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Is it possible, based on a given turnover rate per year (in %), to calculate the average time (in years) an employee stays in an organization?

I have looked at the Coupon Collector's problem:

https://en.wikipedia.org/wiki/Coupon_collector's_problem

but so far haven't been able to figure this out.

Turnover could basically be calculated like this:

Turnover Calculation

Especially Lime
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FloWEffect
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  • No coupons here....this is a straight forward example of a Geometric Distribution – lulu Feb 23 '18 at 11:59
  • @lulu Only if we assume leaving is independent of length of service. But actually the average doesn't depend on who leaves when. – Especially Lime Feb 23 '18 at 12:07
  • @EspeciallyLime Of course. Obviously some model is needed here, but at the level of the OP (introductory, I assume) I think it is clear what assumptions are intended. – lulu Feb 23 '18 at 12:08
  • To clarify: I believe the intended assumption is "each employee has a $5%$ chance of leaving each year independent of their own history and independent of any other employee." – lulu Feb 23 '18 at 12:10
  • I think independence can be assumed in this problem. At least this was how I encountered the problem. No dependence on other variables was mentioned.

    @lulu: Introductory hits the mark all to well

     – FloWEffect Feb 23 '18 at 12:22
  • So, working with that assumption, can you solve the problem? This really is just a geometric distribution...it's the same as saying "I have a coin that comes up $H$ with probability $.05$, what's the expected number of tosses before I see $H$". – lulu Feb 23 '18 at 12:29
  • @lulu No, my point was that no model is needed. If you maintain a constant turnover rate, that determines the average time. – Especially Lime Feb 23 '18 at 12:57
  • @EspeciallyLime Ah, fair point. Of course, no matter what we do we need to take a limit in large time. But, yes. I agree that the large time limit should be model independent. – lulu Feb 23 '18 at 13:28

3 Answers3

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I don't think this has anything to do with the coupon collector problem.

Say the company has $100$ employees. After $1$ year they have employed $100$ people for a total of $100$ person-years. After $2$ years they have employed $105$ people for a total of $200$ person years. After $n$ years they have employed $100+5n$ people for a total of $100n$ person-years. So what is the average number of years worked, if $n$ is very large?

Especially Lime
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This strikes me as application of Little's Law $$L = \lambda W$$ where $L$ is the long-term average of the number of customers in a queuing system, $\lambda$ is the arrival rate, and $W$ is the average time a customer spends in the system. Assuming the company size is stable, the turnover is the arrival rate of employees divided by the number of employees, i.e. turnover is $$\tau = \lambda / L$$ so by Little's Law, $$W = L / \lambda =1/ \tau$$ Put in words, the average time in the company is the reciprocal of the turnover rate.

I don't see any connection with the Coupon Collector's Problem.

awkward
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Trying to apply basics of probability here, not sure whether I got everything right.

When

  • The events are $\Omega = \{L_1, L_2, L_3, ..\}$, where $L_1$ is the employee leaving after the first year, $L_2$ is the employee leaving after the second year, and so on;
  • $f = 0.05$ for a turnover rate of 5%.

Then the probability for leaving after the $i$th year is the probability for staying $i-1$ years and leaving in the $i$th year:

$P(X=L_i) = (1-f)^{i-1}*f$

The expected value is then 20 years for 5%, which would be inline with the answer for Little's law above.

Interestingly, when I draw a sample based on the probability distribution above, and make a boxplot, I get a median of ~13.

filantik
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