# How long does it take to read X number of characters?

How does the time needed to read a sentence scale with the number of characters? Or does this time scaling depend on something more than just character count?

For example, let $X$ be the number of characters and $F(X)$ the seconds to read an entire sentence of $X$ characters long. There are 3 options:

1. Linear $F(X) \in \Theta(X)$: Is the relationship linear? That is, if it takes 0.5 seconds to read "hello", it would take 1.0 second to read "hello world": $F(X) = 0.1 X$

2. Sublinear $F(X) \in o(X)$: Or does it decay? That is, as people read a long sentence, their reading speed increases (crazy example $F(X) = 1 / X$)

3. Super-linear $F(X) \in \omega(X)$: Or is it like quadratic? That is, each additional character ads more time than the previous word? (example: $F(X) = X ^ 2$)

Which of the above 3 options best describes reading speed based on length of text? Is the specific functional form of this relationship known?

### Motivation

I am creating an iPhone app where I need to show transient confirmation messages. For example, when a user submits a comment, I pop up a message saying "thanks for submitting your comment". Shortly after, the message will fade away. There exist many such transient messages all over my iPhone app.

What I would like to do is to calculate the optimal time to show each message based off of the number of characters in that message. I want users to have enough time to comfortably read the message, but not so long that the message annoys them. The purely UX version of this question is here:

How long does it take to read X number of characters?

-
I wrote an android app that takes a note entered by the user and displays it later as a Toast. You are limited to LENGTH.LONG (~3 seconds) and LENGTH.SHORT (~2 seconds). If the note entered exceeds 50 characters, the Toast is of LENGTH.LONG. –  cornbread ninja Jun 15 '12 at 23:07
I reposted my comment on ux.stackexchange.com before some mod migrated my question to cogsci. –  JoJo Jun 16 '12 at 0:35
Honestly this is fine on either site, and will probably get different answers on both; more focused on academic research here, with probably a more practical and simple (possibly less precise but more actionable) on UX. I'm fine with it being on both if Cog Sci welcomes it –  Ben Brocka Jun 16 '12 at 0:42
I made a significant edit to the question to make it focus on the scientific aspect of the question. Please edit the UX version to focus on the UX-part and not be a copy-and-paste of the original Q. If you think I changed the spirit too much, feel free to roll-back my edits, but keep in mind we are not UX and prefer scientific questions. –  Artem Kaznatcheev Jun 16 '12 at 1:05
@Artem: My answer to the question got migrated from EL&U, so I answered from a language perspective. I hope people reading my answer will keep that in mind, should it appear too unscientific for this community. Nice to meet you, though. –  J.R. Jun 16 '12 at 1:19

## migrated from english.stackexchange.comJun 16 '12 at 0:31

This question came from our site for linguists, etymologists, and serious English language enthusiasts.

I've studied this a little bit within the context of timing responses to personality test items. General models of reading speed look at both the time to read the words as well as to comprehend. From memory, eye tracking studies have shown how the eyes will often back track to confusing parts of a sentence (apologies for lack of reference).

Some general findings that I've found in my own data on time to answer personality test items of different lengths:

• Expect an initiation period. I.e., the time between displaying a message and the commencement of reading. In mathematical terms, this will translate to a non-zero constant in your model of reading time as a function of properties of the text.
• The more familiar the content and the words are to the reader, the quicker they will be. This translates both in absolute terms that simple words are quicker to read, and also to interactions between individuals and the content.
• In general, character count, word count, and syllable count tend to be highly correlated in normal prose. From my own data, these indices tend to have a relatively linear effect on reading speed.
• More complex sentences (e.g., ones with negation, multiple clauses, etc.) take longer to read. For example, I found that sentences with "not" took systematically longer to process.
• Individuals differ substantially in reading speed. Expect to see a distribution of individual differences that is a little bit positive skewed from normal. Thus, if you are designing a system where you determine the length of time text is displayed, you may need to pitch the speed of your system to capture a certain percentage of that distribution (e.g., 95, 99% etc.). Various factors explain variation in reading speed including whether English is a second language and education.
• There are also outliers. People get distracted. Thus, there is reading time when people are focussed and there is reading times that incorporate distraction and other processes.

More generally, if you have the resources, you can do some pilot testing on a sample of individuals where you time how long people take to read the text, ideally within the context of the application. This could allow you to develop a domain specific model of reading speed.

Although a little tangential, some articles that discuss speed of item responding in a personality testing context include Casey and Tyron (2001) and Wagner-Menghim (2002).

### References

• Casey, M.M. & Tryon, W.W. (2001). Validating a double-press method for computer administration of personality inventory items.. Psychological assessment, 13, 521.
• Wagner-Menghin, M. (2002). Towards the identification of non-scalable personality questionnaire respondents: Taking response time into account. Psychologische Beiträge, 44(1), 62-77.
-
Your fifth point (that people read at different speeds) might prompt the system designers to have a setup option whereby individual users can tailor how long they would prefer the messages to appear. –  J.R. Jun 16 '12 at 9:31

A few thoughts spring to mind:

1. Part of the answer might depend on the maximum value of X (if all the messages are relatively short, that's a key piece of information).

2. It doesn't decay, I don't think. The more information presented to the user, the more it all has to put into context with each other. But I don't think it's quadratic, either; that seems too steep.

3. If X is the number of characters, and W is the number of words, the time might be more dependent on W than X.

Do you remmeber tihs erxesice? It's fnuny how wlel you can raed wrods as lnog as the fisrt and lsat ltteers saty cnonstat. Wehn redanig, our birans seem iclinned to prcoses by the wrod, mroe tahn by the ltteer.

EDIT: Those were my initial thoughts, when this question was first posed on English Language & Usage. I won't pretend I'm an expert (I'm not), but there does seem to be quite a bit of research that would support my third hypothesis – that readers often parse by word, more so than by letter, so the number of words in a text string might be more significant than the number of letters. I'll cite one paper I managed to track down and peruse:

Inhoff, Albrecht Werner, Integrating information across eye fixations in reading: The role of letter and word units. Acta Psychologica (73:3), April 1990, Pp. 281–297

-
Why did this get downvoted so quickly? LOL. –  Jasper Loy Jun 15 '12 at 22:59
@JasperLoy: Evidently, someone didn't like my answer. It's not the first time that happened, and it probably won't be the last. C'est la vie. :^) P.S. Maybe it's all the typos in my quote. Someone from PETSC? (People for the Ethical Treatment of Spell Checkers). –  J.R. Jun 15 '12 at 23:03
Yes, the downvotes are coming fast and furious and anonymous these days. –  JeffSahol Jun 15 '12 at 23:06
I am not sure if the downvotes came before or after the migration to CogSci... but as it stands now, your answer lacks sufficient scientific evidence to stand as a good answer on this site. Most good answers here need to cite reference papers or studies. –  Josh Gitlin Jun 16 '12 at 13:06
Let's give you a reference then! Your "typo" text is actually called "transposed text": see [ White, S. J., Johnson, R. L., Liversedge, S. P., & Rayner, K. (2008). Eye movements when reading transposed text: the importance of word-beginning letters. Journal of Experimental Psychology: Human Perception and Performance, 34(5), 1261-1276. ] –  vizzero Jun 16 '12 at 18:36

If you really wanted to know you could use models of reading behaviour - e.g. EZ-Reader or Swift. The Rayner reviews are the classic go-to to outlne this kind of thing:

Rayner, K. (2009). Eye movements and attention in reading, scene perception, and visual search. The Quarterly Journal of Experimental Psychology (2006) (Vol. 62, pp. 1457-506).

It will depend on: - Size of text - Language - Word frequency - Word length - Predictability - and a whole host of other factors.

Of all the words put together, plus any occasions where regressions to previous words are made because the observer has failed to understand the text.

In practice, it's probably not worth worrying about to this level of complexity so just beta test it with some people and make sure they don't complain that they don't get enough time to read the messages.

-

This will be short but sweet. In the spirit of @J.R answer- (in particular the block section)- the information content of any "word" is most certainly less than that in an equivalent random amount of characters in that word (if one is writing) or syllables (if one is speaking.) As anyone who reads @J.R. block passage will agree, predictability is assured- providing we are on the same rational basis. Nothing more is required- anyone with any questions: look up information theory.

yes. "Sublinear"

ok so it seems i need to do all the work:

Entropy is defined in the context of a probabilistic model. Independent fair coin flips >have an entropy of 1 bit per flip. A source that always generates a long string of B's has >an entropy of 0, since the next character will always be a 'B'. The entropy rate of a data source means the average number of bits per symbol needed to encode it. Shannon's experiments with human predictors show an information rate of between 0.6 and 1.3 bits per character,[15] depending on the experimental setup; the PPM compression algorithm can achieve a compression ratio of 1.5 bits per character in English text. From the preceding example, note the following points: The amount of entropy is not always an integer number of bits. Many data bits may not convey information. For example, data structures often store information redundantly, or have identical sections regardless of the information in the data structure. Shannon's definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digits (see caveat below in italics). The formula can be derived by calculating the mathematical expectation of the amount of information contained in a digit from the information source. See also Shannon-Hartley theorem. Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. See Markov chain.<

right out of wikipedia: http://en.wikipedia.org/wiki/Entropy_(information_theory)

and seeing as the random generation of an alphabetical character- in english in particular- requires 4.7 bits of information (anyone? anyone?) we simply see that there is no possible way for amount of time it takes "to read X number of characters" to scale linearly or "super-linearly" (whatever that means) with X. In particular note in the block quote of @J.R. above we can remove characters and replace them with others but still not alter the information content of the message. This clearly indicates that- as my own block quote above states- that the brain is engaged in a probabilistic modeling of what the words (spoken or written) actually are- and it is not relying on the input of a certain number of characters to ascertain what is communicated.

to the downvoters: sorry but this is well established- and the question is clearly hinting at some sort of compression algorithm- so perhaps this is not really a neuroscience problem ehh? as my shorter answer was lost somehow on those...

-
and now upvote mine instead –  jaskey13 Jun 17 '12 at 15:40
Sublinear doesn't mean what you think it means. You are right that English text has less entropy/information content than a random sequence of characters, but the information still goes up linearly. Even if for every 10 characters there is only one bit of information, this still makes the information content of an X-character English string be something like 0.1X, which is still a linear function. (Your advice is a good one: "look up information theory". Even in Shannon's original paper he calculates the linear function for various stochastic models of English text.) –  ShreevatsaR Jun 18 '12 at 3:43
$$timeToRead = 1300 + (chars * 65);$$
thanks for your answer travis. did you test other models as well, or did you assume a linear relationship? also, do you know the $r^2$ from your dataset? –  Jeff Dec 8 '12 at 20:56