Averaging Percentages: What's the trend?

cmbarona

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I'm trying to find the best way to present average giving to an organization based on how much had been given in previous years, but I realized that different methods of averaging give me different percentages. I understand why I get different figures, but I'm having a hard time wrapping my head around why that is and what those different figures mean. To give context, I'm comparing year-to-date giving across the five previous years to see where we should be by this point in our current year. So, for example, I should be able to say we should have received x% of our giving by this point in the current year, and therefore make a recommendation of how we should focus future efforts for the rest of the year. I'm applying this problem across a number of different groups of donors, but the core of the problem remains the same.

I'll refer to the raw data in the following way: C=Current (Year-to-date) giving. C1-C5 refer to the five previous years, and CN refers to that giving in the current year (N=Now). T=Total giving, so T1-T5 refer to the total giving in those previous years. What I'm trying to find is PN, or the percentage of total giving I should have in the current year. I'm using the percentage I'll generate to compare to two different figures: the average of total giving in the five previous years, and the goal set at the beginning of the current year. Which of the following methods would be best, or should I use a different method? What are the implications for the different methods?


  1. Method 1: Average the percentages. Find P1-P5. That is, C1/T1=P1. After this is done for all five previous years, (P1+P2+P3+P4+P5)/5=PN
  2. Method 2: Percent of the averages. ((C1+C2+C3+C4+C5)/5)/((T1+T2+T3+T4+T5)/5)=PN
  3. Method 3: Percent of raw figures (no averaging). (C1+C2+C3+C4+C5)/(T1+T2+T3+T4+T5)=PN
 

j-astron

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Hi cmbarona,

Methods 2 and 3 should give you the exact same answer, because the factor of 1/5 cancels from both the numerator and denominator of Method 2, and you just end up with the ratio from Method 3.

Neither of these is what you want. If the question you're trying to answer is "by this point in the year, what percentage of our annual earnings have we typically received?", then Method 1 provides the answer to that question. You're averaging the percentages, which gives you what fraction of the total annual earnings you've earned YTD (year-to-date), on average, over the last 5 years.

To illustrate why Method 2 is not what you want, it's because the average of the absolute YTD earnings in two different years is misleading, since YTD earnings would be very different in different years if the total donation amount were very different in those two years. I think this is best illustrated with a very simplistic example, let's say you got only $1000 in year 1, but $5000 in year two. Furthermore, let's say that by the end of June, you had $750 in year 1 and $2500 in year 2. So, to summarize:

C1 = $750
C2 = $2500

T1 = $1000
T2 = $5000

Using Method 1: P1 = 750/1000 = 0.75, and P2 = 2500/5000 = 0.5, and therefore P_avg = P1+P2/2 = (0.75+0.5)/2 = 1.25/2 = 0.625
Here we got an answer that is representative what's actually happening. By this point in year1 you had 75% of your donations, and by this point in year 2, you had 50% of your donations, so on average, you get about 62.5% of your donations by mid-year

Using Method 2: C_avg = (C1 + C2)/2 = (750 + 2500)/2 = 3250/2 = 1625
T_avg = (T1 + T2)/2 = (1000 + 5000)/2 = 6000/2 = 3000

P_avg = C_avg / T_avg = 1625/3000 = 0.54

So, despite the fact that in year 1, you got 75% of your donations by mid-year (which is a lot more than half) you're reporting an average mid-year percentage earnings of only 54%. That's clearly not right. Your average percentage value is low because the absolute earnings in year 1 were low, illustrating that this is not the right statistic.

One final suggestion: you could considering quantifying this using more than just a single number. If you have YTD earnings for every single month of the year, for several years, you could make a plot (a graph) of the cumulative percentage earnings vs. time for each year, and compare them. Maybe in some years it increases towards 100% very steeply at first, because most donations came in early in the year and then it tapers off. Or maybe, every year, it's close to a straight line (with constant slope), because the rate of donations is steady over the year: you get about the same amount coming in every month). That graph, I think, would be very informative to look at.
 

cmbarona

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Thanks, that makes a lot of sense now.

This brings me to part 2 of my question. Is there a meaningful percentage I can present to show variance from goal, and if so, how do I calculate it? Keep in mind, I'll project this based on two figures, a 5-year average of total yearly giving and goals we set at the beginning of the year.

So, let's say by this point in the year we expect to have received 60% of total yearly donations. And let's say the average giving over the last 5 years is $750k, but we set an optimistic goal for this year of $1M.

Adjusted goal for 5-year average is therefore .6 * 750000 = 450,000. And the adjusted goal for the goal we set is .6 * 1000000 = 600,000. Easy enough, we've got our benchmarks now.

Now, let's compare actual current giving to those benchmarks. Let's say we've raised $500k thus far this year. We're $50k ahead of the average, but $100k behind our goal.

Here's the question. Is there any meaningful way to represent that +$50k and -$100k in their own percentages? How would I calculate it?
 

j-astron

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Now, let's compare actual current giving to those benchmarks. Let's say we've raised $500k thus far this year. We're $50k ahead of the average, but $100k behind our goal.

Here's the question. Is there any meaningful way to represent that +$50k and -$100k in their own percentages? How would I calculate it?
I don't know if I fully understand what you're trying to do, but I could take a stab at it:

500k/450k = 1.111... or 111% of the 5-year YTD giving average. So you could say you're up 11% from the average.

In contrast, 500k/600k is only 0.8333... = 83.3% of your YTD goal for this year. So you're up 11% from the average, but down 16.67% from this year's goal.

Is that the sort of thing you'd like to be able to report? 'Cause it could make equal sense to just report the surplus or shortfall in absolute dollar amounts, depending on the point of all this.
 

cmbarona

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I am reporting absolute dollar amounts, but bear in mind this report is done across different donor groups, and the people who see it - who are not knee-deep in data all day - are more likely to be able to wrap their heads around what the different figures mean if I've got a meaningful percentage I can attach to them. For example: "overall donations are behind 10%, and most donor groups are close to that figure, but donations from graduates are down 30%; therefore, we should refocus efforts to reach graduates and investigate what campaigns have been successful in the past." Using absolute dollar figures is useful, but it's hard to show meaningful deviation from expectations if we just use dollar figures. They vary enough that some other statistical figure would be more useful, I think. What I'm doing now is to compare the average percentage expected by this point in the year to the percent of either the projected 5-year average or the yearly goal.

Examples: We've received $500k. That's 66% of the $750k average year-end giving, and 50% of the $1M goal. Now, we expect to have 60% of all giving in by this point of the year, so that means we're (66%-60%) = 6% ahead of average giving, but (50%-60%) = -10% behind our yearly goals. Again, it's not so much the math that's tripping me up, but what the different methods imply about the data. This method obviously differs substantially from the method you suggested, but what do the different figures mean?
 

j-astron

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Messages
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I am reporting absolute dollar amounts, but bear in mind this report is done across different donor groups, and the people who see it - who are not knee-deep in data all day - are more likely to be able to wrap their heads around what the different figures mean if I've got a meaningful percentage I can attach to them. For example: "overall donations are behind 10%, and most donor groups are close to that figure, but donations from graduates are down 30%; therefore, we should refocus efforts to reach graduates and investigate what campaigns have been successful in the past." Using absolute dollar figures is useful, but it's hard to show meaningful deviation from expectations if we just use dollar figures. They vary enough that some other statistical figure would be more useful, I think. What I'm doing now is to compare the average percentage expected by this point in the year to the percent of either the projected 5-year average or the yearly goal.

Examples: We've received $500k. That's 66% of the $750k average year-end giving, and 50% of the $1M goal. Now, we expect to have 60% of all giving in by this point of the year, so that means we're (66%-60%) = 6% ahead of average giving, but (50%-60%) = -10% behind our yearly goals. Again, it's not so much the math that's tripping me up, but what the different methods imply about the data. This method obviously differs substantially from the method you suggested, but what do the different figures mean?
Yup, the challenging part of stats is often not doing the calculations themselves, but figuring out which calculations to do in the first place in order to accurately represent what you're trying to show.

Your percentages make perfect sense and are meaningful:

You're at 50% of your yearly goal
Normally by this point in the year, you're at 60% of your yearly goal, so you're 10% behind in terms of proportion of received donations YTD
However, your yearly goal is optimistic. You're at 67% of the 5-year average yearly earnings, so you're actually outpacing the average rate of earnings by a bit (66.667% / 60% = 11% higher rate of earnings).

I'm gonna think out loud and make some assumptions here. Let's assume the rate of earning is constant throughout the year. So normally you make $750k/12 = $62,500 monthly. If you like, this is the monthly average. However, to meet your new goal, you'd have to make a monthly average of $1M/12 = $83,333.33. That's 20,833 more per month. It's 33% more per month. It's 33% more in any time interval.

So...in this case, YTD, you normally make $450k, but you actually have made $500k. That's $50k more, which is only 11% more. You need to have made 33% more. So your surplus over the past year's average YTD earnings is only 11/33 = 1/3 of what it needs to be. It needed to be $150k to get you from 450k to 600k. Instead it's only 50k.

So yeah, this is an interesting way of looking at it. Your rate of earnings is so far 1.11x past years, but needs to be 1.33x past years to meet your optimistic goal. If you project this rate of earnings to year end, you'll predict that you'll make 1.11 * 750k = $833k. You'll be at 83.3% of your optimistic goal, and thus fall short of it by 16.7%. (Note that 1.11/1.33 = 0.83). So all the numbers we've computed have come up and mean something. Which ones you use depend on what question you are trying to answer
 
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