{"id":748,"date":"2009-03-04T20:55:07","date_gmt":"2009-03-04T20:55:07","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2009\/03\/04\/basics-significant-figures\/"},"modified":"2009-03-04T20:55:07","modified_gmt":"2009-03-04T20:55:07","slug":"basics-significant-figures","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2009\/03\/04\/basics-significant-figures\/","title":{"rendered":"Basics: Significant Figures"},"content":{"rendered":"

After my post the other day about rounding errors, I got a ton of
\nrequests to explain the idea of significant figures<\/em>. That’s
\nactually a very interesting topic.<\/p>\n

The idea of significant figures is that when you’re doing
\nexperimental work, you’re taking measurements – and measurements
\nalways have a limited precision. The fact that your measurements – the
\ninputs to any calculation or analysis that you do – have limited
\nprecision, means that the results of your calculations likewise have
\nlimited precision. Significant figures (or significant digits, or just “sigfigs” for short) are a method of tracking measurement
\nprecision, in a way that allows you to propagate your precision limits
\nthroughout your calculation.<\/p>\n

Before getting to the rules for sigfigs, it’s helpful to show why
\nthey matter. Suppose that you’re measuring the radius of a circle, in
\norder to compute its area. You take a ruler, and eyeball it, and end
\nup with the circle’s radius as about 6.2 centimeters. Now you go to
\ncompute the area: π=3.141592653589793… So what’s the area of the
\ncircle? If you do it the straightforward way, you’ll end up with a
\nresult of 120.76282160399165 cm2<\/sup>.<\/p>\n

The problem is, your original measurement of the radius was
\nfar too crude to produce a result of that precision. The real
\narea of the circle could easily be as high as 128, or as low as
\n113, assuming typical measurement errors. So claiming that your
\nmeasurements produced an area calculated to 17 digits of precision is
\njust ridiculous.<\/p>\n

<\/p>\n

As I said, sigfigs are a way of describing the precision of a
\nmeasurement. In that example, the measurement of the radius as 6.2
\ncentimeters has two digits of precision – two significant
\ndigits<\/em>. So nothing computed using that measurement can
\nmeaningfully have more than two significant digits – anything beyond
\nthat is in the range of roundoff errors – further digits are artifacts
\nof the calculation, which shouldn’t be treated as meaningful.<\/p>\n

The rules for significant figures are pretty straightforward:<\/p>\n

    \n
  1. Leading zeros are never<\/em> significant digits. So in “0.0000024”, only the “2” and the “4” could be significant; the leading
    \nzeros aren’t.<\/li>\n
  2. Trailing zeros are only significant if they’re measured. So,
    \nfor example, if we used the radius measurement above, but expressed
    \nit in micrometers, it would be 62,000 micrometers. I couldn’t
    \nclaim that as 5 significant figures, because I really only measured
    \ntwo. On the other hand, if I actually measured it as 6.20 centimeters, then I could could three significant digits.<\/li>\n
  3. Digits other than zero in a measurement are always significant
    \ndigits.<\/li>\n
  4. In multiplication and division, the number of the significant
    \nfigures in the result is the smallest<\/em> of the number
    \nof significant figures in the inputs. So, for example,
    \nif you multiple 5 by 3.14, the result will have on significant
    \ndigit; if you multiply 1.41421 by 1.732, the result will have
    \nfour significant digits.<\/li>\n
  5. In addition and subtraction, you keep the number of
    \nsignificant digits in the input with the smallest number of
    \ndecimal places<\/em>.<\/li>\n<\/ol>\n

    That last rule is tricky. The basic idea is, write the numbers
    \nwith the decimal point lined up. The point where the last significant
    \ndigit occurs first is the last digit that can be significant in
    \nthe result. For example, let’s look at 31.4159 plus 0.000254. There
    \nare 6 significant digits in 31.3159; and there are 3 significant digits in 0.000254. Let’s line them up to add:<\/p>\n

    \n31.4159\n+  0.000254\n-------------\n31.4162\n<\/pre>\n
     The “9” in 31.4159 is the significant digit occuring in the
    \nearliest decimal place – so it’s the cutoff line. Nothing
    \nsmaller that 0.0001 can be significant. So we round off
    \n0.000254 to 0.0003; the result still has 5 significant
    \nfigures.<\/p>\n
     Significant figures are a rather crude way of tracking
    \nprecision. They’re largely ad-hoc. There is mathematical reasoning
    \nbehind these rules – so they do work pretty well most of the time. The
    \n“right” way of tracking precision is error bars: every measurement has
    \nan error range, and those error ranges propagate through your
    \ncalculations, so that you have a precise error range for every
    \ncalculated value. That’s a much better way of measuring potential
    \nerrors than significant digits. But most of the time, unless we’re in
    \na very careful, clean, laboratory environment, we don’t really
    \nknow<\/em> the error bars for our measurements. Significant digits
    \nare basically a way of estimating error bars. (And in fact, the
    \nmathematical reasoning underlying these rules is based on how
    \nyou handle error bars.)<\/p>\n
     The beauty of significant figures is that they’re so incredibly
    \neasy to understand and to use. Just look at any<\/em> computation
    \nor analysis result described anywhere, and you can easily see if
    \nthe people describing it are full of shit or not. For example, you
    \ncan see people claiming to earn 2.034523% on some bond; they’re
    \nnot, unless they’ve invested a million dollars, and then those last
    \ndigits are pennies – and it’s almost certain that the calculation
    \nthat produced that figure of 2.034523% was done based on
    \ninputs which had a lot less that 7 significant digits.<\/p>\n
     The way that this affects the discussion of rounding is
    \nsimple. The standard rules I stated for rounding are for
    \nrounding one<\/em> significant digit. If you’re doing a computation
    \nwith three significant digits, and you get a result of
    \n2.43532123311112, anything after the 5 is noise<\/em>. It doesn’t
    \ncount. It’s not really there. So you don’t get to say “But
    \nit’s more<\/em> than 2.435, so you should round up to
    \n2.44.”. It’s not<\/em> more: the stuff that’s making you think it’s
    \nmore is just computational noise. In fact, the “true” value is
    \nprobably somewhere +\/-0.005 of that – so it could be slightly more
    \nthan 2.435, but it could also<\/em> be slightly less. The computed
    \ndigits past the last significant digit are insignificant<\/em> –
    \nthey’re beyond the point at which you can say anything accurate. So
    \n2.43532123311112 is the same<\/em> as 2.4350000000000 if you’re
    \nworking with three significant digits – in both cases, you round off
    \nto 2.44 (assuming even preference). If you count the trailing digits
    \npast the one digit after the last significant one, you’re just using
    \nnoise in a way that’s going to create a subtle upward bias in
    \nyour computations.<\/p>\n
     On the other hand, if you’ve got a measured value of 2.42532, with
    \nsix significant figures, and you need to round it to 3 significant
    \nfigures, then<\/em> you can use the trailing digits in your
    \nrounding. Those digits are real<\/em> and significant<\/em>.
    \nThey’re a meaningful, measured quantity – and so the correct rounding
    \nwill take them into account. So even if you’re working with
    \neven preference rounding, that number should be rounded to three sigfigs as 2.43.<\/p>\n","protected":false},"excerpt":{"rendered":"
    After my post the other day about rounding errors, I got a ton of requests to explain the idea of significant figures. That’s actually a very interesting topic. The idea of significant figures is that when you’re doing experimental work, you’re taking measurements – and measurements always have a limited precision. The fact that your […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74,43],"tags":[],"class_list":["post-748","post","type-post","status-publish","format-standard","hentry","category-basics","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-c4","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/748","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=748"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/748\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=748"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}