I am aware that normally, you “shouldn’t” calculate the arithmetic mean of a list of percentages to find the average percentage, as this will skew the results compared with calculating the arithmetic mean from the raw data. But I have found a couple of sources that suggest that this type of average can be useful occasionally, and I think it might be of use in the application I have in mind - can anyone help me with some examples of where you might use this type of average? Thanks in advance.
Asolutely nothing meaningful.
80% of widows over 70 like Jay Leno, while 10% of people aged 20-40 like him. Does that mean that 45% of people like Jay?
Without the weights, an agglomeration of miscellaneous numbers means nothing. All it does is give you a very rough ballpark number, which can be skewed by the variability in the weights.
Thanks, this is my thought also. But perhaps it would help if I gave the context - this is for work, and we are looking at the increase in business volumes (broken down day-by-day) between 2009 and 2011. This gives a list of percentages (one percentage figure for each business day, showing the percentage increase/decrease in business that day between 2009 and 2011). It is this list of percentages we are looking to average. I am fairly convinced (as you say) that calculating this average from the raw data is the way to go. But me and my boss are wondering if the fact that the arithmetic mean of the percentages is higher than this is significant in any way. Could it, for example, mean that we have some outlier “really busy days”, which would therefore imply that using the higher percentage is the way to go when working out how many more staff we might need (in order to cover the “really busy days” more effectively, even if that means we are slightly over-staffed the rest of the time)?
Don’t average out averages! Don’t average %'s!
Let’s say that Bob takes five at bats this week. He goes 5 for 5. He played in two games. Good week, though! Average: 100% (batting 1.000)
The week before, Bob had a full week of games. He got in 30 at bats, and got 3 hits. Average: 10% (batting .100)
If you review weekly batting averages, Bob batted 100% and 10%. 100 + 10 = 110. Divide by 2 for the average, and Bob is batting an insane .550! Get this guy on my team NOW!!! We gotta have this guy!!!
Well, using raw data, Bob went 8 for 35, and is really batting .229! NOW how does Bob look?
Bob Sucks.
I don’t mean to be ungrateful, but I obviously haven’t been clear; I know this, my question is: is there any application for an average of percentages? I am well aware that in most situations, it produces poor results. Or, to put it another way, does the fact that an average of percentages differs from an average of the raw data tell us anything? I realise the answer to both of these questions could be a flat “no”, but I haven’t seen anyone say that yet.
For most purposes, such as your example of correlating business volume to staff required, then you should be looking at figures for business volume, not percent change from day to day. I would take the actual volume figures and just calculate a basic mean and standard deviation. You can use that to calculate efficiency at various staffing levels.
When the numbers are very close every day/interval/whatver.
In a call center, daily call volume can be so close, that in a spot one can look across 30 days of averages and just average them out.
For example, look over 30 days of average answer speed. If the call volume is just about the same day to day, you can average out the average answer speeds.
if your level of accuracy isn’t a decimal point , you might be good to go (only YOU can answer this for your examples). I do it for answer speeds (averaging the average answer speeds), and say I come up with 45. If I use raw data, I might come up with 47.
Doesn’t matter. I’m close enough. My accuracy can be within 5 seconds. I don’t need it down to a second or half a second.
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It tells you that the data is skewed. Data points with higher weights have different values than data points with lower weights. Outside of that, I don’t think an unweighted average has any particular use. Even that use is pretty limited, you’re better off plotting the two factors and doing a linear regression to see if the skew is significant.
The previous responders are referring to the fact that taking the average of two proportions won’t give you the true overall proportion unless the denominators are equal, but that’s not at all relevant to this problem. If you take the average of the percentage change from each day to the next, the value you get really is the average change in daily volume, and that’s a perfectly legitimate quantity to be interested in.
I actually can think of a situation where this would be useful and probably actually correct. If you have several distinct samples of different sizes, but all of the sizes are quite large, and if you believe that there is a substantial difference between the samples underlying percentages, and you wanted to determine what the average of these underlying percentages were.
Say for example I have 4 different tumor samples.
The first has 2,000/10,000 = 20% cells being malignant
The second has 1,000/100,000 = 1% being malignant
The third has 6,000/20,000 = 30% being malignant
The fourth has 6,000/15,000 =25% being melignant
To describe the malignancy percentage of the average sample it would be better so say
(0.2+0.01+0.3+0.25)/4 = 19%
Than it would be to say it was 15,000/145,000=10.3%
I agree with this at face value. I suppose there are situations where averaging percentages is appropriate but I’ll bet that there are better alternatives for them.
If you have a volume of 500 units one day, then 250, then back to 500, then you first went down 50% then went up 100%, although the units went down and up by the same number. So your average fluctuation is (100-50)/2=25%. Which I don’t think really paints the picture of what’s actually happening.
Other measures will characterize the fluctuations better.
This is an example of what is bad about what you are doing!
There’s some subtlety here. If you want to estimate the expected single period growth, the arithmetic mean is the appropriate quantity. On the other hand, if you want to estimate the expected growth over multiple periods, you should be using the geometric mean. Of course, you can’t estimate anything from two data points…
Both methods are good, but they’re just answers to different questions. The overall proportion of malignant cells is 10.3%, but the average rate of malignancy is 19%. If you’re fitting the equivalent of an ANOVA for binary outcomes, the second quantity really is what you’re interested in. The flaw is in using the wrong method for your problem, not in using the second method.
I don’t think you’re better off using the average of averages.
What you should do (in my opinion) is to calculate averages based on the actual numbers, as you’ve done, and also calculate standard deviation. Then base your staffing on some level of the standard deviation that produces the desired amount of under/over staffing.
If the average says you need 10 people, then you will be understaffed about 50% of the time if you use 10 people. Using the standard deviation extends the usefulness. If the standard deviation is +/-1, then you know that a staff of 11 will be understaffed 16% of the time. A staff of 12 will be understaffed 3% of the time.
I assume you and your boss have already ruled out staffing predictions based on seasonality or cyclical changes? If you ups and downs are linked with days of the week, beginnings of a month, winter, etc. then you can just match staffing to needs at the same point in the previous cycle.
Sorry, I must have been unclear in my previous postings. The percentages being calculated are not the change between (say) 19 September to 20 September, then 20 September to 21 September, then 21 September to 22 September, etc. They are the change between 19 September 2010 to 19 September 2011, 20 September 2010 to 20 September 2011, etc. I presume it is being done this way because the volume from day to day varies, so it is the overall percentage increase in which we are interested, in order to estimate what might happen next year.
Thank you - this is the sort of response I was looking for! But what is the significance of the average change in daily volume (if any) compared with the overall change in volume?
This sounds promising. It’s been a long time since I did any stats - can anyone tell me how to calculate SD from a list of percentages in Excel? Where does the 16% and 3% figures come from? Is this based on normal distribution?
In answer to your final paragraph, this already is seasonal - we require more staff in March and April than any other time of the year, but the business is also growing year-on-year. Hope this makes sense, please ask if I can clarify anything else, and I really appreciate all your help (everyone).
Dead Cat, the plain-text answer is that if you are interested in the relationships of the groups (percentages) as such, and not of the individual (i.e., data points) members of the groups, then your method would be useful.
A lot of people here seem to have gone off on a crusade not having a clue what they were attacking. It’s simply one statistical method which is sometimes useful - if that’s what you want.
Thank you, that clarification is very helpful.
One example where this kind of percentage is measured is in retail year-over-year same-store sales. I do not know if anyone averages those percentages but that would a promising path for research if you wanted to run that down.
I don’t think I’ve seen the answer to your last question yet. If the arithmic mean of all the daily changes (so the mean of 365 percentages) is higher than the change of the of total sales, this means that the cases with a lower weight (ie the not so busy days) on average showed more growth/change than the busier days.
If the arithmic mean of the daily changes is lower than the overall change for the whole period, it is the ‘busy’ days that showed more growth.
So yes, there is definately some interesting (allthough crude) information to be gotten from comparing these two numbers.
Let’s say that Bob takes five at bats this week. He goes 5 for 5.