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Significant digits - QUESTION AND ANSWERS

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  • Significant digits - QUESTION AND ANSWERS

    *** THE QUESTION ***

    Hi everyone,

    This is general question relevant to a lot of our research. It should
    be simple, but I don't recall the answer.

    How many digits should be used in reporting data?

    Here is an example. A biomechanist is trying to show a difference
    between Technique A and Technique B. He/she recruits 36 subjects and
    records each one of them performing both techniques, using equipment
    that measures each person to the nearest 0.1 unit. How many decimal
    places should be used in reporting the data? I can make a "common
    sense" determination, but is there some accepted procedure or guideline?

    Example data:


    Subject 1: 13.3 units

    Subject 2: 35.0 units

    Subject 3: 22.2 units

    Subject 36: 18.4 units


    Subject 1: 12.9 units

    Subject 2: 35.0 units

    Subject 3: 21.9 units

    Subject 36: 18.5 units

    Even though the mean within-subject difference is less than 0.1 units, a
    paired t-test reveals a statistically significant difference.

    Technique A mean: 25.33333 units

    Technique B mean: 25.36170 units

    Average difference : 0.02837 units

    Paired t-test p-value: 0.035

    How many decimal places would you use in reporting these mean values and
    difference in your paper? Why?

    Thanks in advance,

    - Glenn S. Fleisig, Ph.D.

    Glenn S. Fleisig, Ph.D., Smith & Nephew Chair of Research
    American Sports Medicine Institute
    833 St. Vincent's Drive, Suite 100
    Birmingham, AL 35205
    (tel) 205-918-2139


    *** THE ANSWER ***

    Thank you to all who responded. Based on the responses it is clearly a
    community that is bright and polite. :-)

    There were many well-written, insightful responses, which I've posted
    below. From these responses, my own experience, and conversations with
    others (biostatisticians, etc.), I've concluded that there is no hard
    scientific rule for significant digits. Therefore, I am please to share
    my conclusions from all of these thoughts, named (with tongue-in-cheek):

    Fleisig's Four Steps for Significant Digits

    1. In general, do not present individual values, mean values,
    standard deviations, or other calculated values in smaller units than
    the accuracy of your equipment.

    2. If you find a statistical significance with a magnitude less
    than the accuracy of the measurement, then question whether strict
    enough statistics were used. For example, if you find a small
    difference to be significant with a t-test, then perhaps the alpha-level
    was set too high.

    3. If you decide the statistical test was appropriate, then report
    the values with the added decimal place.

    4. If you report such a statistical difference, then explain
    whether or not you believe the difference has practical significance.
    In most cases, the researcher will decide the small difference has no
    practical significance. If you feel the difference is important, then
    you probably should have used more accurate equipment.


    *** THE RESPONSES ***


    From: Michael Cinelli []
    Sent: Tuesday, May 22, 2007 7:25 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Hi Glenn,

    You can only use the number of decimal places that is equivalent to the
    precision of your instrumentation.

    Michael Cinelli

    Postdoctoral Fellow

    Dept. of Cognitive and Linguistic Sciences

    Brown University

    190 Thayer St. Rm. 102F

    Providence, RI


    Every body continues in its state of rest, or of uniform motion in a
    right line, unless it is compelled to change that state by forces
    impressed upon it - Sir Isaac Newton


    From: Chris Kirtley []
    Sent: Tuesday, May 22, 2007 7:32 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Dear Glenn,

    That's a great example - wish I'd had it to hand when I used to nag my
    students about this.

    The key point to understand is of course the accuracy of the measurement
    - just because there is a readout with 3 significant figures doesn't
    mean that the measurement is accurate to 1 part in a thousand.

    The t-test also illustrates the difference between statistical
    signifance and practical significance. Clearly the measured difference
    is practically insignificant even if it is statistically significant
    (even assuming that the t-test was appropriate, which it may not be). It
    would be interesting to recalculate with the input data rounded off to
    integers, though even then it really is the practical signifiacnce of
    the difference that's most important.

    BTW, there's nothing special about decimal places. We should also round
    off integers according to the measurement accuracy. For example, it
    makes no sense to record blood pressure to the nearest mmHg - it should
    be rounded to the nearest 5 mmHg at least (10 is probably more



    -----Original Message-----
    From: []
    Sent: Tuesday, May 22, 2007 7:45 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    In your example, the program calculates a difference, but that
    difference is

    smaller than your ability to differentiate (measure). The general rule

    my chemistry background) is that you cannot report values that are more

    "accurate" than your original measure, because the reporting implies you

    actually measure what you are reporting to the degree you are reporting.

    Additionally, just because something is statistically significant does

    mean it is clinically significant.

    Just my two cents.

    Gary Christopher

    Doctoral Candidate

    Dept. of Kinesiology

    Texas Woman's University


    -----Original Message-----
    From: Sian Lawson []
    Sent: Tuesday, May 22, 2007 8:11 AM
    To: Glenn Fleisig; BIOMCH-L@NIC.SURFNET.NL
    Subject: RE: [BIOMCH-L] Significant digits - a Quiz for our community


    This is a good illustration of the disparity between a statistically

    significant difference and a functionally (or clinically) significant

    difference. Your t-test may show a statistical difference in the

    populations, but the difference is still negligibly small.

    Statistics, even means, cannot create accuracy. If the accuracy of your

    recording is to one decimal place, then your means etc will not be any

    more accurate than that.

    Best regards,


    Dr. Sian E. M. Lawson

    Centre for Rehabilitation and Engineering Studies (CREST)

    School of Mechanical and Systems Engineering

    University of Newcastle upon Tyne, U.K.

    CREST: +44 (0)191 222 6170

    Direct: +44 (0)191 222 8224

    Fax: +44 (0)191 222 8600


    -----Original Message-----
    From: []
    Sent: Tuesday, May 22, 2007 8:16 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community


    A quick answer is that if one is investigating the mean of x_1, ..., x_N
    then the *resolution* improves as stdev{x}/sqrt{N} when the number of
    data N increases. Thus, reporting mean value, enough decimals should be
    included so as to resolve the group standard deviation.

    Regards Frank Borg



    -----Original Message-----
    From: Torrence Welch []
    Sent: Tuesday, May 22, 2007 8:50 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Hi Glenn,

    You should report values with the same number of significant digits as

    least precise measurement in your study. Since the equipment in your

    example reported values with 3 sig figs, your average measurements

    be reported as:

    > Technique A mean: 25.3 units


    > Technique B mean: 25.4 units


    > Average difference : 0.0284 units


    > Paired t-test p-value: 0.0350

    Even if some of your measurements were over 100 units, you would still

    report 3 sig figs. You would move to 4 sig figs only if all measurements

    surpassed 100 units and you maintained 0.1 precision on your equipment.

    Happy researching,


    Torrence D.J. Welch, MSE

    Graduate Research Assistant / PhD Candidate

    Laboratory for Neuroengineering

    Wallace H. Coulter Department of Biomedical Engineering

    Georgia Institute of Technology / Emory University

    313 Ferst Drive

    Atlanta, GA 30332-0535

    (404) 385-4340 [lab]


    From: D. Gordon E. Robertson, PhD []
    Sent: Tuesday, May 22, 2007 9:29 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    For biomechanics courses and elsewhere I use a method originally
    in an engineering mechanics textbook by Beer & Johnston. It is
    ultimately based
    on the accuracy of slide rules but it works quite well in biomechanics
    where we
    can't get highly accurate measures of most body segment parameters.

    The concept is to make numbers accurate to 0.5%. That means all numbers
    are rounded to three significant figures unless the first number is a
    "1" in which
    case use four. Students like it because it is definitive and easy to

    Note that for a final answer to be accurate to 0.5% you must carry an
    extra digit
    in intermediate results.



    -----Original Message-----
    From: David Stodden []
    Sent: Tuesday, May 22, 2007 10:31 AM
    To: Glenn Fleisig
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Greetings Glenn,

    I seem to remember from some distant measurement or statistics class

    you report one decimal place more than your measurement device. Thus, if

    you are measuring degrees (e.g., 0-360 degrees) then you provide your

    statistic to the tenths of degrees - 90.8. If your calculations provide

    with an extra 3 or 4 decimal places, you simply round to the nearest

    decimal that is one decimal from your designated measurement. Thus,

    90.842222 degrees would still be reported as 90.8 degrees. Ultimately, I

    believe it depends not only what you are measuring and the specific

    instrumentation device, but also what type of sensitivity you demand.

    is my best recollection.

    I agree with the practical significance information that was presented

    the other two respondents, but the practical significance is related to

    effect size statistics, which are based on your power estimates and

    previous literature effect sizes, that should be reported in your



    -----Original Message-----
    From: David Smith []
    Sent: Tuesday, May 22, 2007 12:45 PM
    To: Glenn Fleisig; BIOMCH-L@NIC.SURFNET.NL
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Dear Glenn

    I studied statistics for a while but I still don't really understand

    For me they are a tautology in that they use their own rules to prove

    themselves. If you use different rules to prove something different its

    valid because you didn't use the rules of statistics.

    So if your two different groups had intervention =A and no intervention

    and A and B averages had no significant difference but A had a highly

    variable sequence of scores and B was fairly even across the board.

    this might intuitively indicate that there was a significant clinical

    difference in the out comes, which is what we are interested in, isn't

    Also what if the scores recorded had an exponential relationship with

    baseline value how would that effect the statistical significance. Or if

    where measuring mass for instance then 0.01grams in a range of 0-2grams

    be significant but in a range of 1000-10,000 grams it may not be

    What if your sensor has poor linearity or your sample rate was to low

    etc ie if your recording is to several decimal places then you must be

    that the input data is reliable and accurate enough to support that


    At the end of the day is it not down to the reader to decide if the

    and statistics are suitable for the conclusions that they may want to

    from a set of results.

    Cheers Dave


    -----Original Message-----
    From: David Smith []
    Sent: Tuesday, May 22, 2007 1:12 PM
    To: Glenn Fleisig; BIOMCH-L@NIC.SURFNET.NL
    Subject: Re: [BIOMCH-L] Significant digits - a Quiz for our community

    Hello again Glenn

    Just another point

    The P value is the indication of the confidence in rejecting the null


    If your P value is 0.035 (alpha =