# Quality Scores

A quality score (or Q-score) expresses an error probability. In particular, it serves as a convenient and compact way to communicate very small error probabilities.

Given an assertion, A, the quality score, Q(A), expresses the probability that A is not true, P(\~A), according to the relationship:

`Q(A) =-10 log10(P(~A))`

where P(\~A) is the estimated probability of an assertion A being wrong.

The relationship between the quality score and error probability is demonstrated with the following table:

| **Quality score, Q(A)** | **Error probability, P(\~A)** |
| ----------------------- | ----------------------------- |
| 10                      | 0.1                           |
| 20                      | 0.01                          |
| 30                      | 0.001                         |

{% hint style="info" %}
On supported Illumina systems, Q-scores are automatically binned. The specific binning applied depends on the current Q-table. See the white paper [Reducing Whole Genome Data Storage Footprint](https://www.illumina.com/documents/products/whitepapers/whitepaper_datacompression.pdf) for more information, available from the Illumina website.
{% endhint %}

### Quality Score Encoding

In FASTQ files, quality scores are encoded into a compact form, which uses only 1 byte per quality value. In this encoding, the quality score is represented as the character with an ASCII code equal to its value + 33. The following table demonstrates the relationship between the encoding character, its ASCII code, and the quality score represented.

{% hint style="info" %}
When Q-score binning is in use, the subset of Q-scores applied by the bins is displayed.
{% endhint %}

| **Symbol** | **ASCII Code** | **Q-Score** |
| ---------- | -------------- | ----------- |
| !          | 33             | 0           |
| "          | 34             | 1           |
| #          | 35             | 2           |
| $          | 36             | 3           |
| %          | 37             | 4           |
| &          | 38             | 5           |
| '          | 39             | 6           |
| (          | 40             | 7           |
| )          | 41             | 8           |
| \*         | 42             | 9           |
| +          | 43             | 10          |
| ,          | 44             | 11          |
| -          | 45             | 12          |
| .          | 46             | 13          |
| /          | 47             | 14          |
| 0          | 48             | 15          |
| 1          | 49             | 16          |
| 2          | 50             | 17          |
| 3          | 51             | 18          |
| 4          | 52             | 19          |
| 5          | 53             | 20          |
| 6          | 54             | 21          |
| 7          | 55             | 22          |
| 8          | 56             | 23          |
| 9          | 57             | 24          |
| :          | 58             | 25          |
| ;          | 59             | 26          |
| <          | 60             | 27          |
| =          | 61             | 28          |
| >          | 62             | 29          |
| ?          | 63             | 30          |
| @          | 64             | 31          |
| A          | 65             | 32          |
| B          | 66             | 33          |
| C          | 67             | 34          |
| D          | 68             | 35          |
| E          | 69             | 36          |
| F          | 70             | 37          |
| G          | 71             | 38          |
| H          | 72             | 39          |
| I          | 73             | 40          |


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