ^ (-(z ^ 2) / 2))

e is Euler's number, the basis for natural logarithms

z is:    (value being considered) / st dev

Incorporate that into your SQL statements.


STDEV function
Returns the statistical standard deviation of all values in the given expression.

STDEV ( expression )


Is a numeric expression. Aggregate functions and subqueries are not permitted. expression is an expression of the exact numeric or approximate numeric data type category, except for the bit data type.

Return Types

If STDEV is used on all items in a SELECT statement, each value in the result set is included in the calculation. STDEV can be used with numeric columns only. Null values are ignored.

This example returns the standard deviation for all royalty payments in the titles table.

USE pubs
FROM titles

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You also have STDEVP, VAR & VARP
garethmanuelAuthor Commented:
Thanks matthewspatrick. Its been a while since i've done any statistics. This corresponds with the articles i just found:
Normal Distribution

Standard Normal Distribution

Where SND has mean of zero and standard deviation of 1 hence coming up with your simplified equation.

Sorry twoboats, your answer wasn't quite what i was looking for
Patrick MatthewsCommented:

You're welcome, and a slight correction...

z is:    (value being considered) / st dev

should be...

z is:    (value being considered - mean) / st dev


The above formula posted by matthewspatrick is an approximation.  The below functions will calculate the Standard Normal Distribution to 3 decimal places +/-.0002 at the 4th decimal place.

To get a value to match a value in a Table of Standard Normal Probabilites take 1 - Normal((value being considered - mean)/st dev, True).  The True/False entry tells the function if your value is in the left or right tail of the distribution curve.  True = Right Tail, False = Left Tail

Central Region Probability Function
The CRPF calculates values for -2.32 < x < 2.32 (values not in the tails of the distribution curve)

Normal Continued Fraction
The NCF calculates vaues for x < -2.32 or x > 2.32 (values in the tails of the distribution curve)

Public Function Normal(ByVal x As Double, ByVal upper As Boolean) As Double
'This procedure is adapted from Algorithm 304 Normal Curve Integral by I. D. Hill and S. A. Joyce
'which is published in Communications of the ACM, Volume 10, Number 6, June 1967 pages 374-375.  It has been
'adapted from the programming language ALGOL 60 to VBA by Philippe Perrault
    Dim M As Double
    Dim N As Double
    Dim p1 As Double
    Dim p2 As Double
    Dim q1 As Double
    Dim q2 As Double
    Dim S As Double
    Dim t As Double
    Dim x2 As Double
    Dim Y As Double
    Const PI = 3.14159265359
    If x = 0 Then
        Normal = 0.5 'Area under half the curve
        Exit Function
        x = Abs(x)
        upper = upper Eqv x > 0
        x2 = x * x
        Y = 1 / (2 * PI) ^ 0.5 * exp(-0.5 * x2)
        N = Y / x
    End If
    If upper = True And 1 - N = 1 Then
        Normal = 0 'Area under whole curve since x is a large negative number
        Exit Function
    End If
    If upper = False And N = 0 Then
        Normal = 1 'None of the area under the curve since x is a large positive number
        Exit Function
    End If
    q1 = x
    p2 = Y * x
    N = 1
    p1 = Y
    q2 = x2 + 1
    If upper = True And x > 2.32 Then
        M = 1 - p1 / p2
        S = M
        t = p2 / q2
        Normal = NCF(M, N, p1, p2, q1, q2, t, x, upper)
        Exit Function
    End If
    If upper = False And x > 2.32 Then
        M = p1 / p2
        S = M
        Normal = NCF(M, N, p1, p2, q1, q2, t, x, upper)
        Exit Function
    End If
    xy = x * Y
    S = x
    N = 3
    t = 0
    S = CRPF(x)
    If upper = True Then
        Normal = S
        Normal = S
    End If
End Function

Public Function NCF(M, N, p1, p2, q1, q2, t, x As Double, upper As Boolean) As Double
    'Uses the continued fraction method to calculate the Normal Curve Integral at the tails
        S = x * p2 + N * p1
        p1 = p2
        p2 = S
        S = x * q2 + N * q1
        q1 = q2
        q2 = S
        S = M
        M = t
        If upper = True Then
            t = p2 / q2
            t = 1 - p2 / q2
        End If
        N = N + 1
    Loop While M <> t And S <> t
    NCF = t
End Function

Public Function CRPF(x As Double)
'This procedure was adapted from function 26.2.18
'Calculates the Normal Distribution in the central region of the probability function (CRPF)
'accurate to 3 decimal places
    Dim B() As Variant
    Const p = 0.2316419
    Const PI = 3.14159265359
    Dim t As Double
    Dim z As Double
    B = Array(0.196854, 0.115194, 0.000344, 0.019527)
    CRPF = 1 - 0.5 * (1 + B(0) * x + B(1) * x ^ 2 + B(2) * x ^ 3 + B(3) * x ^ 4) ^ -4
    CRPF = 1 - CRPF
End Function

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