Sales growth

[Rassis]

Just to be provocative:

During a corporate presentation of progress in sales, a salesman reports that the sales growth of product X during the past four quarters was 2%, 4%, 6% and 8%. The person making the presentation ends by saying that the mean growth rate was 5% per quarter. Is he right or is he pulling your leg?

[krtxmrtz]

Just to be provocative:

During a corporate presentation of progress in sales, a salesman reports that the sales growth of product X during the past four quarters was 2%, 4%, 6% and 8%. The person making the presentation ends by saying that the mean growth rate was 5% per quarter. Is he right or is he pulling your leg?

I think this is not too far from the correct answer which is, I believe, about 5.36%.
If you call V the volume of sales at the beginning of the first quarter, then:
1.02V is the growth after this first quarter
1.04*1.02V is the growth after the second quarter
1.06*1.04*1.02V, same after the third q.
1.08*1.06*1.04*1.02V = 1.2144V is the overall growth after all 4 quarters.

The mean growth per quarter will be (1.2144 - 1) / 4 = 0.0536 i.e. 5.36%

[Rassis]

Congratulations! You haven’t given the apparently most obvious answer, that is (2 + 4 + 6 + 8) / 4 = 5%. Nevertheless, 5,36% is not the right answer. You treated data has a compound interest problem where tax rates are different each year. That is not the case. The right answer is little less than 5%. Take into consideration that data follow a geometric series. This is also a common situation when it comes to deal with inflation rates. Will you try once more?

If you want the answer right away, please email me. I will post the solution in this forum within a few more days. Meanwhile, I hope to get some more results.

Have fun.

[krtxmrtz]

You treated data has a compound interest problem where tax rates are different each year. That is not the case.

Then I have probably not understood what the salesman exactly meant with his statement, i.e. I don't know how I must interpret what he says, can you set forth an example with actual numbers?

[Rassis]

Let me put the problem a little bit different, though keeping the same foundation. Suppose you are told that the inflation rate in your economy was 2%, 4%, 6%, 8% during the past four years. If you are asked what the mean inflation rate was, what would your answer be?

[krtxmrtz]

I'm not sure I get you but this is another way to interpret it: if 2%, 4%, 6% and 8% growth refer to the increments after each quarter over the initial amount in that quarter, then, if we start from an amount V:

After the first quarter we have 1.02V (growth = 1.02)
After the second quarter we have 1.04V (growth = 1.04/1.02 = 1.0196)
After the third quarter we have 1.06V (growth = 1.06/1.04 = 1.0192)
After the fourth quarter we have 1.08V (growth = 1.08/1.06 = 1.0189)

So, the mean over the 4 quarters will be (1.02 + 1.0196 + 1.0192 + 1.0189)/4 = 1.0194 (i.e. 1.94%)

[Rassis]

If you use index numbers you get:

Period 0: 1
Period 1: 1 x (1 + 0,02) = 1,02
Period 2: 1,02 x (1 + 0,04) = 1,0608
Period 3: 1,0608 x (1 + 0,06) = 1,1244
Period 4: 1,1244 x (1 + 0,08) = 1,2144

Now, let i be the mean growth rate.

Then: (1 + i)^4 = 1,2144 yielding: i = 0,04976 or 4,976%

Another way considering the fact that data follow a geometric series:

Geomean = [(1 + 0,02) x (1 + 0,04) x (1 + 0,06) x (1 + 0,08)]^(1/4) -1 = 0,04976 or 4,976%.

Or, still, in EXCEL: GEOMEAN(1,02; 1,04; 1,06; 1,08) = 0,04976 or 4,976%.

[krtxmrtz]

I guess I'll never get hired by a bank...