Written by Colin+ in ask uncle colin.

*Ask Uncle Colin* is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I work in a call centre, which is run by the numbers. Every employee has to stay above a certain conversion rate, calculated as the number of successful deals resulting from their calls, divided by the number of times they have a call that doesn't go to voicemail.

My conversion rate isn't all that good, even though I make more than 100 calls a day; the 'best' employee makes fewer than 50.

I have a theory that the harder you work, the lower your conversion rate is - failures (invalid numbers and flat refusal are a lot quicker to deal with than deals). My boss thinks this is rubbish. How can I prove it?

-- Telemarketing Has A Less Equal System

Hi, THALES, and thanks for your message.

I can't immediately see an argument for your theory, I'm afraid!

To simplify the maths, I'm going to pretend that there are three classes of people who you might have to call:

- definite customers, who are waiting anxiously for your call and who sign up after (say) 10 minutes;
- definite non-customers, who have put their number (or someone else's) in because they feel like they have to and hang up after one minute; and
- maybe-maybe-nots, who might go either way depending on how good the person they talk to is. If they say yes, it takes 20 minutes, and if not, 5 minutes.

I'm also going to assume the three classes are equally likely -- it doesn't change the underlying thought process if you use different probabilities and timings, as long as "NO!" is quicker than "Um... no", which is quicker than "YES!", which is quicker than "Um... yes".

Suppose you're a perfect, silver-tongued salesperson. In your first 30 calls, you get:

- 10 definite customers, who take 100 minutes altogether;
- 10 non-customers, who take 10 minutes in total; and
- 10 maybes, who take 200 minutes to become customers.

These 30 calls take 310 minutes - a bit more than 5 hours1.

Suppose instead you're as good at telesales as I would be if I tried it. In my first 30 calls, I'd get:

- 10 customers (100 minutes);
- 10 non-customers (10 minutes); and
- 10 maybes who would get fed up in five minutes each (50 minutes altogether).

That's 160 minutes for 30 calls, so I can manage roughly twice as many calls as Mr Perfect up there.

So far as I can make out, good salespeople make fewer calls because they wind up spending longer on them, not (or at least not necessarily) because they're messing around.

There is one thing about the structure that does strike me as unfair: if someone's entered a bogus number, the lack of a deal clearly isn't the fault of the person calling it, and it seems odd that a 'number not recognised' would count against you.

So, sorry I can't support your theory, THALES: I don't think making few calls is a cause of a high conversion rate; rather, it's a symptom.

-- Uncle Colin

- I know, my numbers aren't realistic - they're just for the sake of an example. [↩]