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Estimating Probability Distributions: Part 2 [Statistics
Posted on July 18, 2013 @ 04:17:00 PM by Paul Meagher

In my last blog, I discussed some useful probability distributions for representing our uncertainty about a parameter; the uniform and the triangular distributions.

Our uncertainty about a parameter θ such as "the price of gas next week" can be represented using a uniform distribution where the gas price could be anywhere between some low estimate and some high estimate of the price next week. If we also want to hazard a guess as to the most likely value, then we would be using a triangular distribution to represent our uncertainty about the price of gas.

There are other simple techniques for eliciting a probability distribution to represent our uncertainty about a parameter. In today's blog I want to discuss a simple technique called "Merit Scoring".

The Future Price of Corn

The easiest way to explain this technique is if you look at the table below.

Corn Price (per bushel)Merit Score

The table has future corn prices ranging from $4.50 to $5.50 per bushel (see quotecorn.com for current price). Now, I might ask you to assign a relative merit score to each price point in this range. A merit score can range between, say, 1 and 10. If you assign a merit score of 1 to a price point, that means you think the future price will not be nearest to that price - the price estimate has low merit. Conversely, a merit score of 10 means that you think the future price will be nearest to that price - the price estimate has high merit. My merit scores for the price of corn on Sept 1, 2013, looks like this.

Corn Price (per bushel)Merit Score

In this example, we are not directly assigning a probability to each possible price point. Instead we are supplying a merit score to each possible price point. We can easily convert each merit score to a corresponding probability by summing all the merit scores and then dividing each merit score by this sum. The result is a probability assignment for each price point with probabilities for each price point summing to 1. This gives as a probability distribution for our parameter which is the price of corn on Sept 1, 2013.

To demonstrate how merit scores can be converted to probabilities and how this forms a probability distribution I have devised a PHP-based script that shows how the calculation is done, what the calculated price probabilities are, and that these probabilities sum to 1.


* @script merit_scoring.php
* @author Paul Meagher
* @purpose: Convert merit scores to a probability distribution.

// Enter merit labels and merit scores here.

$merit_scores = array(

// Compute sum of scores so we can compute probabilities below.

$merit_sum array_sum($merit_scores);

// Compute the corresponding probability of each label.

foreach($merit_scores AS $merit_label=>$merit_score
$prob_dist[$merit_label] = $merit_score/$merit_sum;

// Dump the contents of the $prob_dist array to the screen.

echo "<pre>";

// Verify that the sum of each probability is 1

$total_prob array_sum($prob_dist);

"Total probability is $total_prob";


    [4.25] => 0.037037037037037
    [4.50] => 0.18518518518519
    [4.75] => 0.37037037037037
    [5.00] => 0.2962962962963
    [5.25] => 0.11111111111111

Total probability is 1 



The merit scoring technique and script can be used to estimate a probability distribution for any parameter that interests you. One limitation of this technique is that it is discrete in nature so can't give you probabilties for prices that might fall between two price points (e.g., $4.85). This may be of concern if you think you should be trying to estimate the future price of corn with more resolution (e.g., 10 cent increments) and/or the daily variability in corn prices is not that high. The daily price of corn is actually quite high so being correct to within 25 cents might be a good goal for your predictions.


Statistic Brain [Statistics
Posted on April 5, 2013 @ 06:57:00 AM by Paul Meagher

There are some University of Tennessee Research statistics on startup failure rates by industry at Statistic Brain.

Statistic Brain looks to be quite a useful and entertaining resource for those seeking a thrill with numbers.

From the site:

Statistic Brain is a group of passionate number people. We love numbers, their purity, and what they represent. Numbers can bring humans together, they tell us how we are alike and how we are beautifully unique. Numbers are a way to reflect on how far we’ve come and give us hope for the future.

Our goal is to bring you accurate and timely statistics. We will never become number analysts because we believe numbers should only be interpreted by the reader. We want to educate, assist, and sometimes entertain with numbers on every subject.

We hope that today you learn something new, find inspiration for tomorrow, and use your knowledge for something good.

Seth Harden
CEO / Founder


Characteristics of high income Canadians [Statistics
Posted on February 6, 2013 @ 06:18:00 AM by Paul Meagher

Statistics Canada has release a recent update on the composition of high-income Canadians. The cutoff for defining a high income Canadian is someone with an annual income of $201,400 or higher. This conveniently splits the distribution of Tax filers into the 1% earning this amount or above, and the 99% of Canadians earning less than this amount. Given this definition of a high-income Canadian, here is a table giving the composition of these high-income Canadians:

One interesting observation from this report is that the "income of top filers was increasingly dependent on their jobs, rather than on investments". The fact that Statistics Canada feels the need to point this out is probably because in other countries high income earners derive the majority of their income from investments rather than wages. It is possible that policy in Canada needs to change to incentivise these top income earners to invest in small businesses, a position that the National Angel Capital Association has been advocating for many years.

Another interesting observation is where in Canada these high income earners are located:

In 2010, four provinces – Ontario, Alberta, Quebec and British Columbia – accounted for 92% of the 254,700 people in the top 1%.

Ontario had 110,300, followed by Alberta with 52,200, Quebec at 42,600 and British Columbia with 29,500.

Between 1990 and 2010, Alberta's share of the top 1% of filers doubled from 10% to 20%, while Ontario's proportion fell from 51% to 43%.

The areas of strongest angel investment activity in Canada corresponds to areas where high-income earners are most concentrated. In recent years, Alberta has become a leader in the angel investment industry with one of the leading organizations being Venture Alberta.




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