How do I calculate weighted mean with negative weights?

Question

This might be very trivial for you guys.
If :
I buy 2 oranges for 5 dollars each then
I buy 3 oranges for 6 dollars each and then
I sell 3 oranges for 4 dollars each

How do I calculate the average price of an orange ? thanks.

Edit 1

I will elaborate :
I need to show in software the amount of the product and the avg price.
So , if lets say I bought 4 ranges for 100 usd each. and I sold 2 oranges for 110 usd each.

I now present in my systems 2 oranges - and in what price?

Option A: (4 * 110 + 2 * 100)/6 = 106.667 or Option B: (4 * 110 - 2 * 100)/2 = 120 (doesnt think so )

If I do go with option A then if I sold 2 more for 110 I would get 0 amount but average (4*100+2*110+2*110)/8 - 105 . 0 stocks for 105 doesn't make any sense.

(I beleive I want to be able to say amount*price - this is what I have, cause I want to show "How Much Do I Worth") Thanks A lot

Answer 1

You're a bit vague about what you mean by "average" price. Do you mean the average price you paid for oranges or the average price oranges were transacted on the market. If the first, you account for the ones you sold as an income instead of a loss, so you should give it the opposite sign. But if you just are interested in the average price at which transactions occurred, not accounting for whether they were sold or bought, then the simple formula with absolute values is OK. What do you want?

Suppose you're interested in the average price you paid for the oranges.

You have used an amount of money equal to $2\times 5+3\times 6-3\times 4=16$ dollars to pay for oranges. You are left with two oranges, so you could say the average price per orange you paid is $8$ dollars.

On the other hand, if you're interested in the average transaction price, then you might need:

$$\frac{2\times 5+3\times 6+3\times 4}{2+3+3}=5 \text{ dollars.}$$

In your edited version, you say you want to evaluate "worth", I think averages are not really useful. Totals would be more useful. In fact, a good double-entry bookkeeping system that keeps track of your amount of oranges and your capital would be more useful here.

Answer 2

I am going to interpret the question as: "At what price should I sell my remaining oranges if I am to break even?"

Total amount spent buying oranges $B=5 \times 2 + 6 \times 3 = 10+18=28$.

If there are $n_i$ oranges bought at a cost of $b_i$ in each transaction, then $B=\Sigma b_i n_i$.

I then sell my oranges, gaining $S=4 \times 3=12$.

If there are $m_j$ oranges bought at a cost of $s_j$ in each transaction, then $S=\Sigma s_j m_j$.

For simplicity, we will assume that we have sold fewer oranges than we originally bought: $\Sigma m_j < \Sigma n_i$.

So I still have some remaining unsold oranges. The quantity is $\Sigma n_i - \Sigma m_j$. In your example the quantity is $5-3=2$.

I want to sell these remaining oranges at a price that will allow me to at least break even. At the moment there is a difference $D$ between my buying cost $B$ and my sales $S$. $D=B-S$.

If $D=0$ then I have already recouped my initial costs. I can sell the remaining oranges at any price I choose. I can even give them away free!

If $D<0$ then I have recouped my initial costs and I have made a profit $P=S-B$. Again I can sell the remaining oranges at any price I choose. I can even pay someone to take them off my hands...

If $D>0$ then I still need to sell the remaining oranges in such a way as to brek even. This means that the price I sell them for $p$ must be such that $p\left (\Sigma n_i-\Sigma m_j \right)=D$.

This gives us a required price $p=\frac{\Sigma b_i n_i - \Sigma s_j m_j}{\Sigma n_i - \Sigma m_j}$

In your example, this comes to $p=\frac{28 - 12}{5 - 3}=\frac {16}{2}=8$.

This value agrees with the price calculated by @Raskolnikov.