how to curve the exam score by the power function

When a class has poor percentage grades, "take the square root and multiply by 10" is a common technique for raising the grades of the entire class. For finer control of the raising curve, an extension from the square root function to the power/exponential function with an extra RATIO parameter is discussed here. The RATIO parameter can take a recommended value between 0 and 1. It can be roughly seen as the proportion by which the original passing threshold of 60 points is reduced. 1 means that the passing threshold is not reduced, and 0.5 means that it is roughly reduced to 60*0.5=30 points to pass.

Assume the original grade is Old_Score in the range of [0,100] points and the adjusted grade is New_Score in the range of [0,100] points as well. Then using the parameter Ratio in the range of [0,1], the following EXCEL formula for exam score adjustment can be defined:
  

New_Score = ROUNDUP(POWER(Old_Score,Ratio)*100/POWER(100,Ratio),0) 

where ROUNDUP(X,Y) will round X up to the Y decimal places and POWER(X,Y) will compute the result of X raised to the power of Y. Both are EXCEL built-in functions.

Some comments about the above exam score curving formula can be made:

1. When Ratio is equal to 1,  New_Score is equal to Old_Score which means no adjustment.
2. When Ratio is equal to 0.5, New_Score is equal to the square root of Old_Score multiplied by 10 which means the traditional square root curving.
3. When Ratio is equal to 0, New_Score is equal to 100 for all Old_Score except 0 which means the extreme adjustment.
4. For all Ratio in the range of [0,1], this formula will only give non-decreasing adjustment. It means that all adjusted grades will be greater than or equal to the original grades and never exceed the full score of 100 points.

Therefore by properly using Ratio from high (1) to low (0), a spectrum of curving from no adjustment to extreme upward adjustment can be fine tuned. A graph of the curves for different Ratios is shown below. The traditional square root curve is labeled with adjusted scores.

A table for the curving formula is listed below. Each column represents a curving result for a specific Ratio from 0.1 to 1.1 in steps of 0.1. Note that grades in pink color are below 60, the usual passing threshold. Also, when the ratio is 1.1, some scores will decrease, resulting in the undesirable effect of passing grades becoming failing grades.

Ratio	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	1.1
0	0	0	0	0	0	0	0	0	0	0	0
1	64	40	26	16	10	7	4	3	2	1	1
2	68	46	31	21	15	10	7	5	3	2	2
3	71	50	35	25	18	13	9	7	5	3	3
4	73	53	39	28	20	15	11	8	6	4	3
5	75	55	41	31	23	17	13	10	7	5	4
6	76	57	43	33	25	19	14	11	8	6	5
7	77	59	46	35	27	21	16	12	10	7	6
8	78	61	47	37	29	22	18	14	11	8	7
9	79	62	49	39	30	24	19	15	12	9	8
10	80	64	51	40	32	26	20	16	13	10	8
11	81	65	52	42	34	27	22	18	14	11	9
12	81	66	53	43	35	29	23	19	15	12	10
13	82	67	55	45	37	30	24	20	16	13	11
14	83	68	56	46	38	31	26	21	18	14	12
15	83	69	57	47	39	33	27	22	19	15	13
16	84	70	58	49	40	34	28	24	20	16	14
17	84	71	59	50	42	35	29	25	21	17	15
18	85	71	60	51	43	36	31	26	22	18	16
19	85	72	61	52	44	37	32	27	23	19	17
20	86	73	62	53	45	39	33	28	24	20	18
21	86	74	63	54	46	40	34	29	25	21	18
22	86	74	64	55	47	41	35	30	26	22	19
23	87	75	65	56	48	42	36	31	27	23	20
24	87	76	66	57	49	43	37	32	28	24	21
25	88	76	66	58	50	44	38	33	29	25	22
26	88	77	67	59	51	45	39	35	30	26	23
27	88	77	68	60	52	46	40	36	31	27	24
28	89	78	69	61	53	47	42	37	32	28	25
29	89	79	69	61	54	48	43	38	33	29	26
30	89	79	70	62	55	49	44	39	34	30	27
31	89	80	71	63	56	50	45	40	35	31	28
32	90	80	72	64	57	51	46	41	36	32	29
33	90	81	72	65	58	52	47	42	37	33	30
34	90	81	73	65	59	53	47	43	38	34	31
35	91	82	73	66	60	54	48	44	39	35	32
36	91	82	74	67	60	55	49	45	40	36	33
37	91	82	75	68	61	56	50	46	41	37	34
38	91	83	75	68	62	56	51	47	42	38	35
39	92	83	76	69	63	57	52	48	43	39	36
40	92	84	76	70	64	58	53	49	44	40	37
41	92	84	77	71	65	59	54	50	45	41	38
42	92	85	78	71	65	60	55	50	46	42	39
43	92	85	78	72	66	61	56	51	47	43	40
44	93	85	79	73	67	62	57	52	48	44	41
45	93	86	79	73	68	62	58	53	49	45	42
46	93	86	80	74	68	63	59	54	50	46	43
47	93	86	80	74	69	64	59	55	51	47	44
48	93	87	81	75	70	65	60	56	52	48	45
49	94	87	81	76	70	66	61	57	53	49	46
50	94	88	82	76	71	66	62	58	54	50	47
51	94	88	82	77	72	67	63	59	55	51	48
52	94	88	83	77	73	68	64	60	56	52	49
53	94	89	83	78	73	69	65	61	57	53	50
54	95	89	84	79	74	70	65	62	58	54	51
55	95	89	84	79	75	70	66	62	59	55	52
56	95	90	85	80	75	71	67	63	60	56	53
57	95	90	85	80	76	72	68	64	61	57	54
58	95	90	85	81	77	73	69	65	62	58	55
59	95	90	86	81	77	73	70	66	63	59	56
60	96	91	86	82	78	74	70	67	64	60	58
61	96	91	87	83	79	75	71	68	65	61	59
62	96	91	87	83	79	76	72	69	66	62	60
63	96	92	88	84	80	76	73	70	66	63	61
64	96	92	88	84	80	77	74	70	67	64	62
65	96	92	88	85	81	78	74	71	68	65	63
66	96	93	89	85	82	78	75	72	69	66	64
67	97	93	89	86	82	79	76	73	70	67	65
68	97	93	90	86	83	80	77	74	71	68	66
69	97	93	90	87	84	81	78	75	72	69	67
70	97	94	90	87	84	81	78	76	73	70	68
71	97	94	91	88	85	82	79	77	74	71	69
72	97	94	91	88	85	83	80	77	75	72	70
73	97	94	91	89	86	83	81	78	76	73	71
74	98	95	92	89	87	84	81	79	77	74	72
75	98	95	92	90	87	85	82	80	78	75	73
76	98	95	93	90	88	85	83	81	79	76	74
77	98	95	93	91	88	86	84	82	80	77	76
78	98	96	93	91	89	87	85	82	80	78	77
79	98	96	94	92	89	87	85	83	81	79	78
80	98	96	94	92	90	88	86	84	82	80	79
81	98	96	94	92	90	89	87	85	83	81	80
82	99	97	95	93	91	89	88	86	84	82	81
83	99	97	95	93	92	90	88	87	85	83	82
84	99	97	95	94	92	91	89	87	86	84	83
85	99	97	96	94	93	91	90	88	87	85	84
86	99	98	96	95	93	92	90	89	88	86	85
87	99	98	96	95	94	92	91	90	89	87	86
88	99	98	97	96	94	93	92	91	90	88	87
89	99	98	97	96	95	94	93	92	91	89	88
90	99	98	97	96	95	94	93	92	91	90	90
91	100	99	98	97	96	95	94	93	92	91	91
92	100	99	98	97	96	96	95	94	93	92	92
93	100	99	98	98	97	96	96	95	94	93	93
94	100	99	99	98	97	97	96	96	95	94	94
95	100	99	99	98	98	97	97	96	96	95	95
96	100	100	99	99	98	98	98	97	97	96	96
97	100	100	100	99	99	99	98	98	98	97	97
98	100	100	100	100	99	99	99	99	99	98	98
99	100	100	100	100	100	100	100	100	100	99	99
100	100	100	100	100	100	100	100	100	100	100	100

To see the distribution of boost, a graph and a table showing the score boost (or difference) before and after the adjustment are given below. It can be seen that as Ratio increases, the maximum score boost decreases and occurs at greater original scores.  For example, at Ratio=0.3, a maximum score boost of 42 occurs in the interval of Old_Score=[13,24]. At Ratio=0.5, a maximum score boost of 25 occurs in the interval of Old_Score=[17,35]. And at Ratio=0.7, a maximum score boost of 14 occurs in the interval of Old_Score=[28,33].

In the following table, each column represents the score boost  for a specific Ratio. Maximum score boost in each column is marked in pink color.

Ratio	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	1.1
0	0	0	0	0	0	0	0	0	0	0	0
1	63	39	25	15	9	6	3	2	1	0	0
2	66	44	29	19	13	8	5	3	1	0	0
3	68	47	32	22	15	10	6	4	2	0	0
4	69	49	35	24	16	11	7	4	2	0	-1
5	70	50	36	26	18	12	8	5	2	0	-1
6	70	51	37	27	19	13	8	5	2	0	-1
7	70	52	39	28	20	14	9	5	3	0	-1
8	70	53	39	29	21	14	10	6	3	0	-1
9	70	53	40	30	21	15	10	6	3	0	-1
10	70	54	41	30	22	16	10	6	3	0	-2
11	70	54	41	31	23	16	11	7	3	0	-2
12	69	54	41	31	23	17	11	7	3	0	-2
13	69	54	42	32	24	17	11	7	3	0	-2
14	69	54	42	32	24	17	12	7	4	0	-2
15	68	54	42	32	24	18	12	7	4	0	-2
16	68	54	42	33	24	18	12	8	4	0	-2
17	67	54	42	33	25	18	12	8	4	0	-2
18	67	53	42	33	25	18	13	8	4	0	-2
19	66	53	42	33	25	18	13	8	4	0	-2
20	66	53	42	33	25	19	13	8	4	0	-2
21	65	53	42	33	25	19	13	8	4	0	-3
22	64	52	42	33	25	19	13	8	4	0	-3
23	64	52	42	33	25	19	13	8	4	0	-3
24	63	52	42	33	25	19	13	8	4	0	-3
25	63	51	41	33	25	19	13	8	4	0	-3
26	62	51	41	33	25	19	13	9	4	0	-3
27	61	50	41	33	25	19	13	9	4	0	-3
28	61	50	41	33	25	19	14	9	4	0	-3
29	60	50	40	32	25	19	14	9	4	0	-3
30	59	49	40	32	25	19	14	9	4	0	-3
31	58	49	40	32	25	19	14	9	4	0	-3
32	58	48	40	32	25	19	14	9	4	0	-3
33	57	48	39	32	25	19	14	9	4	0	-3
34	56	47	39	31	25	19	13	9	4	0	-3
35	56	47	38	31	25	19	13	9	4	0	-3
36	55	46	38	31	24	19	13	9	4	0	-3
37	54	45	38	31	24	19	13	9	4	0	-3
38	53	45	37	30	24	18	13	9	4	0	-3
39	53	44	37	30	24	18	13	9	4	0	-3
40	52	44	36	30	24	18	13	9	4	0	-3
41	51	43	36	30	24	18	13	9	4	0	-3
42	50	43	36	29	23	18	13	8	4	0	-3
43	49	42	35	29	23	18	13	8	4	0	-3
44	49	41	35	29	23	18	13	8	4	0	-3
45	48	41	34	28	23	17	13	8	4	0	-3
46	47	40	34	28	22	17	13	8	4	0	-3
47	46	39	33	27	22	17	12	8	4	0	-3
48	45	39	33	27	22	17	12	8	4	0	-3
49	45	38	32	27	21	17	12	8	4	0	-3
50	44	38	32	26	21	16	12	8	4	0	-3
51	43	37	31	26	21	16	12	8	4	0	-3
52	42	36	31	25	21	16	12	8	4	0	-3
53	41	36	30	25	20	16	12	8	4	0	-3
54	41	35	30	25	20	16	11	8	4	0	-3
55	40	34	29	24	20	15	11	7	4	0	-3
56	39	34	29	24	19	15	11	7	4	0	-3
57	38	33	28	23	19	15	11	7	4	0	-3
58	37	32	27	23	19	15	11	7	4	0	-3
59	36	31	27	22	18	14	11	7	4	0	-3
60	36	31	26	22	18	14	10	7	4	0	-2
61	35	30	26	22	18	14	10	7	4	0	-2
62	34	29	25	21	17	14	10	7	4	0	-2
63	33	29	25	21	17	13	10	7	3	0	-2
64	32	28	24	20	16	13	10	6	3	0	-2
65	31	27	23	20	16	13	9	6	3	0	-2
66	30	27	23	19	16	12	9	6	3	0	-2
67	30	26	22	19	15	12	9	6	3	0	-2
68	29	25	22	18	15	12	9	6	3	0	-2
69	28	24	21	18	15	12	9	6	3	0	-2
70	27	24	20	17	14	11	8	6	3	0	-2
71	26	23	20	17	14	11	8	6	3	0	-2
72	25	22	19	16	13	11	8	5	3	0	-2
73	24	21	18	16	13	10	8	5	3	0	-2
74	24	21	18	15	13	10	7	5	3	0	-2
75	23	20	17	15	12	10	7	5	3	0	-2
76	22	19	17	14	12	9	7	5	3	0	-2
77	21	18	16	14	11	9	7	5	3	0	-1
78	20	18	15	13	11	9	7	4	2	0	-1
79	19	17	15	13	10	8	6	4	2	0	-1
80	18	16	14	12	10	8	6	4	2	0	-1
81	17	15	13	11	9	8	6	4	2	0	-1
82	17	15	13	11	9	7	6	4	2	0	-1
83	16	14	12	10	9	7	5	4	2	0	-1
84	15	13	11	10	8	7	5	3	2	0	-1
85	14	12	11	9	8	6	5	3	2	0	-1
86	13	12	10	9	7	6	4	3	2	0	-1
87	12	11	9	8	7	5	4	3	2	0	-1
88	11	10	9	8	6	5	4	3	2	0	-1
89	10	9	8	7	6	5	4	3	2	0	-1
90	9	8	7	6	5	4	3	2	1	0	0
91	9	8	7	6	5	4	3	2	1	0	0
92	8	7	6	5	4	4	3	2	1	0	0
93	7	6	5	5	4	3	3	2	1	0	0
94	6	5	5	4	3	3	2	2	1	0	0
95	5	4	4	3	3	2	2	1	1	0	0
96	4	4	3	3	2	2	2	1	1	0	0
97	3	3	3	2	2	2	1	1	1	0	0
98	2	2	2	2	1	1	1	1	1	0	0
99	1	1	1	1	1	1	1	1	1	0	0
100	0	0	0	0	0	0	0	0	0	0	0

In conclusion, the traditional square root curving method is a special case of the discussed power/exponential curving method. Compared to the square root curving method, the power/exponential curving method with a single tuning parameter can give you an easier and finer control of the entire class grades to meet your desired statistics (say, the average or the flunk ratio) requirement.

References:
1.Richeson (2008), How to curve an exam and assign grades, root functions
2.Nicolai (2008), How to Curve an Exam