Sanger (And Illumina 1.3+ (And Solexa)) Phred Score (Q) ASCII Glyph Base Error Conversion Tables

Given the importance of the use of these scores both in FASTQ and MAQ (for MAQ (for me), specifically using alignment quality scores from Illumina sequencing runs to monitor run and sample quality), I was a bit surprised to not find some complete work-up of the meanings, the scores, the glyphs coordinated to the scores, and the encoding interpretations of these scores in one location. The two (three) tables shown here hopefully provide a meaningful summary.

I should qualify that much of the background for this page was taken from four key places. First is the wikipedia entry for FASTQ. Second is the wikipedia entry for Phred quality score. Third is the Rosetta Stone of Phred Score interpretation in the form of the open access article: P. J. A. Cock, C. J. Fields, N. Goto, M. L. Heuer and P. M. Rice, "The Sanger FASTQ file format for sequences with quality scores, and the Solexa/Illumina FASTQ variants." Nucleic Acids Research, 2010, Vol. 38, No. 6, 1767-1771 doi:10.1093/nar/gkp1137. Fourth is seqanswers.com in various forms.

(Sanger) Phred Quality Scores

I refer you to the two wikipedia articles on FASTQ and Phred Quality Scores for historical content (and for a brief discussion of the processing of chromatogram data for the production of quality scores). Table 1 shows the Q[Phred] (Phred Q) from P[Phred] values (Probability (P) Of Wrong Base), then adds the ASCII glyph codes (Sanger "Q + 33" Shift) and characters (Sanger "Q + 33" ASCII GLYPH) for the original Phred scores (Phred scores 0-to-93 use ASCII characters 33-to-126 in the Sanger method – this is performed to keep the single-character associated letters readable) and the Illumina 1.3+ codes (Illumina 1.3+ "Q + 64" Shift, using ASCII glyphs 64-to-126 to score from 0-to-62 on the "P" scale) and corresponding ASCII glyphs (Illumina 1.3+ "Q + 64" ASCII GLYPH). This is all likely completely self-explanatory (or hopefully will be by the bottom of the post). For review, the relationship between Phred quality score Q[Sanger] and the base-calling error probability P is

Q[Sanger]= -10 * log₁₀P

or, re-written for the logarithmically challenged…

P = 10^[-Q/10]

Table 1. Phred Quality Scores (Q), Wrong Base Probabilities, And Sanger And Illumina 1.3+ ASCII Glyphs.
Phred Q	Probability (P) Of Wrong Base	Sanger "Q + 33" Shift	Sanger "Q + 33" ASCII GLYPH	Illumina 1.3+ "Q + 64" Shift	Illumina 1.3+ "Q + 64" ASCII GLYPH
00	1.0000000000	033	!	064	@
01	0.7943282347	034	"	065	A
02	0.6309573445	035	#	066	B
03	0.5011872336	036	$	067	C
04	0.3981071706	037	%	068	D
05	0.3162277660	038	&	069	E
06	0.2511886432	039	'	070	F
07	0.1995262315	040	(	071	G
08	0.1584893192	041	)	072	H
09	0.1258925412	042	*	073	I
10	0.1000000000	043	+	074	J
11	0.0794328235	044	,	075	K
12	0.0630957344	045	–	076	L
13	0.0501187234	046	.	077	M
14	0.0398107171	047	/	078	N
15	0.0316227766	048	0	079	O
16	0.0251188643	049	1	080	P
17	0.0199526231	050	2	081	Q
18	0.0158489319	051	3	082	R
19	0.0125892541	052	4	083	S
20	0.0100000000	053	5	084	T
21	0.0079432823	054	6	085	U
22	0.0063095734	055	7	086	V
23	0.0050118723	056	8	087	W
24	0.0039810717	057	9	088	X
25	0.0031622777	058	:	089	Y
26	0.0025118864	059	;	090	Z
27	0.0019952623	060	<	091	[
28	0.0015848932	061	=	092	\
29	0.0012589254	062	>	093	]
30	0.0010000000	063	?	094	^
31	0.0007943282	064	@	095	_
32	0.0006309573	065	A	096	`
33	0.0005011872	066	B	097	a
34	0.0003981072	067	C	098	b
35	0.0003162278	068	D	099	c
36	0.0002511886	069	E	100	d
37	0.0001995262	070	F	101	e
38	0.0001584893	071	G	102	f
39	0.0001258925	072	H	103	g
40	0.0001000000	073	I	104	h
41	0.0000794328	074	J	105	i
42	0.0000630957	075	K	106	j
43	0.0000501187	076	L	107	k
44	0.0000398107	077	M	108	l
45	0.0000316228	078	N	109	m
46	0.0000251189	079	O	110	n
47	0.0000199526	080	P	111	o
48	0.0000158489	081	Q	112	p
49	0.0000125893	082	R	113	q
50	0.0000100000	083	S	114	r
51	0.0000079433	084	T	115	s
52	0.0000063096	085	U	116	t
53	0.0000050119	086	V	117	u
54	0.0000039811	087	W	118	v
55	0.0000031623	088	X	119	w
56	0.0000025119	089	Y	120	x
57	0.0000019953	090	Z	121	y
58	0.0000015849	091	[	122	z
59	0.0000012589	092	\	123	{
60	0.0000010000	093	]	124	|
61	0.0000007943	094	^	125	}
62	0.0000006310	095	_	126	~
63	0.0000005012	096	`
64	0.0000003981	097	a
65	0.0000003162	098	b
66	0.0000002512	099	c
67	0.0000001995	100	d
68	0.0000001585	101	e
69	0.0000001259	102	f
70	0.0000001000	103	g
71	0.0000000794	104	h
72	0.0000000631	105	i
73	0.0000000501	106	j
74	0.0000000398	107	k
75	0.0000000316	108	l
76	0.0000000251	109	m
77	0.0000000200	110	n
78	0.0000000158	111	o
79	0.0000000126	112	p
80	0.0000000100	113	q
81	0.0000000079	114	r
82	0.0000000063	115	s
83	0.0000000050	116	t
84	0.0000000040	117	u
85	0.0000000032	118	v
86	0.0000000025	119	w
87	0.0000000020	120	x
88	0.0000000016	121	y
89	0.0000000013	122	z
90	0.0000000010	123	{
91	0.0000000008	124	|
92	0.0000000006	125	}
93	0.0000000005	126	~

An assumption going in when I was producing plots from the Q[Sanger] and Q[Solexa] data was that the "P" was the same value and the Solexa system simply opted to use the Odds (P/(1-P)) as their metric. A proper two-second consideration of the shape of the form of P and P/(1-P) would have lead to the immediate conclusion that something was afoot. The table columns on the left of the black bar in Table 2 (2A) are the Q[Solexa] values based on the use of the Q[Sanger] probabilities. This is here simply to show that they are, in fact, not the same and if you've spent any time wondering why you can't adequately… manipulate Excel's rounding tools to reproduce the Q[Solexa] integer values, this is why.

The probabilities obtained for Q[Solexa] were, in fact, worked backwards from the integer values of Q[Solexa] (having found no table online that gives a number-by-number summary of the probability or odds). For background, the Q[Solexa] values are obtained from:

Q[Solexa] = -10 * log₁₀[(P/1-P)]

Table 2A: Q[Solexa] from P[Sanger]			Table 2B: Q[Solexa] and associated odds (P/(1-P)).
Probability (P) Of Wrong Base	Associated Sanger Odds [P/(1-P)]	Q[Solexa] Based On Phred Probability	Solexa Q [-5 to 62]	Solexa Probability (P) Of Wrong Base	Solexa Odds [P/(1-P)]	Solexa "Q + 64" Q Shift	Solexa "Q + 64" ASCII GLYPH
0.7943282	3.8621161	-5.8682532	-5	0.7597469	3.1622774	59	;
0.6309573	1.7097139	-2.3292343	-4	0.7152527	2.5118860	60	<
0.5011872	1.0047602	-0.0206244	-3	0.6661394	1.9952619	61	=
0.3981072	0.6614253	1.7951917	-2	0.6131368	1.5848929	62	>
0.3162278	0.4624753	3.3491146	-1	0.5573117	1.2589255	63	?
0.2511886	0.3354498	4.7437242	0	0.5000000	1.0000000	64	@
0.1995262	0.2492602	6.0334710	1	0.4426884	0.7943284	65	A
0.1584893	0.1883390	7.2505963	2	0.3868632	0.6309575	66	B
0.1258925	0.1440241	8.4156483	3	0.3338606	0.5011873	67	C
0.1000000	0.1111111	9.5424251	4	0.2847473	0.3981072	68	D
0.0794328	0.0862868	10.6405549	5	0.2402531	0.3162278	69	E
0.0630957	0.0673449	11.7169522	6	0.2007600	0.2511887	70	F
0.0501187	0.0527631	12.7766933	7	0.1663376	0.1995263	71	G
0.0398107	0.0414613	13.8235685	8	0.1368069	0.1584893	72	H
0.0316228	0.0326554	14.8604457	9	0.1118158	0.1258926	73	I
0.0251189	0.0257661	15.8895167	10	0.0909091	0.1000000	74	J
0.0199526	0.0203588	16.9124707	11	0.0735876	0.0794328	75	K
0.0158489	0.0161042	17.9306177	12	0.0593509	0.0630957	76	L
0.0125893	0.0127498	18.9449785	13	0.0477267	0.0501187	77	M
0.0100000	0.0101010	19.9563519	14	0.0382865	0.0398107	78	N
0.0079433	0.0080069	20.9653650	15	0.0306534	0.0316228	79	O
0.0063096	0.0063496	21.9725111	16	0.0245034	0.0251189	80	P
0.0050119	0.0050371	22.9781790	17	0.0195623	0.0199526	81	Q
0.0039811	0.0039970	23.9826759	18	0.0156017	0.0158489	82	R
0.0031623	0.0031723	24.9862446	19	0.0124327	0.0125893	83	S
0.0025119	0.0025182	25.9890773	20	0.0099010	0.0100000	84	T
0.0019953	0.0019993	26.9913260	21	0.0078807	0.0079433	85	U
0.0015849	0.0015874	27.9931114	22	0.0062700	0.0063096	86	V
0.0012589	0.0012605	28.9945291	23	0.0049869	0.0050119	87	W
0.0010000	0.0010010	29.9956549	24	0.0039653	0.0039811	88	X
0.0007943	0.0007950	30.9965489	25	0.0031523	0.0031623	89	Y
0.0006310	0.0006314	31.9972589	26	0.0025056	0.0025119	90	Z
0.0005012	0.0005014	32.9978228	27	0.0019913	0.0019953	91	[
0.0003981	0.0003983	33.9982707	28	0.0015824	0.0015849	92	\
0.0003162	0.0003163	34.9986264	29	0.0012573	0.0012589	93	]
0.0002512	0.0002513	35.9989090	30	0.0009990	0.0010000	94	^
0.0001995	0.0001996	36.9991334	31	0.0007937	0.0007943	95	_
0.0001585	0.0001585	37.9993116	32	0.0006306	0.0006310	96	`
0.0001259	0.0001259	38.9994532	33	0.0005009	0.0005012	97	a
0.0001000	0.0001000	39.9995657	34	0.0003979	0.0003981	98	b
0.0000794	0.0000794	40.9996550	35	0.0003161	0.0003162	99	c
0.0000631	0.0000631	41.9997260	36	0.0002511	0.0002512	100	d
0.0000501	0.0000501	42.9997823	37	0.0001995	0.0001995	101	e
0.0000398	0.0000398	43.9998271	38	0.0001585	0.0001585	102	f
0.0000316	0.0000316	44.9998627	39	0.0001259	0.0001259	103	g
0.0000251	0.0000251	45.9998909	40	0.0001000	0.0001000	104	h
0.0000200	0.0000200	46.9999133	41	0.0000794	0.0000794	105	i
0.0000158	0.0000158	47.9999312	42	0.0000631	0.0000631	106	j
0.0000126	0.0000126	48.9999453	43	0.0000501	0.0000501	107	k
0.0000100	0.0000100	49.9999566	44	0.0000398	0.0000398	108	l
0.0000079	0.0000079	50.9999655	45	0.0000316	0.0000316	109	m
0.0000063	0.0000063	51.9999726	46	0.0000251	0.0000251	110	n
0.0000050	0.0000050	52.9999782	47	0.0000200	0.0000200	111	o
0.0000040	0.0000040	53.9999827	48	0.0000158	0.0000158	112	p
0.0000032	0.0000032	54.9999863	49	0.0000126	0.0000126	113	q
0.0000025	0.0000025	55.9999891	50	0.0000100	0.0000100	114	r
0.0000020	0.0000020	56.9999913	51	0.0000079	0.0000079	115	s
0.0000016	0.0000016	57.9999931	52	0.0000063	0.0000063	116	t
0.0000013	0.0000013	58.9999945	53	0.0000050	0.0000050	117	u
0.0000010	0.0000010	59.9999957	54	0.0000040	0.0000040	118	v
0.0000008	0.0000008	60.9999966	55	0.0000032	0.0000032	119	w
0.0000006	0.0000006	61.9999973	56	0.0000025	0.0000025	120	x
0.0000005	0.0000005	62.9999978	57	0.0000020	0.0000020	121	y
0.0000004	0.0000004	63.9999983	58	0.0000016	0.0000016	122	z
0.0000003	0.0000003	64.9999986	59	0.0000013	0.0000013	123	{
0.0000003	0.0000003	65.9999989	60	0.0000010	0.0000010	124	|
0.0000002	0.0000002	66.9999991	61	0.0000008	0.0000008	125	}
0.0000002	0.0000002	67.9999993	62	0.0000006	0.0000006	126	~

With all three data sets, I reproduce a plot familiar to the FASTQ community below, showing the asymptotic behavior of the Q[Solexa] and Q[Sanger] values at high Q (which represent the lowest read errors. They approach one another because the numbers are simply too damn small on the plot). Also obvious from the plot is that the plots show poor agreement with each other in the range where the error probability is highest (so the entire analysis goes to pot as the data quality goes to pot [ed. Note for the international reader: "pot" refers to the device found in the water-closet). The grey line is a good plot of the wrong data (that in Table 2A).

The presentation of this data is likely complete overkill, but I have found it useful in discussion. Hopefully your having tables in front of someone during an explanation will help clarify that explanation.