Average – Sum of Averages vs Average of Sums

average

I have essentially a table of numbers — a time series of measurements. Each row in the table has 5 values for the 5 different categories, and a sum row for the total of all categories.

If I take the average of each column and sum the averages together, should it equal the average of the rows' sums (ignoring rounding error, of course)?

(I've got a case where the two values keep coming out different by about 30% and I'm wondering just how crazy I am.)

Update: See below — I was (slightly) crazy and had an error in my code.

Best Answer

The average of the entries in a column is the sum of the entries in that column, divided by the number of entries. The number of entries is the number of rows. So the sum of the averages is the sum of all the entries in the table, divided by the number of rows.

The average of the row sums is the sum of all entries in the table divided by the number of rows, so you should get the same number either way.

Example

Let say you compute the two indicators for summer:

strength_sum summer = n_june + n_july + n_august = 62
strength_avg summer = (n_june + n_july + n_august) / 3 = 20.67

And for winter:

strength_sum winter = n_december + n_january + n_february = 70
strength_avg winter = (n_december + n_january + n_february) / 3 = 23.33

Then strength_sum summer < strength_sum winter is equivalent to strength_avg summer < strength_avg winter (just by multiplying by 3). In both cases, the summer had less animals than the winter.

But what if...

If your seasons has a different number of months, or if you want to generalize to other time periods, I think that using the average of averages is more meaningful, regarding your issue.

Imagine you were in the Alps mountains, where the summer and autumn are way shorter than winter. It makes sense to favor the average of averages in order to correctly compare a 2-month long summer with a 5-month long winter.

For example, if you use the sum of average criterion: if strength_sum summer = 20 * 2 = 40 (n = 20 for each month of summer) and strength_sum winter = 10 * 5 = 50 (n = 10 for each month of winter), you don't want the winter to "win" simply because it's more than twice as long as summer : strength_sum summer = 40 < 50 = strength_sum winter, but strength_avg summer = 20 > 10 = strength_avg winter. It seems to me than strength_avg makes more sense here.

[Math] What are the different types of “averages” and when to use each one

I did some testing using what you cal "MSR" but I call Square Mean Root (SMR). It looks useful for dealing with data where large measurements indicate the opposite of an effect. For example in reaction time measurements, large reaction times indicate the participant was not paying attention. Unfortunately the "tyranny of large numbers" means those large reaction times have an inaapropriate effect on the results.

OTHER AVERAGES 1. Normalized Averages. a) Convert each data set to relative values. b) find the mean relative value c) restore the mean to actual value by multiplying and adding back the means of the subtracted constants and mean fo the divisors. NB this method wcan also produce realistic standard deviations.

Kalman Averages These weight means inversely according to their standard deviations. Thus less reliable measurements have less influence. This type of average can be found on the net and in some statistics books.
Winsorized Means These set limits to outliers, but do not eliminate them. The 5% and95% values of a set of measurments are taken as maximum values. Any measurments above or below these values are set to these max and minimum values. A mean is calculated from these modified measurements in the usual way. This is less questionable than simply eliminating the "outliers".

When using any "average" suh as mean, median, RMS, .., it is important to keep in mind what has been measured and what the size of the measurement numbers mean in terms of the effect producing those numerical values.

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[Math] average of averages vs sum of averages

Example

But what if...

[Math] What are the different types of “averages” and when to use each one

Related Question