Date: Tue, 6 Jun 2023 00:26:55 +0000 (UTC) Message-ID: <1794179423.610.1686011215518@localhost> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_609_458626529.1686011215516" ------=_Part_609_458626529.1686011215516 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Cumulative Ascending

# Cumulative Ascending

The Cumulative Ascending Distribution can be constructed with Crystal Ball's Custom distribution<= /p>

Cumulative ascending equations

The Cumulative distribution in Crystal Ball can be cons= tructed using Crystal Ball's Custom distribu= tion, and requires two arrays of data, {xi} and {Pi}) where {xi} is an array of x-values w= ith cumulative probabilities {Pi} and w= here the distribution falls between the minimum and maximum. The figure bel= ow shows the Cumulative distribution using data ({0,1,4,6,10},{0,0.1,0.6,0.= 8,1.0}) as it is defined in its cumulative form and how it looks as a relat= ive frequency plot. #### Uses

##### 1. Empirical d= istribution of data

The Cumulative distribution is very useful for converti= ng a set of data values into a first or second order empirical distribut= ion that can be sampled by Crystal Ball.

##### 2. = Modeling expert opinion

The Cumulative distribution can be used to construct un= certainty distributions when using some classical statistical methods. Exam= ples: p in a Binomial process; l in a Poisson process.<= /p>

##### 3. = Modeling expert opinion

The Cumulative distribution is used in some texts to mo= del expert opinion. The expert is asked for a minimum, maximum and a few pe= rcentiles (e.g. 25%, 50%, 75%). However, we have found it largely unsatisfa= ctory because of the insensitivity of its probability scale. A small change= in the shape of the Cumulative distribution that would pass unnoticed prod= uces a radical change in the corresponding relative frequency plot that wou= ld not be acceptable. The figure below provides an illustration:

= A smooth and natural relative frequency plot (A) is con= verted to a cumulative frequency plot (B) and then altered slightly (C). Co= nverting back to a relative frequency plot (D) shows that the modified dist= ribution is dramatically different from the original, though this would alm= ost certainly not have been appreciated by comparing the cumulative frequen= cy plots. For this reason, we usually prefer to model expert opinion lookin= g at the relative frequency distribution instead.

The cumulative distribution is however very us= eful to model an expert's opinion of a variable whose range covers several = orders of magnitude in some sort of exponential way. For example, the n= umber of bacteria in a kg of meat will increase exponentially with time. Th= e meat may contain 100 units of bacteria or 1 million. In such circumstance= s, it is fruitless to attempt to use a General distribution = directly.