# What do quality scores mean?¶

## Phred Quality Scores¶

We can calculate the phred quality score from the probability of sequencing error (i.e. the base call is wrong) using:

\begin{equation*} Q = -10 \log_{10} p \end{equation*}

Alternatively, we can rearrange to calculate the probability of error from the phred quality score using:

\begin{equation*} p = 10^{\frac{Q}{-10}} \end{equation*}

Where $$Q$$ is the phred quality score and $$p$$ is the probability of error (i.e. the base call is wrong)

In :

def error_prob(quality):
qval = quality
return 10**(qval/-10.0)

print ("{0:^5}  {1:^10}".format("Phred", "Prob of"))
print ("{0:^5}  {1:^10}".format("score", "Error"))
for phred in range(0,42):
print ("{0:^5}  {1:03.5f}".format(phred, error_prob(phred)))

Phred   Prob of
score    Error
0    1.00000
1    0.79433
2    0.63096
3    0.50119
4    0.39811
5    0.31623
6    0.25119
7    0.19953
8    0.15849
9    0.12589
10    0.10000
11    0.07943
12    0.06310
13    0.05012
14    0.03981
15    0.03162
16    0.02512
17    0.01995
18    0.01585
19    0.01259
20    0.01000
21    0.00794
22    0.00631
23    0.00501
24    0.00398
25    0.00316
26    0.00251
27    0.00200
28    0.00158
29    0.00126
30    0.00100
31    0.00079
32    0.00063
33    0.00050
34    0.00040
35    0.00032
36    0.00025
37    0.00020
38    0.00016
39    0.00013
40    0.00010
41    0.00008


## Ascii Codes¶

In FASTQ files, phred scores are represented using characters. Each character on the keyboard can be represented by a number, called an ascii code.

In :

print ("{0:^8}  {1:^8}".format("Character", "ASCII #"))
for i in range(33,90):
print("{0:^8}  {1:^8}".format(chr(i),i))

Character  ASCII #
!         33
"         34
#         35
$36 % 37 & 38 ' 39 ( 40 ) 41 * 42 + 43 , 44 - 45 . 46 / 47 0 48 1 49 2 50 3 51 4 52 5 53 6 54 7 55 8 56 9 57 : 58 ; 59 < 60 = 61 > 62 ? 63 @ 64 A 65 B 66 C 67 D 68 E 69 F 70 G 71 H 72 I 73 J 74 K 75 L 76 M 77 N 78 O 79 P 80 Q 81 R 82 S 83 T 84 U 85 V 86 W 87 X 88 Y 89  ## Phred Encodings¶ There are several different ways to encode phred scores with ascii characters. The two most common are called phred+33 and phred+64. The names are strange until you understand how then encoding works. ### Phred+33¶ To use the phred+33 encoding, take the phred quality score, add 33 to it, then use the ascii character corresponding to the sum. For example, using the phred+33 encoding, a quality score of 30 would be represented with the ascii character with the ascii code of 63 (30 + 33), which is ‘?’. ### Phred+64¶ The phred+64 encoding works the same as the phred+33 encoding, except you add 64 to the phred score to determine the ascii code of the quality character. You will only find phred+64 encoding on older data, which was sequenced several years ago. The tricky part is that there is no indication in the FASTQ file as to which encoding was used, you have to make an educated guess. In :  print ("{0:^5} {1:^8} {2:^8} {3:^8}".format("Phred", "Prob of", "Phred+33", "Phred+64")) print ("{0:^5} {1:^8} {2:^8} {3:^8}".format("score", "Error", "Ascii", "Ascii")) for phred in range(0,42): print ("{0:^5} {1:03.5f} {2:^8} {3:^8}".format(phred, error_prob(phred), chr(phred+33), chr(phred+64)))  Phred Prob of Phred+33 Phred+64 score Error Ascii Ascii 0 1.00000 ! @ 1 0.79433 " A 2 0.63096 # B 3 0.50119$         C
4    0.39811     %         D
5    0.31623     &         E
6    0.25119     '         F
7    0.19953     (         G
8    0.15849     )         H
9    0.12589     *         I
10    0.10000     +         J
11    0.07943     ,         K
12    0.06310     -         L
13    0.05012     .         M
14    0.03981     /         N
15    0.03162     0         O
16    0.02512     1         P
17    0.01995     2         Q
18    0.01585     3         R
19    0.01259     4         S
20    0.01000     5         T
21    0.00794     6         U
22    0.00631     7         V
23    0.00501     8         W
24    0.00398     9         X
25    0.00316     :         Y
26    0.00251     ;         Z
27    0.00200     <         [
28    0.00158     =         \
29    0.00126     >         ]
30    0.00100     ?         ^
31    0.00079     @         _
32    0.00063     A
33    0.00050     B         a
34    0.00040     C         b
35    0.00032     D         c
36    0.00025     E         d
37    0.00020     F         e
38    0.00016     G         f
39    0.00013     H         g
40    0.00010     I         h
41    0.00008     J         i


## Why +33?¶

ASCII 33 is the first “normal” ASCII character that. 1 through 32 include whitespace and non-printing characters, which cannot be identified by eye)

In :

print ("{0:^8}  {1:^8}".format("Character", "ASCII #"))
for i in range(0,40):
print("{0:^8}  {1:^8}".format(chr(i),i))

Character  ASCII #
0
         1
         2
         3
         4
         5
         6
         7
         8
9

10
11
12
13
         14
         15
         16
         17
         18
         19
         20
         21
         22
         23
         24
         25
         26
         27

28

29

30
31
32
!         33
"         34
#         35
\$         36
%         37
&         38
'         39
`