PHP 7, 79 bytes

As I don't have enough reputation to comment on cb0's answer to provide him hints to improve his answer (mainly using the char index method in a string and while to loop through a string). But I still bothered to post it due to the large reduction in byte count.

<?php $i=$b=0;while($a=ord($argv[1][$i++]??'')){$b+=$a;echo chr(round($b/$i));}

I didn't find a way though to use the <?= shortcut and avoid undefined errors without using command line flags (this is why i used $i=$b=0;). This answer does not work in PHP 5 due to the ?? syntax.

Ruby, 27 bytes

putc gets.sum.quo(~/$/)+0.5

Solution produced through some collaborative golfing at work.

How this works

• Kernel#putc prints a single character (e.g. putc "Hello" outputs H). Passing an object to it prints the character whose code is the least-significant byte of the object. So putc 72 outputs H.
• Kernel#gets reads stdin and returns a string.
• String#sum returns a checksum. The result is the sum of the binary value of each byte in the string modulo 2**16 - 1. Since ASCII bytes comfortably fit in this space, this just returns the byte total.
• Because the byte total is an integer, we can’t do floating point calculations (in Ruby 10 / 3 is 3 whereas 10.to_f / 3 is 3.33…). Here’s where Numeric#quo comes in – 10.quo(3) returns the Rational (10/3).
• Regexp#~ matches a regular expression against the contents of $_ (the last string read by gets). It returns index of the match.
• The Regexp /$/ matches the end of the string, so ~/$/ means “print the index of the end of the string” – in other words, the string length. This gives us the final is a short-hand for the length of the string read by gets.
• This gives us the Rational (byte total/string length). Now we need to round it into an integer. +0.5 casts the Rational to a float. putc floors the float into an integer.

PowerShell, 59 bytes

-join($args|%{[char][int][Math]::Floor(($s+=+$_)/++$i+.5)})

Try it online!

Zig 0.6.0, 84 bytes

fn a(b:[]u8)void{var t:u64=0;for(b)|c,i|{t+=c;b[i]=@intCast(u8,(t+(i+1)/2)/(i+1));}}

Try it Online

Formatted:

fn a(b: []u8) void {
    var t: u64 = 0;
    for (b) |c, i| {
        t += c;
        b[i] = @intCast(u8, (t + (i+1)/2) / (i+1));
    }
}

Zsh, 52 bytes

for c (${(s::)1})printf ${(#)$((.5+(n+=0.+#c)/++i))}

Try it online!

In arithmetic mode, #c gets the code of the first character of $c. The parameter expansion ${(#) } prints the character associated with the code.

Python 2, 65 bytes

f=lambda s:s and f(s[:-1])+chr(int(sum(map(ord,s))*1./len(s)+.5))

Try it online!

J, 23 bytes

(0.5<.@++/%#)&.(a.&i.)

Try it online!

How it works

                        on prefixes
               (   i.)   index of the first occurence
               (  &  )   in
               (a.   )   the character set
            x&.y         apply y, then x, then the inverse of y,                            
(0.5        )              that is the element of a. with a given index
(       +/  )            sum
(          #)            number of elements
(         % )            division
(   <.@+    )            add, then floor 

Husk, 10 bytes

zȯci/Nt∫mc

Try it online!

                                                    | "tuna"
         mc  -- map each character to ASCII value   | [116,117,110,97]
       t∫    -- prefix sums & drop leading 0        | [116,233,343,440]
z(   )N      -- zip the list with [1..] using       | 
    /        --   divide                            | [116/1,233/2,343/3,440/4] == [116.0,116.5,114.̅3,110.0]
   i         --   round                             | [116,117,114,110]
  c          --   convert to character              | "turn"

Jelly, 8 bytes

OÆmƤ+.ḞỌ

Try it online!

PHP, 176 byte

<?=(implode('',array_reduce(str_split($argv[1]),function($c,$k){array_push($c[1],chr(floor(((ord($k)+$c[0])/(count($c[1])+1))+0.5)));return[ord($k)+$c[0],$c[1]];},[0,[]])[1]));

Example:

>php cg.php Hello!
HW^adY
>php cg.php test  
tmop
>php cg.php 42
43

The biggest solution so far, but based on php it can't get much shorter I think. 2 bytes could be saved by removing the newlines.

Clojure, 69 bytes

#(for[c(map /(reductions +(map int %))(rest(range)))](char(+ c 0.5)))

Returns a sequence of characters, arguably a string-like construct. Would need an #(apply str(for[...]...)) to convert it into a string.

Japt, 11 bytes

c@V±X /°T r

Try it

SmileBASIC, 65 bytes

LINPUT S$FOR I=1TO LEN(S$)A=A-A/I+ASC(S$[I-1])/I?CHR$(A+.5);
NEXT

JavaScript (ES6), 75 bytes

let f =
s=>s.replace(/./g,x=>String.fromCharCode((t+=x.charCodeAt())/++i+.5),i=t=0)

<input oninput="O.value=f(this.value)" value="Hello!"><br>
<input id=O value="HW^adY" disabled>

I can't believe there's no JS answer with this technique yet...

Japt, 13 bytes (non-competing)

£T±Xc)/°Y r d

Test it online!

How it works

£   T± Xc)/° Y r d
mXY{T+=Xc)/++Y r d}
                     // Implicit: U = input string, T = 0
mXY{              }  // Replace each char X and index Y in the string by this function:
    T+=Xc            //   Add X.charCodeAt() to T.
         )/++Y       //   Take T / (Y + 1).
               r d   //   Round, and convert to a character.
                     // Implicit: output result of last expression

Jelly, 16 bytes (non-competing)

S÷L+.Ḟ
OµḣLR$Ç€Ọ

Try it online!

Only if there were Average and Round built-ins...

05AB1E, 15 bytes [Non-competing?]

.pvyDSÇOsg/îç}J

Try it online!

Factor, 80 bytes

[ cum-sum [ dup zero? 1 0 ? + / ] map-index [ .5 + floor >fixnum ] map >string ]

><>, 30 bytes

i:0(?v
v &l~<
+l2(?
&,12,+o;

• The first line reads from stdin and puts the characters on the stack
• The second will remove the EOL char, take the size of the stack and put it in the & register
• The third line will add numbers on the stack while there are two or more of them
• The fourth line will divide the resulting number by the register's value, then add 1/2, output the value as a character and stop. When faced with a float value when displaying a char, ><> will floor it, which is why we added 1/2

You can try it on the online interpreter but then you need to use the following version, because the online interpreter pads the code box to a rectangle and applies ? to spaces.

i:0(?v
v &l~<
+l2(   ?
&,12,+o;

Mathcad, 60 "bytes"

Mathcad is mathematical application based on 2D worksheets comprised of "regions" each of which can be text, a mathematical expression, program, plot or scripted component.

A mathematical or programming instruction is picked from a palette toolbar or entered using a keyboard shortcut. For golfing purposes, an operation ("byte") is taken to be the number of keyboard operations necessary to create a name or expression (for example, to set the variable a to 3, we would write a:=3. The definition operator := is a single keypress ":", as are a and 3 giving a total of 3 "bytes". The programming for operator requires typing ctl-shft-# (or a single click on the programming toolbar) so again is equivalent to 1 byte.

In Mathcad the user enters programming language commands using keyboard shortcuts (or picking them from the Programming Toolbar) rather than writing them in text. For example, typing ctl-] creates a while-loop operator that has two "placeholders" for entering the condition and a single line of the body, respectively. Typing = at the end of a Mathcad expressions causes Mathcad to evaluate the expression.

(Count bytes) By looking at it from a user input perspective and equating one Mathcad input operation (keyboard usually, mouse-click on toolbar if no kbd shortcut) to a character and interpreting this as a byte. csort = 5 bytes as it's typed char-by-char as are other variable/function names. The for operator is a special construct that occupies 11 characters (including 3 blank "placeholders" and 3 spaces) but is entered by ctl-shft-#, hence = 1 byte (similar to tokens in some languages). Typing ' (quote) creates balanced parentheses (usually) so counts as 1 byte. Indexing v = 3 bytes (type v[k).

TSQL, 118 bytes

DECLARE @ varchar(400) = 'StackExchange'

SELECT
top(len(@))char(avg(ascii(stuff(@,1,number,''))+.5)over(order by number))FROM
master..spt_values
WHERE'P'=type

Returning characters vertical

S
d
c
c
d
_
c
c
c
c
d
d
d

Perl 5, 41 bytes

say map{$s+=ord;chr($s/++$c+.5)}pop=~/./g

run as

$ perl -E 'say map{$s+=ord;chr($s/++$c+.5)}pop=~/./g' StackExchange
Sdccd_ccccddd

Ruby, 46

s=0.0
$<.bytes{|b|s+=b;$><<'%c'%(0.5+s/$.+=1)}

ideone.

With apologies to w0lf, my answer ended up different enough that it seemed worth posting.

$<.bytes iterates over each byte in stdin, so we print the rolling average in each loop. '%c' converts a float to a character by rounding down and taking the ASCII, so all we have to do is add 0.5 to make it round properly. $. is a magic variable that starts off initialized to 0--it's supposed to store the line count, but since here we want byte count we just increment it manually.

JavaScript ES6, 111 bytes

w=>w.replace(/./g,(_,i)=>String.fromCharCode([for(f of w.slice(0,++i))f.charCodeAt()].reduce((a,b)=>a+b)/i+.5))

This is annoyingly long thanks in part to JavaScript's long String.fromCharCode and charCodeAt functions. The stack snippet contains ungolfed, commented, testable code.

f=function(w){
  return w.replace(/./g,function(e,i){
    return String.fromCharCode(w.slice(0,++i).split('').map(function(f){
      return f.charCodeAt()
    }).reduce(function(a,b){
      // Adds all numbers in the array
      return a+b
      // String.fromCharCode automatically floors numbers, so we add .5 to round up
    })/i+.5)
  })
}

run=function(){document.getElementById('output').innerHTML=f(document.getElementById('input').value)};document.getElementById('run').onclick=run;run()

<input type="text" id="input" value="Hello!" /><button id="run">Run</button><br />
<pre id="output"></pre>

Q, 15 12 bytes

12 bytes as an expression

"c"$avgs"i"$

q)"c"$avgs"i"$"Hello!"
"HW^adY"
q)"c"$avgs"i"$"test"
"tmop"
q)"c"$avgs"i"$"42"
"43"
q)"c"$avgs"i"$"StackExchange"
"Sdccd_ccccddd"

or 15 bytes as a function

{"c"$avgs"i"$x}

q){"c"$avgs"i"$x} "Hello!"
"HW^adY"
q){"c"$avgs"i"$x} "test"
"tmop"
q){"c"$avgs"i"$x} "42"
"43"
q){"c"$avgs"i"$x} "StackExchange"
"Sdccd_ccccddd"

takes advantage of

1. the "i"$ cast to convert a string (list of characters) to a list of integers
2. the avgs function, which computes the running average of a list as a list of floats
3. the "c"$ cast to convert a list of floats to a list of characters, and which automatically rounds each float to the nearest integer before doing so [i.e. ("c"$99.5) = ("c"$100) and ("c"$99.4) = ("c"$99) ]

Python 3, 65 bytes

n=t=0
for c in input():n+=1;t+=ord(c);print(end=chr(int(.5+t/n)))

Try it online!

If I use round() instead of int(.5+ etc., it saves one character but is technically not in compliance with the challenge: Python's round() rounds halves to the nearest even integer, not upwards. However, it works correctly on all sample inputs.

Java, 100

Much like many other answers here, I'm summing and averaging in a loop. Just here to represent Java :)

void f(char[]z){float s=0;for(int i=0;i<z.length;System.out.print((char)Math.round(s/++i)))s+=z[i];}

My original code is a 97, but it only returns the modified char[] rather than printing it:

char[]g(char[]z){float s=0;for(int i=0;i<z.length;z[i]=(char)Math.round(s/++i))s+=z[i];return z;}

Now, it's just long enough for scrollbars to appear for me, so here's a version with some line breaks, just because:

void f(char[]z){
    float s=0;
    for(int i=0;
            i<z.length;
            System.out.print((char)Math.round(s/++i)))
        s+=z[i];
}

Mathematica, 75 bytes

FromCharacterCode@Floor[.5+Accumulate@#/Range@Length@#]&@ToCharacterCode@#&

Matlab, 43

Using an anonymous function:

f=@(s)char(round(cumsum(+s)./(1:numel(s))))

Examples:

>> f=@(s)char(round(cumsum(+s)./(1:numel(s))))
f = 
    @(s)char(round(cumsum(+s)./(1:numel(s))))

>> f('Hello!')
ans =
HW^adY

>> f('test')
ans =
tmop

>> f('42')
ans =
43

>> f('StackExchange')
ans =
Sdccd_ccccddd

Perl: 31 30 characters

(29 characters code + 1 character command line option.)

s!.!chr.5+($s+=ord$&)/++$c!ge

Sample run:

bash-4.3$ perl -pe 's!.!chr.5+($s+=ord$&)/++$c!ge' <<< 'StackExchange'
Sdccd_ccccddd

C# 189 135 134 106 Bytes

var x=s.Select((t,i)=>Math.Round(s.Select(a=>(int)a).Take(i+1).Average())).Aggregate("",(m,c)=>m+(char)c);

Can be seen here

First time golfer

Brainfuck 106 bytes

,[>,<.[->+<]>>++<[>[->+>+<<]>[-<<-[>]>>>[<[>>>-<<<[-]]>>]<<]>>>+<<[-<<+>>]<<<]>[-]>>>>[-<<<<<+>>>>>]<<<<<]

This is my first participation in a code-golf, please be gentle! It does work but brainfuck can't handle floats (not that i know of) so the rounded value is always the bottom one (might fix my algorithm later).

Also, the algorithm averages the values 2 by 2, meaning it could be innacurate in some spots. And I need to fix a bug that is printing a number at the end of the output too.

Python 2, 71

i=s=0
r=''
for c in input():s+=ord(c);i+=1.;r+=chr(int(s/i+.5))
print r

With each new character, updates the character sum s and the number of characters i to compute and append the average character.

Python 2, 106 bytes

It's not short enough. Since it's python it's way too verbose, you can even read what it does by looking code. But it's working.

a=[.0]+[ord(i)for i in raw_input()]
print"".join([chr(int(.5+(sum(a[:i+1])/i)))for i in range(1,len(a))])

Ruby 59 61

->w{s=c=0.0;w.chars.map{|l|s+=l.ord;(s/c+=1).round.chr}*''}

Test: http://ideone.com/dT7orT

R, 135 127 Bytes

This got long real quick and I really got it wrong the first time:) Need to read the questions properly.

cat(sapply(substring(a<-scan(,''),1,1:nchar(a)),function(x)rawToChar(as.raw(round(mean(as.integer(charToRaw(x)))+.5)))),sep='')

Test Run

cat(sapply(substring(a<-scan(,''),1,1:nchar(a)),function(x)rawToChar(as.raw(round(mean(as.integer(charToRaw(x)))+.5)))),sep='')
1: Hello!
2: 
Read 1 item
HW^adY

Pyth, 16 bytes

smCs+.5csaYCdlYz

Pretty straightforward. Using s+.5 instead of rounding, because for some reason round(0.5, 0) is 0 in Python.

C, 62 bytes

c;t;main(n){for(;(c=getchar())>0;n++)putchar(((t+=c)+n/2)/n);}

(ideone link)

The results are slightly different from the OP's examples, but only because this code rounds 0.5 down instead of up. Not any more!

K, 36 bytes

`0:_ci_.5+{(+/x)%#x}'.0+1_|(-1_)_ic

Usage:

  `0:_ci_.5+{(+/x)%#x}'.0+1_|(-1_)_ic"Hello!"
HW^adY
  `0:_ci_.5+{(+/x)%#x}'.0+1_|(-1_)_ic"test"
tmop
  `0:_ci_.5+{(+/x)%#x}'.0+1_|(-1_)_ic"42"
43
  `0:_ci_.5+{(+/x)%#x}'.0+1_|(-1_)_ic"StackExchange"
Sdccd_ccccddd

_ci and _ic convert ascii to chars and vice versa, respectively. {(+/x)%#x} is a classic K idiom for calculating a mean. Pretty straightforward overall.

Edit: oh, misread the spec. `0: is needed to print the result to stdout. Waiting for clarification on input re. Dennis' question.

Julia, 85 81 bytes

s->(i=[int(c)for c=s];print(join([char(iround(mean(i[1:j])))for j=1:length(i)])))

This creates an unnamed function that accepts a string and creates a vector of its ASCII code points. Means are taken for each sequential group, rounded to integers, converted to characters, joined into a string, and printed to STDOUT.

JavaScript ES7, 122 bytes

s=>String.fromCharCode(...[for(i of s)i.charCodeAt()].map((l,i,a)=>Math.round(eval((t=a.slice(0,++i)).join`+`)/t.length)))

Mostly everything is happening in this bit

eval((t=a.slice(0,++i)).join`+`)/t.length)

The rest is looping / character code conversion

Split up:

s=> 
 String.fromCharCode(...                        ) // Converts average character code array to string, ... allows it to take an array
   [for(i of s)i.charCodeAt()]                    // Converts string to char code array
   .map((l,i,a)=>                             )   // Loops through each character
     Math.round(                    /t.length)    // Rounds sum of previous char codes, divides by position + 1
       eval(                       )              // evals string of char codes seperated with +
            (                ).join`+`            // joins previous char codes with +
             t=a.slice(0,++i)                     // creates an array with all the char codes

If functions aren't allowed:

alert(String.fromCharCode(...[for(i of prompt())i.charCodeAt()].map((l,i,a)=>Math.round(eval((t=a.slice(0,++i)).join`+`)/t.length))))

_{133 bytes}

ES5 Snippet:

function _toConsumableArray(r){if(Array.isArray(r)){for(var e=0,t=Array(r.length);e<r.length;e++)t[e]=r[e];return t}return Array.from(r)}function _taggedTemplateLiteral(r,e){return Object.freeze(Object.defineProperties(r,{raw:{value:Object.freeze(e)}}))}var _templateObject=_taggedTemplateLiteral(["+"],["+"]),f,t=function t(s){return String.fromCharCode.apply(String,_toConsumableArray(function(){var r=[],e=!0,t=!1,a=void 0;try{for(var n,i=s[Symbol.iterator]();!(e=(n=i.next()).done);e=!0){var o=n.value;r.push(o.charCodeAt())}}catch(l){t=!0,a=l}finally{try{!e&&i["return"]&&i["return"]()}finally{if(t)throw a}}return r}().map(function(l,i,a){return Math.round(eval((f=a.slice(0,++i)).join(_templateObject))/f.length)})))};

// Demo
document.getElementById('go').onclick=function(){
  document.getElementById('output').innerHTML = t(document.getElementById('input').value)
};

<div style="padding-left:5px;padding-right:5px;"><h2 style="font-family:sans-serif">Average of Words Snippet</h2><div><div  style="background-color:#EFEFEF;border-radius:4px;padding:10px;"><input placeholder="Text here..." style="resize:none;border:1px solid #DDD;" id="input"><button id='go'>Run!</button></div><br><div style="background-color:#EFEFEF;border-radius:4px;padding:10px;"><span style="font-family:sans-serif;">Output:</span><br><pre id="output" style="background-color:#DEDEDE;padding:1em;border-radius:2px;overflow-x:auto;"></pre></div></div></div>

CJam, 19 bytes

Uq{i+_U):Ud/moco}/;

Try it online in the CJam interpreter.