The Law of Averages

The Law of Averages is not actually a law, but really just a lay term for some mathematical principles either completely unfounded in any mathematical law or borrowed from the Law of Large Numbers.  For more information, see this Wikipedia article.  However, this post is titled 'The Law of Averages' because I wanted to go into some detail about averaging averages.

In particular, I was asked this question: "If I have a data point for every minute of the data, I also have an average of 15 data points for every 15 minute period of the day, and I wanted to calculate the daily average, it would be better to use the 1 minute data versus the 15 minute averages (aka rollup data).  The average of the 15 minute averages would really be an average of averages and is therefore not accurate."

The reality is actually contrary to that thinking.  The average of averages of data is exactly the same as the average of the same source data.  To illustrate this, let's look at some data.

Below, we have random numbers between 0 and 100 to represent our data.  We have one data point for each minute of the day.  Every 15 minutes, an average is calculated.  For example, at 0:30 an average is calculated averaging the points from 0:30-0:44.

Time	1 Minute Data	15 Minute Average
0:00	37	51.6666666666667
0:01	74
0:02	49
0:03	21
0:04	59
0:05	15
0:06	32
0:07	69
0:08	40
0:09	26
0:10	87
0:11	38
0:12	96
0:13	79
0:14	53
0:15	60	25.8
0:16	35
0:17	7
0:18	24
0:19	46
0:20	3
0:21	22
0:22	1
0:23	39
0:24	9
0:25	34
0:26	0
0:27	42
0:28	34
0:29	31
0:30	4	33.8
0:31	5
0:32	81
0:33	36
0:34	35
0:35	9
0:36	36
0:37	41
0:38	76
0:39	93
0:40	21
0:41	12
0:42	10
0:43	21
0:44	27
0:45	5	48.8
0:46	42
0:47	83
0:48	48
0:49	18
0:50	79
0:51	24
0:52	78
0:53	19
0:54	5
0:55	52
0:56	100
0:57	39
0:58	70
0:59	70
1:00	14	44.0666666666667
1:01	45
1:02	82
1:03	34
1:04	74
1:05	90
1:06	33
1:07	10
1:08	43
1:09	24
1:10	96
1:11	12
1:12	17
1:13	62
1:14	25
1:15	96	54.7333333333333
1:16	19
1:17	91
1:18	58
1:19	76
1:20	24
1:21	86
1:22	35
1:23	48
1:24	31
1:25	67
1:26	72
1:27	9
1:28	45
1:29	64
1:30	79	40.4
1:31	68
1:32	0
1:33	10
1:34	30
1:35	23
1:36	65
1:37	74
1:38	53
1:39	78
1:40	4
1:41	3
1:42	38
1:43	78
1:44	3
1:45	91	47.6
1:46	85
1:47	44
1:48	4
1:49	6
1:50	13
1:51	39
1:52	93
1:53	52
1:54	35
1:55	49
1:56	69
1:57	67
1:58	64
1:59	3
2:00	11	56.8666666666667
2:01	72
2:02	54
2:03	45
2:04	16
2:05	17
2:06	75
2:07	96
2:08	96
2:09	62
2:10	80
2:11	55
2:12	96
2:13	34
2:14	44
2:15	73	46.6666666666667
2:16	57
2:17	21
2:18	61
2:19	28
2:20	11
2:21	16
2:22	1
2:23	86
2:24	23
2:25	64
2:26	89
2:27	46
2:28	38
2:29	86
2:30	42	45.8
2:31	15
2:32	7
2:33	3
2:34	40
2:35	41
2:36	8
2:37	66
2:38	1
2:39	85
2:40	59
2:41	89
2:42	94
2:43	42
2:44	95
2:45	64	57.6
2:46	58
2:47	6
2:48	14
2:49	17
2:50	100
2:51	85
2:52	72
2:53	93
2:54	58
2:55	100
2:56	32
2:57	53
2:58	36
2:59	76
3:00	4	38.7333333333333
3:01	19
3:02	32
3:03	14
3:04	84
3:05	26
3:06	17
3:07	65
3:08	20
3:09	15
3:10	26
3:11	52
3:12	79
3:13	31
3:14	97
3:15	8	55.6666666666667
3:16	47
3:17	87
3:18	66
3:19	23
3:20	88
3:21	28
3:22	53
3:23	87
3:24	66
3:25	32
3:26	71
3:27	87
3:28	10
3:29	82
3:30	84	54.9333333333333
3:31	81
3:32	54
3:33	88
3:34	77
3:35	98
3:36	2
3:37	15
3:38	92
3:39	72
3:40	16
3:41	45
3:42	4
3:43	60
3:44	36
3:45	84	63.4666666666667
3:46	97
3:47	87
3:48	29
3:49	53
3:50	11
3:51	84
3:52	83
3:53	76
3:54	64
3:55	50
3:56	36
3:57	83
3:58	38
3:59	77
4:00	3	64.1333333333333
4:01	79
4:02	90
4:03	91
4:04	46
4:05	53
4:06	26
4:07	77
4:08	46
4:09	62
4:10	60
4:11	93
4:12	78
4:13	92
4:14	66
4:15	35	37.1333333333333
4:16	81
4:17	42
4:18	50
4:19	14
4:20	26
4:21	72
4:22	4
4:23	15
4:24	65
4:25	14
4:26	9
4:27	10
4:28	38
4:29	82
4:30	14	51
4:31	48
4:32	62
4:33	36
4:34	89
4:35	3
4:36	53
4:37	58
4:38	86
4:39	71
4:40	49
4:41	67
4:42	20
4:43	96
4:44	13
4:45	0	47.3333333333333
4:46	91
4:47	64
4:48	28
4:49	96
4:50	23
4:51	37
4:52	37
4:53	77
4:54	15
4:55	13
4:56	98
4:57	7
4:58	58
4:59	66
5:00	48	48.6
5:01	22
5:02	5
5:03	76
5:04	96
5:05	5
5:06	10
5:07	3
5:08	34
5:09	56
5:10	82
5:11	96
5:12	58
5:13	80
5:14	58
5:15	48	48.1333333333333
5:16	30
5:17	50
5:18	71
5:19	12
5:20	12
5:21	98
5:22	24
5:23	41
5:24	21
5:25	66
5:26	24
5:27	99
5:28	73
5:29	53
5:30	8	37.3333333333333
5:31	67
5:32	17
5:33	0
5:34	0
5:35	37
5:36	51
5:37	33
5:38	28
5:39	58
5:40	3
5:41	74
5:42	79
5:43	90
5:44	15
5:45	56	55.7333333333333
5:46	82
5:47	66
5:48	64
5:49	69
5:50	94
5:51	81
5:52	23
5:53	22
5:54	41
5:55	16
5:56	89
5:57	47
5:58	74
5:59	12
6:00	70	49.2666666666667
6:01	72
6:02	76
6:03	28
6:04	62
6:05	28
6:06	46
6:07	62
6:08	61
6:09	34
6:10	52
6:11	69
6:12	12
6:13	37
6:14	30
6:15	79	51.8666666666667
6:16	35
6:17	74
6:18	71
6:19	12
6:20	52
6:21	72
6:22	6
6:23	89
6:24	16
6:25	17
6:26	30
6:27	64
6:28	76
6:29	85
6:30	47	42.8666666666667
6:31	44
6:32	46
6:33	46
6:34	2
6:35	83
6:36	12
6:37	24
6:38	47
6:39	94
6:40	32
6:41	70
6:42	70
6:43	24
6:44	2
6:45	77	54.2
6:46	1
6:47	27
6:48	47
6:49	18
6:50	75
6:51	46
6:52	83
6:53	58
6:54	24
6:55	58
6:56	81
6:57	65
6:58	57
6:59	96
7:00	34	66.8666666666667
7:01	99
7:02	79
7:03	75
7:04	89
7:05	92
7:06	63
7:07	47
7:08	49
7:09	93
7:10	45
7:11	40
7:12	97
7:13	41
7:14	60
7:15	56	52.0666666666667
7:16	71
7:17	92
7:18	31
7:19	68
7:20	1
7:21	3
7:22	64
7:23	21
7:24	7
7:25	82
7:26	8
7:27	99
7:28	96
7:29	82
7:30	97	51.4
7:31	5
7:32	1
7:33	82
7:34	67
7:35	30
7:36	29
7:37	89
7:38	53
7:39	80
7:40	14
7:41	88
7:42	73
7:43	36
7:44	27
7:45	53	47.4666666666667
7:46	47
7:47	55
7:48	4
7:49	46
7:50	65
7:51	47
7:52	21
7:53	25
7:54	51
7:55	87
7:56	24
7:57	46
7:58	75
7:59	66
8:00	63	41.5333333333333
8:01	46
8:02	69
8:03	58
8:04	14
8:05	21
8:06	83
8:07	75
8:08	23
8:09	12
8:10	60
8:11	63
8:12	6
8:13	9
8:14	21
8:15	53	40.6
8:16	42
8:17	22
8:18	0
8:19	65
8:20	6
8:21	66
8:22	16
8:23	85
8:24	8
8:25	78
8:26	44
8:27	86
8:28	30
8:29	8
8:30	98	68.3333333333333
8:31	81
8:32	93
8:33	66
8:34	60
8:35	27
8:36	84
8:37	62
8:38	81
8:39	76
8:40	94
8:41	19
8:42	60
8:43	38
8:44	86
8:45	92	47.7333333333333
8:46	39
8:47	18
8:48	49
8:49	36
8:50	57
8:51	59
8:52	84
8:53	8
8:54	36
8:55	14
8:56	48
8:57	51
8:58	67
8:59	58
9:00	73	44.7333333333333
9:01	90
9:02	59
9:03	2
9:04	1
9:05	2
9:06	33
9:07	51
9:08	33
9:09	81
9:10	2
9:11	93
9:12	79
9:13	18
9:14	54
9:15	87	48.8666666666667
9:16	75
9:17	33
9:18	19
9:19	24
9:20	14
9:21	45
9:22	13
9:23	93
9:24	54
9:25	69
9:26	12
9:27	2
9:28	99
9:29	94
9:30	42	44.4666666666667
9:31	70
9:32	83
9:33	20
9:34	66
9:35	10
9:36	35
9:37	74
9:38	18
9:39	35
9:40	7
9:41	64
9:42	67
9:43	55
9:44	21
9:45	48	53.4
9:46	91
9:47	88
9:48	3
9:49	11
9:50	45
9:51	8
9:52	20
9:53	96
9:54	17
9:55	76
9:56	78
9:57	54
9:58	99
9:59	67
10:00	18	53.5333333333333
10:01	52
10:02	33
10:03	19
10:04	65
10:05	88
10:06	56
10:07	99
10:08	18
10:09	98
10:10	50
10:11	46
10:12	42
10:13	53
10:14	66
10:15	61	58.5333333333333
10:16	84
10:17	66
10:18	44
10:19	43
10:20	8
10:21	26
10:22	77
10:23	55
10:24	85
10:25	6
10:26	91
10:27	66
10:28	66
10:29	100
10:30	74	46.3333333333333
10:31	70
10:32	10
10:33	72
10:34	35
10:35	19
10:36	86
10:37	72
10:38	7
10:39	31
10:40	95
10:41	35
10:42	46
10:43	2
10:44	41
10:45	45	46.3333333333333
10:46	85
10:47	3
10:48	11
10:49	48
10:50	90
10:51	59
10:52	40
10:53	26
10:54	9
10:55	94
10:56	7
10:57	70
10:58	31
10:59	77
11:00	67	47
11:01	78
11:02	59
11:03	0
11:04	70
11:05	77
11:06	71
11:07	49
11:08	51
11:09	63
11:10	21
11:11	21
11:12	50
11:13	11
11:14	17
11:15	4	46.9333333333333
11:16	54
11:17	9
11:18	49
11:19	30
11:20	82
11:21	41
11:22	22
11:23	33
11:24	29
11:25	78
11:26	88
11:27	37
11:28	84
11:29	64
11:30	23	47.4666666666667
11:31	50
11:32	8
11:33	50
11:34	24
11:35	84
11:36	53
11:37	78
11:38	80
11:39	2
11:40	61
11:41	89
11:42	77
11:43	16
11:44	17
11:45	19	36.2666666666667
11:46	64
11:47	85
11:48	9
11:49	30
11:50	24
11:51	5
11:52	10
11:53	59
11:54	7
11:55	42
11:56	21
11:57	69
11:58	55
11:59	45
12:00	37	48.0666666666667
12:01	60
12:02	85
12:03	44
12:04	28
12:05	89
12:06	35
12:07	18
12:08	48
12:09	64
12:10	3
12:11	38
12:12	96
12:13	5
12:14	71
12:15	71	46.4
12:16	93
12:17	37
12:18	7
12:19	38
12:20	85
12:21	28
12:22	86
12:23	12
12:24	65
12:25	38
12:26	6
12:27	66
12:28	24
12:29	40
12:30	26	42
12:31	98
12:32	74
12:33	13
12:34	46
12:35	52
12:36	39
12:37	14
12:38	38
12:39	25
12:40	28
12:41	30
12:42	44
12:43	63
12:44	40
12:45	96	41.0666666666667
12:46	13
12:47	50
12:48	8
12:49	43
12:50	27
12:51	38
12:52	97
12:53	0
12:54	11
12:55	84
12:56	73
12:57	11
12:58	20
12:59	45
13:00	39	60.1333333333333
13:01	94
13:02	95
13:03	82
13:04	39
13:05	26
13:06	80
13:07	70
13:08	19
13:09	79
13:10	74
13:11	78
13:12	16
13:13	20
13:14	91
13:15	58	46
13:16	67
13:17	52
13:18	55
13:19	47
13:20	25
13:21	8
13:22	78
13:23	53
13:24	55
13:25	43
13:26	9
13:27	28
13:28	56
13:29	56
13:30	26	27.7333333333333
13:31	5
13:32	17
13:33	2
13:34	53
13:35	9
13:36	5
13:37	63
13:38	91
13:39	46
13:40	7
13:41	23
13:42	25
13:43	3
13:44	41
13:45	94	49.0666666666667
13:46	70
13:47	49
13:48	60
13:49	59
13:50	51
13:51	71
13:52	16
13:53	37
13:54	31
13:55	13
13:56	64
13:57	76
13:58	30
13:59	15
14:00	0	32.5333333333333
14:01	6
14:02	7
14:03	13
14:04	52
14:05	25
14:06	53
14:07	15
14:08	65
14:09	70
14:10	72
14:11	12
14:12	23
14:13	23
14:14	52
14:15	87	50.9333333333333
14:16	98
14:17	24
14:18	60
14:19	32
14:20	31
14:21	10
14:22	63
14:23	28
14:24	20
14:25	42
14:26	90
14:27	43
14:28	81
14:29	55
14:30	60	53.7333333333333
14:31	45
14:32	98
14:33	96
14:34	11
14:35	14
14:36	3
14:37	97
14:38	63
14:39	29
14:40	23
14:41	96
14:42	35
14:43	89
14:44	47
14:45	67	58.4666666666667
14:46	41
14:47	66
14:48	70
14:49	98
14:50	47
14:51	37
14:52	68
14:53	61
14:54	16
14:55	70
14:56	19
14:57	46
14:58	100
14:59	71
15:00	97	56.7333333333333
15:01	65
15:02	99
15:03	60
15:04	54
15:05	3
15:06	60
15:07	79
15:08	30
15:09	50
15:10	37
15:11	54
15:12	51
15:13	63
15:14	49
15:15	27	43.0666666666667
15:16	10
15:17	77
15:18	82
15:19	15
15:20	55
15:21	68
15:22	74
15:23	11
15:24	0
15:25	72
15:26	49
15:27	62
15:28	41
15:29	3
15:30	54	64.8666666666667
15:31	86
15:32	95
15:33	31
15:34	45
15:35	71
15:36	100
15:37	59
15:38	43
15:39	15
15:40	88
15:41	3
15:42	97
15:43	88
15:44	98
15:45	76	49.7333333333333
15:46	17
15:47	82
15:48	34
15:49	64
15:50	85
15:51	52
15:52	29
15:53	60
15:54	76
15:55	16
15:56	12
15:57	64
15:58	52
15:59	27
16:00	84	42.8
16:01	10
16:02	11
16:03	13
16:04	99
16:05	29
16:06	64
16:07	29
16:08	20
16:09	64
16:10	36
16:11	14
16:12	38
16:13	69
16:14	62
16:15	96	52.0666666666667
16:16	39
16:17	10
16:18	80
16:19	88
16:20	14
16:21	2
16:22	4
16:23	54
16:24	25
16:25	78
16:26	88
16:27	99
16:28	89
16:29	15
16:30	73	56
16:31	68
16:32	29
16:33	43
16:34	26
16:35	54
16:36	13
16:37	100
16:38	89
16:39	33
16:40	48
16:41	35
16:42	92
16:43	56
16:44	81
16:45	30	50.1333333333333
16:46	99
16:47	4
16:48	100
16:49	5
16:50	10
16:51	37
16:52	12
16:53	92
16:54	99
16:55	95
16:56	69
16:57	20
16:58	52
16:59	28
17:00	45	44.3333333333333
17:01	27
17:02	20
17:03	35
17:04	57
17:05	88
17:06	91
17:07	7
17:08	66
17:09	41
17:10	20
17:11	32
17:12	77
17:13	8
17:14	51
17:15	38	53.6
17:16	36
17:17	39
17:18	40
17:19	15
17:20	91
17:21	45
17:22	37
17:23	99
17:24	98
17:25	54
17:26	64
17:27	97
17:28	48
17:29	3
17:30	55	50.6
17:31	33
17:32	71
17:33	40
17:34	65
17:35	80
17:36	46
17:37	57
17:38	43
17:39	3
17:40	70
17:41	30
17:42	73
17:43	63
17:44	30
17:45	79	67.4
17:46	67
17:47	49
17:48	75
17:49	98
17:50	78
17:51	83
17:52	83
17:53	54
17:54	77
17:55	46
17:56	52
17:57	66
17:58	64
17:59	40
18:00	95	58.8666666666667
18:01	100
18:02	83
18:03	61
18:04	9
18:05	11
18:06	35
18:07	40
18:08	75
18:09	80
18:10	14
18:11	82
18:12	37
18:13	72
18:14	89
18:15	48	45.6666666666667
18:16	40
18:17	90
18:18	27
18:19	10
18:20	65
18:21	52
18:22	4
18:23	29
18:24	23
18:25	29
18:26	90
18:27	92
18:28	1
18:29	85
18:30	62	63.2
18:31	46
18:32	99
18:33	50
18:34	61
18:35	74
18:36	90
18:37	45
18:38	91
18:39	100
18:40	93
18:41	67
18:42	19
18:43	16
18:44	35
18:45	86	58.9333333333333
18:46	64
18:47	99
18:48	26
18:49	99
18:50	19
18:51	31
18:52	93
18:53	11
18:54	74
18:55	88
18:56	36
18:57	56
18:58	68
18:59	34
19:00	18	69.3333333333333
19:01	97
19:02	26
19:03	84
19:04	98
19:05	53
19:06	79
19:07	95
19:08	72
19:09	92
19:10	24
19:11	95
19:12	86
19:13	96
19:14	25
19:15	50	58.8
19:16	88
19:17	95
19:18	20
19:19	85
19:20	41
19:21	39
19:22	45
19:23	67
19:24	39
19:25	63
19:26	98
19:27	2
19:28	61
19:29	89
19:30	90	57.8666666666667
19:31	74
19:32	84
19:33	98
19:34	83
19:35	25
19:36	92
19:37	72
19:38	26
19:39	34
19:40	48
19:41	2
19:42	70
19:43	7
19:44	63
19:45	38	53.7333333333333
19:46	12
19:47	44
19:48	23
19:49	0
19:50	80
19:51	88
19:52	88
19:53	93
19:54	34
19:55	69
19:56	15
19:57	95
19:58	85
19:59	42
20:00	37	35
20:01	100
20:02	3
20:03	17
20:04	5
20:05	44
20:06	99
20:07	29
20:08	100
20:09	2
20:10	5
20:11	11
20:12	25
20:13	24
20:14	24
20:15	63	40.7333333333333
20:16	81
20:17	90
20:18	44
20:19	61
20:20	18
20:21	37
20:22	6
20:23	85
20:24	1
20:25	53
20:26	3
20:27	54
20:28	5
20:29	10
20:30	9	54.4666666666667
20:31	68
20:32	89
20:33	69
20:34	74
20:35	50
20:36	84
20:37	15
20:38	58
20:39	80
20:40	30
20:41	69
20:42	86
20:43	11
20:44	25
20:45	0	45.4666666666667
20:46	49
20:47	29
20:48	65
20:49	26
20:50	62
20:51	41
20:52	65
20:53	27
20:54	78
20:55	70
20:56	17
20:57	90
20:58	5
20:59	58
21:00	24	49.4
21:01	95
21:02	23
21:03	16
21:04	0
21:05	25
21:06	70
21:07	70
21:08	85
21:09	81
21:10	81
21:11	90
21:12	44
21:13	9
21:14	28
21:15	67	53.1333333333333
21:16	89
21:17	54
21:18	11
21:19	17
21:20	63
21:21	97
21:22	88
21:23	95
21:24	63
21:25	65
21:26	56
21:27	13
21:28	3
21:29	16
21:30	14	55.8
21:31	83
21:32	2
21:33	73
21:34	81
21:35	70
21:36	5
21:37	84
21:38	63
21:39	88
21:40	33
21:41	49
21:42	90
21:43	75
21:44	27
21:45	43	53.7333333333333
21:46	67
21:47	61
21:48	17
21:49	49
21:50	32
21:51	100
21:52	93
21:53	93
21:54	53
21:55	23
21:56	16
21:57	4
21:58	76
21:59	79
22:00	47	39.6
22:01	46
22:02	9
22:03	92
22:04	67
22:05	2
22:06	39
22:07	38
22:08	52
22:09	1
22:10	15
22:11	17
22:12	78
22:13	14
22:14	77
22:15	70	43.8666666666667
22:16	58
22:17	54
22:18	71
22:19	0
22:20	68
22:21	10
22:22	79
22:23	50
22:24	57
22:25	49
22:26	0
22:27	50
22:28	5
22:29	37
22:30	50	46.8
22:31	27
22:32	54
22:33	39
22:34	1
22:35	88
22:36	23
22:37	81
22:38	70
22:39	54
22:40	39
22:41	52
22:42	33
22:43	2
22:44	89
22:45	5	58.9333333333333
22:46	78
22:47	39
22:48	82
22:49	79
22:50	73
22:51	36
22:52	75
22:53	81
22:54	71
22:55	40
22:56	33
22:57	90
22:58	38
22:59	64
23:00	14	51.2666666666667
23:01	27
23:02	93
23:03	80
23:04	12
23:05	98
23:06	9
23:07	67
23:08	87
23:09	10
23:10	15
23:11	69
23:12	60
23:13	60
23:14	68
23:15	21	59.4666666666667
23:16	86
23:17	76
23:18	21
23:19	81
23:20	31
23:21	71
23:22	68
23:23	90
23:24	63
23:25	8
23:26	85
23:27	95
23:28	29
23:29	67
23:30	45	46.2666666666667
23:31	71
23:32	73
23:33	58
23:34	23
23:35	16
23:36	16
23:37	79
23:38	84
23:39	100
23:40	4
23:41	23
23:42	55
23:43	25
23:44	22
23:45	8	45.4
23:46	27
23:47	49
23:48	22
23:49	84
23:50	46
23:51	24
23:52	19
23:53	53
23:54	32
23:55	72
23:56	69
23:57	96
23:58	20
23:59	60

Here's the funny part.  The question that was posited essentially asked if the average of the 1 minute data would be more accurate than the average of the 15 minute averages.  Here are the results:

Average of all the 1 minute data points: 49.76319444
Average of all the 15 minute averages: 49.76319444

They're exactly the same!  The reason for this comes from the associative property of addition.  Think about the formula for an average:
x̄ = (a₁+a₂+a₃+...+a_n)/n

If you had multiple sets (i.e. 15 minute averages):
x̄ = {(a₁+a₂+a₃+...+a_n)/n_a + (b₁+b₂+b₃+...+b_n)/n_b + (c₁+c₂+c₃+...+c_n)/n_c}/3

Since in this case n_a=n_b=n_c, then substitue n and factor out:
x̄ = {(a₁+a₂+a₃+...+a_n) + (b₁+b₂+b₃+...+b_n) + (c₁+c₂+c₃+...+c_n)}/3n

Associative property of addition:
x̄ = (a₁+a₂+a₃+...+a_n+b₁+b₂+b₃+...+b_n+c₁+c₂+c₃+...+c_n)/3n

Since in this case 3n = n + n + n and n_a=n_b=n_c=n, then substitue for n:
x̄ = (a₁+a₂+a₃+...+a_n+b₁+b₂+b₃+...+b_n+c₁+c₂+c₃+...+c_n)/(n + n + n)
x̄ = (a₁+a₂+a₃+...+a_n+b₁+b₂+b₃+...+b_n+c₁+c₂+c₃+...+c_n)/(n_a + n_b + n_c)

Knowing that n_a + n_b + n_c is actually the total number of original points gets us back to the original equation for calculating the average of the 1 minute data.  So the average of all the data points is the same as the average of evenly spaced averages of all the data points.  If you're wondering if this works, go ahead and mock it up in Excel.  I did.