Wow! Thanks for all of the responses. It's been 14 years since I took
a class that involved floating point issues, and since many of us in the
statistics community (myself included) trash Excel's stat package, I was
guilty of (conciously/subconciously) wanting to trash Excel instead of
remembering what I learned in my Fortran 77 class (boy, am I feeling
middle-aged). Anyway, a number of you responded with very nice
descriptions of the issues in floating point mathematics. Here are the
responses that I received.
By the way, a few of you noted that you tried to access my web page and
received a notice that the page could not be located. Thanks to all of
the viruses that have been floating around, attempting to access our web
servers this semester can be modelled as a Bernoulli trial with p=0.03.
this is not a "bug" but a "feature" of floating point arithmetic.
To avoid seeing it, you need to use Excel's "ROUND" function to
limit your numbers to some significant number of digits (say 10
or 12). The other solution is to use something, like APL or J
(see www.jsoftware.com) with the notion of comparison tolerance
In Excel, for example, instead of the entering the function
"=(A2+4)*(A2-2)" (for (X+4)(X-2)), instead enter
"=ROUND((A2+4)*(A2-2),10)" to round to 10 places past
the decimal point.
The problem is unavoidable in the sense that there are decimal
numbers which have no exact, finite representation in binary
but they have to be stored with a limited number of digits.
^me^ at acm.org
Unless I am misreading this, it sounds like all you need to do is
the cells. Highlight the column in which you plan to type the x-values
right click. Click on Format Cells and select the Number Tab and then
highlight Number, type in 1 for the number of decimal places and then
The problem is that the binary representation of the decimal 0.1 is
0.0001100110011... where the 0011 repeats forever. The floating point
representation of 0.1 is inherently inaccurate. In a more familiar
setting, if we had a machine that could store a two decimal digit
then 1/3 would be stored as .33 and when added to itself two times
yield .99 rather than 1.0. Displayed values are often rounded so that
last couple of digits are not displayed so that the underlying
representation is again different from the displayed value.
The solution to your original problem would be to create the integers
to 60, which can be created and stored exactly in binary. And then
all of these numbers by 10. The underlying representations of some of
the numbers will not exactly match the numbers intended, but the errors
should be a single round-off rather than an accumulation of round-off
There's no bug. That's normal roundoff error.
0.1 is only approximated in floating point. Only rational numbers that
be expressed with a power of 2 in the denominator can be represented
in any binary floating point format.
Excel's strategy for formatting numbers for output (e.g. if it's close
enough to x.xx, it shows only x.xx, not x.xx00000000002) might be doing
some strange stuff there, but if I do one of your sequences such that 0
ought to be there, it almost always isn't, and I'm not seeing evidence
that anything's wrong with the internal floating point values.
I haven't been able to replicate the problem on my Mac G4. Maybe the
work-around is to get a Macintosh.
-- Paul Velleman
I'm running Excel 2002 on Windows XP and looked into this a bit.
Personally, I'm a fan of using Excel in Intro Stats AND understanding
it's shortcomings as is the case here.
I was able to get "rounding" error with a bunch of different
For example, starting with -4 and incrementing by 0.5, after line 27
rounding was occurring. The funny thing is that in some cases, the
rounding error is "corrected". For example, starting at -4 and using
0.1 as the increment, gets off after line 27, but Excel corrects
starting at line 51.
The number of lines before rounding error occurs differs if you use a
formula like $A$1 - 0.1 versus getting a series started with -4, -3.9,
-3.8, and then copying down. My empirical evidence suggests getting a
series started and copying it down is better in terms of controlling
rounding error. Probably most distributing is, if you get a series
started and copy it down you don't necessarily get the same results as
if you start copying a series, stop, and then restart copying again.
The "only" good news is that the rounding errors appear to be in
well beyond the limits that I care about. But, it is still a problem
Good luck on finding a "fix",
Hi. Here are two things to note.
First of all, you may not be seeing any "problem" with Excel. You may
just be exhibiting rounding error. You see, computers store floating
point numbers (non-integers) using a binary rather than a decimal
scheme. In this scheme, numbers such as .5, .25 and .125 would be exact,
but normal increments of 0.1 would be rounded. Adding 0.1 multiple
times will yield errors at about the level of 10^(-15) which is the
limit of floating point accuracy.
In short, it may not be a "bug". If you'd like to remove the problem
"cosmetically", try changing the cell information so that it just shows
fewer decimal digits.
P.S. I can generate similar problems using Matlab (math software).
This rounding issue is even more obvious when you use a programming
language. I see it commonly in Python. My point is that this facet of
floating point arithmetic arises in advanced mathematics software and in
common programming languages.
I guess I don't understand why we think we have found a bug, rather
than just the ever-present effects of finite precision.
I replicated the calculation I think you were doing (repeatedly
adding .1 to -6) in R on a unix workstation, with the follwoing
results... it computes zero as -4.5796699765787707e-15. Hmmm,
that looks a lot like the computed value of zero in the
[One might think the computations were exact for a while if the number
digits printed is too few.]
dx <- rep(.1,120)
dx <- -6
x <- cumsum(dx)
 -6.0000000000000000e+00 -5.9000000000000004e+00
 -5.7000000000000011e+00 -5.6000000000000014e+00
 -5.4000000000000021e+00 -5.3000000000000025e+00
 -5.1000000000000032e+00 -5.0000000000000036e+00
 -4.8000000000000043e+00 -4.7000000000000046e+00
 -4.5000000000000053e+00 -4.4000000000000057e+00
 -4.2000000000000064e+00 -4.1000000000000068e+00
 -3.9000000000000070e+00 -3.8000000000000069e+00
 -3.6000000000000068e+00 -3.5000000000000067e+00
 -3.3000000000000065e+00 -3.2000000000000064e+00
 -3.0000000000000062e+00 -2.9000000000000061e+00
 -2.7000000000000060e+00 -2.6000000000000059e+00
 -2.4000000000000057e+00 -2.3000000000000056e+00
 -2.1000000000000054e+00 -2.0000000000000053e+00
 -1.8000000000000052e+00 -1.7000000000000051e+00
 -1.5000000000000049e+00 -1.4000000000000048e+00
 -1.2000000000000046e+00 -1.1000000000000045e+00
 -9.0000000000000446e-01 -8.0000000000000449e-01
 -6.0000000000000453e-01 -5.0000000000000455e-01
 -3.0000000000000460e-01 -2.0000000000000459e-01
 -4.5796699765787707e-15 9.9999999999995426e-02
 2.9999999999999544e-01 3.9999999999999547e-01
 5.9999999999999543e-01 6.9999999999999540e-01
 8.9999999999999536e-01 9.9999999999999534e-01
 1.1999999999999955e+00 1.2999999999999956e+00
 1.4999999999999958e+00 1.5999999999999959e+00
 1.7999999999999960e+00 1.8999999999999961e+00
 2.0999999999999961e+00 2.1999999999999962e+00
 2.3999999999999964e+00 2.4999999999999964e+00
 2.6999999999999966e+00 2.7999999999999967e+00
 2.9999999999999969e+00 3.0999999999999970e+00
 3.2999999999999972e+00 3.3999999999999972e+00
 3.5999999999999974e+00 3.6999999999999975e+00
 3.8999999999999977e+00 3.9999999999999978e+00
 4.1999999999999975e+00 4.2999999999999972e+00
 4.4999999999999964e+00 4.5999999999999961e+00
 4.7999999999999954e+00 4.8999999999999950e+00
 5.0999999999999943e+00 5.1999999999999940e+00
 5.3999999999999932e+00 5.4999999999999929e+00
 5.6999999999999922e+00 5.7999999999999918e+00
I was able to replicate the problem in part: I got the same values at
(-2.0428E-14 and -4.57967E-15). I have a mac G4, running OSX (10.2.6)
Excel:mac v. X.
How about this...
If I simply make a column of numbers 10 9.9 9.8... and copy the
pattern, by the time I get to 8.5,which just happens to be 16 rows
later, I then have 8.5000000000000100.
Now, when computers do subtraction they usually to 2's complement and
addition. The problem may arise when we try to represent 0.10 in
binary. We have to represent it as a sum of bits whose values are
1/(2^1), 1/(2^2) + 1/(2^3)...etc. No matter how many bits we have (23 in
IEEE Floating point format) that number looks like 11111111111.... Its
two's complement looks like 000000000000...1. We simply cannot represent
Maybe Jeff, our resident CS grad student, can shed additional light on
-- Alan Scrivner through Charles Antoniak
Christopher Jay Lacke
Glassboro, NJ 08028
(856) 256-4500 x3897
(856) 256-4816 (fax)
"There are two types of people in the world - those who
think that there are two types and those who don't"
- author unknown