Summing Percentages

Here is another write-up on summing percentages. Click here to download or view the calculations in Excel.
Example 1: Using the sum of all the numerators as the denominator for all the percentages to be summed yields a valid result.
Numerator	Numerator (Running Total)	Denominator	Denominator (Running Total)	Percentage	Running Sum of %	Running Sum of % divided by the number of percentages summed	Running Percentage
1	1	210	210	0.48%	0.48%	0.48%	0.48%
2	3	210	420	0.95%	1.43%	0.71%	0.71%
3	6	210	630	1.43%	2.86%	0.95%	0.95%
4	10	210	840	1.90%	4.76%	1.19%	1.19%
5	15	210	1050	2.38%	7.14%	1.43%	1.43%
6	21	210	1260	2.86%	10.00%	1.67%	1.67%
7	28	210	1470	3.33%	13.33%	1.90%	1.90%
8	36	210	1680	3.81%	17.14%	2.14%	2.14%
9	45	210	1890	4.29%	21.43%	2.38%	2.38%
10	55	210	2100	4.76%	26.19%	2.62%	2.62%
11	66	210	2310	5.24%	31.43%	2.86%	2.86%
12	78	210	2520	5.71%	37.14%	3.10%	3.10%
13	91	210	2730	6.19%	43.33%	3.33%	3.33%
14	105	210	2940	6.67%	50.00%	3.57%	3.57%
15	120	210	3150	7.14%	57.14%	3.81%	3.81%
16	136	210	3360	7.62%	64.76%	4.05%	4.05%
17	153	210	3570	8.10%	72.86%	4.29%	4.29%
18	171	210	3780	8.57%	81.43%	4.52%	4.52%
19	190	210	3990	9.05%	90.48%	4.76%	4.76%
20	210	210	4200	9.52%	100.00%	5.00%	5.00%
Example 2: Using 100 as the denominator for all the percentages to be summed does not yield a valid result.
Numerator	Numerator (Running Total)	Denominator	Denominator (Running Total)	Percentage	Running Sum of %	Running Sum of % divided by the number of percentages summed	Running Percentage
1	1	100	100	1.00%	1.00%	1.00%	1.00%
2	3	100	200	2.00%	3.00%	1.50%	1.50%
3	6	100	300	3.00%	6.00%	2.00%	2.00%
4	10	100	400	4.00%	10.00%	2.50%	2.50%
5	15	100	500	5.00%	15.00%	3.00%	3.00%
6	21	100	600	6.00%	21.00%	3.50%	3.50%
7	28	100	700	7.00%	28.00%	4.00%	4.00%
8	36	100	800	8.00%	36.00%	4.50%	4.50%
9	45	100	900	9.00%	45.00%	5.00%	5.00%
10	55	100	1000	10.00%	55.00%	5.50%	5.50%
11	66	100	1100	11.00%	66.00%	6.00%	6.00%
12	78	100	1200	12.00%	78.00%	6.50%	6.50%
13	91	100	1300	13.00%	91.00%	7.00%	7.00%
14	105	100	1400	14.00%	105.00%	7.50%	7.50%
15	120	100	1500	15.00%	120.00%	8.00%	8.00%
16	136	100	1600	16.00%	136.00%	8.50%	8.50%
17	153	100	1700	17.00%	153.00%	9.00%	9.00%
18	171	100	1800	18.00%	171.00%	9.50%	9.50%
19	190	100	1900	19.00%	190.00%	10.00%	10.00%
20	210	100	2000	20.00%	210.00%	10.50%	10.50%
Example 3: summing percentages calculated with different denominators does not yield meaningful data.
Numerator	Numerator (Running Total)	Denominator	Denominator (Running Total)	Percentage	Running Sum of %	Running Sum of % divided by the number of percentages summed	Running Percentage
1	1	10	10	10.00%	10.00%	10.00%	10.00%
2	3	25	35	8.00%	18.00%	9.00%	8.57%
3	6	12	47	25.00%	43.00%	14.33%	12.77%
4	10	15	62	26.67%	69.67%	17.42%	16.13%
5	15	17	79	29.41%	99.08%	19.82%	18.99%
6	21	88	167	6.82%	105.90%	17.65%	12.57%
7	28	398	565	1.76%	107.66%	15.38%	4.96%
8	36	49	614	16.33%	123.98%	15.50%	5.86%
9	45	50	664	18.00%	141.98%	15.78%	6.78%
10	55	66	730	15.15%	157.13%	15.71%	7.53%
11	66	5	735	220.00%	377.13%	34.28%	8.98%
12	78	75	810	16.00%	393.13%	32.76%	9.63%
13	91	49	859	26.53%	419.66%	32.28%	10.59%
14	105	30	889	46.67%	466.33%	33.31%	11.81%
15	120	31	920	48.39%	514.72%	34.31%	13.04%
16	136	32	952	50.00%	564.72%	35.29%	14.29%
17	153	33	985	51.52%	616.23%	36.25%	15.53%
18	171	34	1019	52.94%	669.17%	37.18%	16.78%
19	190	35	1054	54.29%	723.46%	38.08%	18.03%
20	210	36	1090	55.56%	779.02%	38.95%	19.27%