This is a question that is asked very often: What is the odds of somebody having the same strengths as I?
First, let me tell you that this is just for fun. When coming up with the 34 talent themes, Donald Clifton chose them as they were emerging from the data and bundled the ones that were somewhat alike each other.
Origin
In short, the 34 were found by taking some 5000 items and narrowing them down to select the items with the strongest psychometric properties. That is, the ones that were best in distinguishing personality traits without being too limiting. Items excluded were either correlated to others, not adding to the results, or too selecting.
1999 – The remaining items were bundled in 35 themes of similar items and data was collected for several months, revising it to 180 items bundled in 34 themes.
2006 – another revision took place based on more than 1 million assessments to shape StrengthsFinder 2.0 with 34 themes and 177 items.
The base question for primary collection of the 5000 items was: how can people work together more efficiently.
While MBTI and Big 5 and such are theory based, CliftonStrengths is evidence based, that is constructed through data analysis: make tons of interviews and see what patterns emerge.
Talent Theme Odds
This, among other things, means that the different talent themes do not occur with the same odds, most probably.
Even the theme frequency data Gallup provides for certified coaches points into that direction:
All over the world, when 20’002’953 assessments were recorded, there were big differences.
Achiever was in 31.22% of people’s top 5, while self-assurance only appeared in 4.52% of the cases.
Yet, those assessments are, even in their great number, not a statistically valid sample as, among other reasons,
- most have to take the assessment through their employer,
- for the others, people with some talent themes might be less likely to take such an assessment in the first place.
The probabilities I am going to calculate are based on normal distribution, that is, assuming that all talent themes appear with the same odds of 1/34.
Why 1/34? When I randomly draw one strength of the 34 talent themes, I have 34 possible talent themes and therefore 34 possible results.
Formulas
When I want to draw 2 talent themes, what is the odds of that combination?
For the first I draw I have 34 possible result, and for the second, the remaining 33, as I cannot have the same talent theme twice. There are 34 * 33 possible combinations of to talent themes then. So the odds of a combination of 2 is 1 : 34 * 33 = 1 : 1122.
For top 5, that would be then 1 : (34 * 33 * 32 * 31 * 30) or 1 : 33’390’720. That is the famous 1 in 33 million often stated.
There is a shorthand this can be written in – math has a shorthand for everything. Let’s develop what is called faculty and the binomial formula.
Fixed order – Faculty
Let’s calculate the number of combinations of all 34 talent themes. First place 34, second place 33, and so on until for the last place, there is only 1 talent theme left. That is 34 * 33 * 21 * … * 2 * 1. And that is the definition of 34! (faculty of 34), the product of all numbers from 1 to 34.
There are 34! possible combinations of talent themes. That is 2.952327990396041e38 or roughly a 3 with 38 zeroes. The odds of another person having exactly your order of talent themes is 1 : 34!
There are roughly 8 billion people on the planet. There are 3.69e28 times as many possible combinations of talent themes than people!
Let’s get back to the top 5. How could this be written in shorthand?
Ok, there are 34! combinations total. When I take away five strengths, there are 29 left. I can combine those 29 in 29! ways with the same math I have done for the 34.
So for the 5 talent themes I have selected, there are 29! combinations on how the other talent themes can be combined. If I now divide the total of possible combinations with this number, I get the number of possible combinations of top5.
Let’s illustrate this:
34!/29! = 34!/(34-5)! = (34 * 33 * 32 * … * 1) / (29 * 28 * 27 * … * 1) = 34 * 33 * 32 * 31 * 30.
Top 5 in fixed order: 34!/29! = 33’390’720
So, again, but with a more usable formula, we have roughly one in 33 million sharing your top 5.
This is the formula we used above.
For top 10, use 34!/(34-10)!
Random Order – Binomial Formula
In all of these formulas, when we draw the first talent theme, it will stay in first rank.
What now when we want to know how many combinations we can build if the talent themes just have to show up in any order?
Well, when I have 5 talent themes, how many combinations are possible for those 5? Could it be 5! ? Most certainly it is, according to the theory above.
We have 34!/29! possible combinations of top 5, and 5! possible combinations of those top 5 in any order. If I draw any top 5, 5! combinations will give me the same top 5.
We have found the binomial formula:
Top 5 in random order: 34!/(29! * 5!) = 278’256.
So, one in 278’256 people share your top 5 in random order.
Or, if going for top 10: 34!/ ((34-10)!(10!)).
Four out of Five
There are now 34!/(29!5!) possible combinations of top 5. What if we only care for 4 of them?
There are 34!/(30!4!) possible combinations of top 4 in any order.
If you look at that, there are 150 more possible combinations for top 5 that top 4.
There are 5!/(4!1!) possible combinations of 4 talent themes within a set of 5. That is, surprisingly enough, 5 possibilities or 5 five spots where the talent theme can show up we do not care about.
Thus, there are 34!/(30!4!) * 5!/(4!1!) possibilities of 4 out of 5 of a set of 34.
That is 5’565’120.
So, roughly one in 5.5 million people have 4 of your top 5 in common with you.
Thus, there are 34!/(31!3!) * 5!/(3!2!) possibilities of 3 out of 5 of a set of 34.
That is 59’840.
So, one in 59’840 people have 3 of your top 5 in common with you.
World Formula
The world formula to calculate m out of n talent themes in common from a set of 34 talent themes therefore is (tada): (0! = 1)
World formula in exact order: 34!/((34-m)!) * n!/((n-m)!m!)
World formula in any order: 34!/((34-m)!m!) * n!/((n-m)!m!)
I am sure I lost most of my readers on the way, so: congratulations, if you made it that far!