too many digits the presentation of numerical data

Too many digits: The presentation of numerical data

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Emperor Joseph II: My dear young man, don't take it too hard. Your work is ingenious. It's quality work. And there are simply too many notes, that's all. Just cut a few and it will be perfect.

Mozart: Which few did you have in mind, Majesty?

Emperor Joseph II: Well, there it is.

Quotation from the film Amadeus (1984)

As a statistical reviewer for Archives and BMJ I am interested in the presentation of numerical data. 1 It concerns me that numbers are often reported to excessive precision, because too many digits can swamp the reader, overcomplicate the story and obscure the message.

A number's precision relates to its decimal places or significant figures (or as preferred here, significant digits ). The number of decimal places is the number of digits to the right of the decimal point, while the number of significant digits is the number of all digits ignoring the decimal point, and ignoring all leading zeros and some trailing zeros (for a fuller definition see 'significant figures' on Wikipedia).

Ideally data should be rounded appropriately, not too much and not too little (one might call it Goldilocks rounding). 2 The European Association of Science Editors guidelines include the useful rule of thumb: "numbers should be given in (sic) 2-3 effective digits". 3

Take as an example the odds ratio (OR) of 22.68 (95% CI 7.51 to 73.67) comparing beta mimetics with placebo for side effects requiring a change of medication. 4 Its two decimal places and four significant digits are excessive when the effect size and confidence interval (CI) are so large. Reporting it rounded to two significant digits, as 23 (7.5 to 74), or even as 23 (8 to 70), with one significant digit for the CI, would be simpler and clearer.

There are several published recommendations (or reporting rules ) about rounding numbers, some of which relate to decimal places (eg, the Cochrane Style Guide 5 or APA Style 6 to round to two decimal places), some to significant digits (eg, the European Association of Science Editors guideline above 3 ) and some to a combination of the two (eg, setting the number of decimal places to ensure two significant digits for the standard deviation (SD) 7 ). However,...

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• DOI: 10.1136/archdischild-2014-307149
• Corpus ID: 9550329

Too many digits: the presentation of numerical data

• Published in Archives of Disease in… 15 April 2015
• Mathematics

97 Citations

Goldilocks rounding: achieving balance between accuracy and parsimony in the reporting of relative effect estimates, missing the point: are journals using the ideal number of decimal places, how many decimals rounding descriptive and inferential statistics based on measurement precision, the bias and precision of reporting the average age of human participants., merging graphics and text to better convey experimental results: designing an “enhanced bar graph”, systemisers are better at maths, the principles of biomedical scientific writing: results, toward a more reliable characterization of fractal properties of the cerebral cortex of healthy subjects during the lifespan, setting number of decimal places for reporting risk ratios: rule of four, interrater and intrarater agreement and reliability of ratings made using the zaidi–dayal and richards–jabbour scales for the shape of the foramen magnum, 15 references, statistics notes: presentation of numerical data, publication manual of the american psychological association, improving tabular displays, with naep tables as examples and inspirations, ease guidelines for authors and translators of scientific articles to be published in english, the effects of weathering demonstrated by maternal age on low birth weight outcome in babies, speed of updating online evidence based point of care summaries: prospective cohort analysis, tocolytic therapy for preterm delivery: systematic review and network meta-analysis, effect of in vitro culture period on birthweight of singleton newborns., related papers.

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• Malays J Med Sci
• v.23(5); 2016 Sep

Reporting Statistical Results in Medical Journals

Wan nor arifin.

1 Statistical Editors, Malaysian Journal of Medical Sciences, Penerbit Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia

2 Unit of Biostatistics and Research Methodology, School of Medical Sciences, Universiti Sains Malaysia, 16150 Kubang Kerian, Kelantan, Malaysia

Abdullah Sarimah

Bachok norsa’adah, yaacob najib majdi, ab hamid siti-azrin, musa kamarul imran.

3 Department of Community Medicine, School of Medical Sciences, Universiti Sains Malaysia,16150 Kubang Kerian, Kelantan, Malaysia

Abd Aziz Aniza

4 Unit of Community Medicine, Faculty of Medicine, Universiti Sultan Zainal Abidin, 20400 Kuala Terengganu, Terengganu, Malaysia

5 PAPRSB Institute of Health Sciences, Universiti Brunei Darussalam, Gadong BE 1410, Brunei

Statistical editors of the Malaysian Journal of Medical Sciences (MJMS) must go through many submitted manuscripts, focusing on the statistical aspect of the manuscripts. However, the editors notice myriad styles of reporting the statistical results, which are not standardised among the authors. This could be due to the lack of clear written instructions on reporting statistics in the guidelines for authors. The aim of this editorial is to briefly outline reporting methods for several important and common statistical results. It will also address a number of common mistakes made by the authors. The editorial will serve as a guideline for authors aiming to publish in the MJMS as well as in other medical journals.

Introduction

Year over year, statistical editors of the Malaysian Journal of Medical Sciences (MJMS) must go through many submitted manuscripts, scrutinising the statistical and methodological soundness of the manuscripts. In 2015 alone, the MJMS received 272 manuscripts from many different countries, 52% of which were original articles ( 1 ). However, the editors have noted many different styles of reporting statistical results, and these styles are not standardised among authors. This has caused unnecessary difficulties for the editors as they have to comment not only on the methods and statistics used, but also on the technical and formatting aspects of the manuscripts. This lack of standardised reporting also causes delay in reviewing and accepting submitted articles. Admittedly, this could be due to a lack of clear written instructions on reporting statistics in the guidelines for authors. Although there are a number of guidelines available on reporting statistical results, for examples in Lang and Altman ( 2 ) and Cummings and Rivara ( 3 ), the editors of the MJMS found them incomplete as guidelines for authors.

The aim of this editorial is to outline reporting methods for several important and common statistical results. It will also address a number of common mistakes made by authors.

Presentation Forms

Statistical results can be presented in text, table, or figure form. The decision depends very much on the amount of information the authors want to present to the readers.

The text form is suitable for brief results, for example, the description of a sample (“A total of 100 patients were recruited,” “Most of the respondents were female...”). Text form is also used to highlight important results in tables that might be missed by readers given the amount of information commonly summarised in tables, for example, “Among all the studied factors, only gender and salary were found to be significantly associated with...”.

The table form is suitable for presentation of detailed statistical results. Common examples are detailed demographic profiles of study participants, results of a multiple logistic regression analysis, and cross-tabulation of factors with outcomes. It is very important to note that the table description is placed at the top of the table, while the list of abbreviations and additional relevant descriptions (especially related to statistical analysis) are placed below the table as footnotes. The footnotes should be indicated by superscript Roman letters (a, b, c, ...) instead of symbols or numbers. All abbreviations used in the table must be described again in the footnotes, although the abbreviations were already described in text or earlier tables.

The figure form includes charts, graphs, and other images. It should be reserved for results that are more presentable in this form, for example, trends or geographical distribution of disease, histopathological or radiological images, and comparison of means over time. Figure descriptions are placed below the figure.

Descriptive Statistics

The descriptive statistics summarise data from a sample, for example, demographic profiles. Whenever there are a number of groups, it is useful to provide the descriptive statistics by group and for the overall sample. This gives a visual impression of the comparability of the groups in term of their baseline characteristics. It is not necessary to report statistical tests and P -values in such a summary because the main concern is the comparability of the participants (which reflects the sampling), not the populations.

Depending on the types of variables, authors should present the appropriate descriptive statistics. For numerical variables, if the variable is normally distributed, the mean and standard deviation (SD) are presented. In the text, this is reported as mean (SD = value), for example, “the mean age was 46.5 (SD = 3.0).” In a table, the “mean (SD)” statement is included in the header. Whenever the variable is not normally distributed, the median and inter-quartile range (IQR) are reported instead. The use of “±” symbol between a mean and an SD must be avoided because the mathematical symbol has its own specific meaning. For the categorical variable, count ( n ) and percentage (%) are presented. In addition, authors must report the group size and total sample size, written as n = size in the table headers and the table description, respectively. The use of a capital N in place of n must be avoided as it refers to population size instead of sample size. A typical demographic table is presented in Table 1 .

Patient demographics ( n = 95)

Confidence Interval

Precision of the estimates, for example, single mean and proportion, are presented in the form “estimate (95% CI: lower limit, upper limit)”. In writing, for the single mean, “the mean body mass index (BMI) was 22.5 (95% CI: 21.5, 23.5)” and for the single proportion/percentage “the prevalence of obesity was 34.5% (95% CI: 30.5%, 38.5%)”. Other common examples are the reporting of mean difference (independent t -test) and odds ratio (logistic regression), which are presented under the specific statistical tests section below.

Common Statistical Tests

In order to standardise the reporting and presenting of statistical results in the MJMS, the editors offer the suggested forms of presentation summarised in Table 2 as general guidelines.

Presenting statistical results

Additional Concerns

In text, the P -value is written as an italic capital P followed by the value, while as a table header, it should be written as P -value. The authors should write the value instead of reporting the result as “not significant” or “NS” ( 3 ). For example, “the comparison was significant, with P = 0.003”. Three decimal places are preferred in the MJMS for all ranges of P -values. The editors are aware of different guidelines on the number of decimal places of P -values, for example, as given in Cummings and Rivara ( 3 ) and Cole ( 4 ).

Italic formatting of statistical tests and coefficients

Statistical tests that are named after the statistical distributions on which they are based, for example, t -test, F -test, and χ 2 -test, are italicised. In addition, coefficients, for example, r (Pearson’s correlation coefficient), R 2 ( R -squared), and α (Cronbach’s alpha) are also italicised.

Statistical analysis

Computer programs used for statistical analysis should be described, specifically, the name of the program and the version should be given as well as the specific add-on packages if applicable. For example, “IBM SPSS for Windows version 22.0 was...” and “ psych version 1.5.8 and lavaan version 0.5–20 packages were used in the R software environment.” The statistical analysis used should be described in sufficient detail to reproduce the analysis ( 2 ), particularly the name of the analysis, its relation to the aims of the study, and the dependent and independent variables. In addition, Lang and Altman ( 2 ) outlined in greater detail general principles of reporting statistical methods.

Formatting and presentation of numbers

In general, one decimal place is used for percentage values. Use two or more decimal places for percentage values less than 1.0%. For descriptive statistics of numerical data, add one additional decimal place to the original data. For example, if cholesterol level is reported with one decimal place (e.g. 4.8 mmol/L), the mean and SD should be reported with two decimal places (e.g. mean = 4.82, SD = 2.11 mmol/L). Use two decimal places for test statistics values, for example, values of t -statistic, F -statistic, and χ 2 -statistic.

Using a dash “−” in between any two numbers must be avoided as it could be mistaken for a minus or negative sign. For example, authors should write “the age ranges between 20 to 29 years old” instead of “the age ranges between 20 – 29 years old”. In relation to formatting of numbers in tables, the last digits of numbers must be right-aligned. The formatting is demonstrated in Table 1 and ​ and2 2 .

Closing Remarks

This editorial outlines the basics of reporting statistical results in medical journals. This editorial will serve as a guide to authors aiming to publish in the MJMS. Given the availability of the guidelines on reporting statistical results, it is hoped that the authors follow the guidelines to ensure standardisation of the submitted manuscripts. This will shorten the process of reviewing and accepting manuscripts submitted to the MJMS.

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Statistics Notes: Presentation of numerical data

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• Peer review
• Douglas G Altman , head a ,
• J Martin Bland , professor of medical statistics b
• a IRCF Medical Statistics Group, Centre for Statistics in Medicine, Institute of Health Sciences, PO Box 777, Oxford OX3 7LF
• b Department of Public Health Sciences, St George's Hospital Medical School, London SW17 0RE
• Correspondence to: Mr Altman.

The purpose of a scientific paper is to communicate, and within the paper this applies especially to the presentation of data.

Continuous data, such as serum cholesterol concentration or triceps skinfold thickness, can be summarised numerically either in the text or in tables or plotted in a graph. When numbers are given there is the problem of how precisely to specify them. As far as possible the numerical precision used should be consistent throughout a paper and especially within a table. In general, summary statistics such as means should not be given to more than one extra decimal place over the raw data. The same usually applies to measures of variability or uncertainty such as the standard deviation or standard error, though greater precision may be warranted for these quantities as they are often used in further calculations. Similar comments apply to the results of regression analyses, where spurious precision should be avoided. For example, the regression equation 1

birth weight=-3.0983527 + 0.142088xchest circumf + 0.158039 x midarm circumf, purports to predict birth weight to 1/1000000 g.

Categorical data, such as disease group or presence or absence of symptoms, can be summarised as frequencies and percentages. It can be confusing to give percentages alone, as the denominator may be unclear. Also, giving frequencies allows percentages to be given as integers, such as 22%, rather than more precisely. Percentages to one decimal place may sometimes be reasonable, but not in small samples; greater precision is unwarranted. Such data rarely need to be shown graphically.

Test statistics, such as values of t or χ 2 , and correlation coefficients should be given to no more than two decimal places. Confidence intervals are better presented as, say, “12.4 to 52.9” because the format “12.4-52.9” is confusing when one or both numbers are negative. P values should be given to one or two significant figures. P values are always greater than zero. Because computer output is often to a fixed number of decimal places P=0.0000 really means P<0.00005—such values should be converted to P<0.0001. P values always used to be quoted as P<0.05, P<0.01, and so on because results were compared with tabulated values of statistical distributions. Now that most P values are produced by computer they should be given more exactly, even for non-significant results—for example, P=0.2. Values such as P=0.0027 can be rounded up to P=0.003, but not in general to P<0.01 or P<0.05. In particular, the use of P<0.05 (or, even worse, P=NS) may conceal important information: there is minimal difference between P=0.06 and P=0.04. In tables, however, it may be necessary to use symbols to denote degrees of significance; a common system is to use *, **, and *** to mean P<0.05, 0.01, and 0.001 respectively. Mosteller gives a more extensive discussion of numerical presentation. 2

The choice between using a table or figure is not easy, nor is it easy to offer much general guidance. Tables are suitable for displaying information about a large number of variables at once, and graphs are good for showing multiple observations on individuals or groups, but between these cases lie a wide range of situations where the best format is not obvious. One point to consider when contemplating using a figure is the amount of numerical information contained. A figure that displays only two means with their standard errors or confidence intervals is a waste of space as a figure; either more information should be added, such as the raw data (a really useful feature of a figure), or the summary values should be put in the text.

In tables information about different variables or quantities is easier to assimilate if the columns (rather than the rows) contain like information, such as means or standard deviations. Interpretation of tables showing data for individuals (or perhaps for many groups) is aided by having the data ordered by one of the variables—for example, by the baseline value of the measurement of interest or by some important prognostic characteristic.

• Bhargava SK ,
• Mohan MAN ,
• Sachdev HPS
• Bailar JC ,
• Mosteller F

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The art of reporting numerical data

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Jonathan A Cook, Dongquan Bi, Jonas Ranstam, The art of reporting numerical data, British Journal of Surgery , Volume 109, Issue 6, 16 May 2022, Pages 548–549, https://doi.org/10.1093/bjs/znac028

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It is unfortunate that in English and a number of other languages, we use the same term ‘statistics’ to refer to both numerical data (e.g. national statistics) and also the science of collecting and analysing data (e.g. a degree in statistics). The former can be very mundane and routine, whereas the latter, which utilizes the former to understand variability and carry out inference, is enlightening and often sobering. The reporting of numerical data should be informed by statistical principles (the science of statistics). One area where this can be counterintuitive is the level of numerical precision to report data (both of individual values and summaries like means and standard deviations). A review of three recent BJS articles identified over 1000 statistics across the articles, which either summarized data from the respective study or reported an output from an analysis of study data. Often in manuscripts submitted to BJS , values are reported to an excessive level of numerical precision.

Simplicity, neatness of presentation, a desire to report fully numerical calculations, or a false understanding of the value of the data collected, can all lead to reporting data to an excessive level of precision. For example, it is tempting but somewhat misleading to report 13 out of 39 as (33.33 per cent). The use of two decimal points (four significant figures) gives the false impression of greater precision than really occurred. In this example, each observation accounted for over 3 per cent of the final percentage. There is a quirk of numerical data that percentages will often not add up to exactly 100 per cent where there are three or more categories. However, as long as the number of observations within each category is accurately reported with the percentage in the group, this apparent discrepancy is but a small price to pay for greater clarity and a more honest representation of the data. A simple rule of thumb is that if there are less than 100 observations in a sample, reporting percentages to fractions of a per cent is not helpful. Arguably even for larger samples, it is rarely necessary except perhaps for reporting the extremes (e.g. 0.1 versus 0 per cent). Many people will find tables filled with redundant decimal points more difficult to read and understand, and find themselves lost in a sea of unnecessary figures. The number of people who are able to differentiate between tenths of a per cent, let alone 100 of a per cent, are, in our view rather miniscule. Equivalent arguments can be made for reporting proportions where two decimal points should be the standard level of precision.

Continuous data are perhaps the hardest to fairly report; to do so requires an understanding of the quantity measured, how well it can be measured, how it is or might be used, and how well the original data were recorded. While an operation time reported as 134.682341 min can be occur and could be recorded as such, it is of no greater value (and certainly of lesser clarity) to do so than recording it as simply 135 min. You would be a brave person to believe an operation time is accurately recorded to this level of precision (microseconds) in practice. For the poor reader of papers, this is even more so the case when it is recognized that operating practices and the recording of the operation time tend to vary greatly between surgeons, surgical teams, and institutions. My 135 min could easily be your 140 min. Reporting mean operation times beyond minutes is of little clinical value irrespective of how large the study sample size is.

Another area where excessive decimal points are commonly used is in reporting P -values as if more zeros create stronger evidence. This probably reflects both some misunderstanding of what a P -value is, which is an indirect measure of evidence against a null hypothesis, and not a measure of the strength, nor the magnitude 1 of statistical disagreement (only of the unlikeliness of compatibility assuming the null hypothesis is true). If the conventional statistical approach is being used with a prior statistical significance level specified (say, the typical two-sided 5 per cent level), then reporting it to a sufficient level to see how the P -value relates to that marker of statistical significance is what is important. As a consequence, reporting a P -value as ‘ P < 0.05’ or ‘n.s.’ does not provide sufficient detail as it is unclear how close or not the P -value was to the cut-off point. However, once the value is some distance from the cut-off marker, further precision is of little consequence. P -values above 0.1 can happily be reported to one decimal point (e.g. 0.2), and if 0.05 is the significance level of interest, then the differences between P -values <0.001 are of little interest. Where large numbers of hypotheses are tested (e.g. analyses of genetic data), the significance level of interest should be reduced 2 , but the same principle can be applied to the corresponding lower significance level. Here, as with reporting other outputs of statistical tests and models such as a mean difference or an odds ratio with confidence intervals, the level of precision that is appropriate should be determined by the question we are seeking to answer and how the findings might be applied, as well as the level of precision the data can bear.

Helpful guidance on reporting numbers is available elsewhere 3 for a range of statistical metrics beyond those covered here. It is difficult to be too prescriptive as surgery and science are wonderfully complex. Nevertheless, it would benefit our research and the readers of it if we were more circumspect in reporting our data and a bit more humble in our data presentation.

Disclosure . The authors declare no conflict of interest.

Cook   JA , Ranstam   J . Statistical methods that provide an effect size are to be preferred . Br J Surg   2016 ; 103 : 1365

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Cook   JA , Ranstam   J . Spurious findings . Br J Surg   2017 ; 104 : 97

Cole   TJ . Too many digits: the presentation of numerical data . Arch Dis Child   2015 ; 100 : 608 – 609

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Presentation of numerical data

Affiliation.

• 1 IRCF Medical Statistics Group, Centre for Statistics in Medicine, Institute of Health Sciences, Oxford.
• PMID: 8595293
• PMCID: PMC2350327
• DOI: 10.1136/bmj.312.7030.572

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• [Presentation of statistical data in nursing 12: with reflection and estimation]. Ostermann R, Wilhelm AF, Wolf-Ostermann K. Ostermann R, et al. Pflege Z. 2004 Dec;57(12):862-4. Pflege Z. 2004. PMID: 15646111 German. No abstract available.
• Statistical tools in published articles of a public health journal in 2013 and 2014: bibliometric cross-sectional study. Arcila Quiceno V, García Restrepo E, Gómez Rúa N, Montenegro Martínez G, Silva Ayçaguer LC. Arcila Quiceno V, et al. Medwave. 2015 Aug 31;15(7):e6238. doi: 10.5867/medwave.2015.07.6238. Medwave. 2015. PMID: 26460577 English, Spanish.
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Should percentages be reported with decimal places?

When presenting data using a percentage, is it a good thing to have decimal places, say 2 decimal places instead of rounding off to whole numbers?

For example, instead of 23.43%, you round off to 23%.

I am looking at this from the perspective of whether the 2 decimal places accuracy will make much difference since we are dealing with percentage and not raw data value.

• 1 $\begingroup$ There are many fields where small percents are so small that people use parts per million, per billion and so forth. Either that's a different question -- because people do or should know that citing numbers like 0.000001 or even 0.0001% is in that circumstance silly and one should use different units -- or it's another answer to this question. When some or all of the numbers of interest are very small, large numbers of decimal places may be essential as well as informative. $\endgroup$ –  Nick Cox Commented Nov 8, 2018 at 17:47
• $\begingroup$ This is a special case of the issues discussed at stats.stackexchange.com/questions/8734 . Note that this question concerns precision: accuracy is a different matter altogether. See gis.stackexchange.com/a/8674/664 for the distinction. $\endgroup$ –  whuber ♦ Commented Nov 8, 2018 at 18:10

4 Answers 4

It depends on the size of the differences between classes. In most applications, saying the 73% prefer option A and 27% prefer option B is perfectly acceptable. But if you're dealing in an election where candidate X has 50.15% of votes and candidate Y has 49.86%, the decimal places are very much necessary.

Of course, you need to take care to make sure that all classes add up to 100%. In my electoral example above, they add up to 100.01%. In that case you might even consider adding a third decimal place.

• 1 $\begingroup$ I'd also say it depends on your data and your goals. If I were briefing an executive on the sources of customer complaints, no way I'd be talking about fractions of a percent. You also have to consider the margins of error: saying 50.15% of voters prefer candidate X +/- 5% is suspicious. $\endgroup$ –  Wayne Commented Mar 6, 2014 at 15:22
• 2 $\begingroup$ If there are just two percents, rounding and adding to 100% are completely compatible. See e.g. statweb.stanford.edu/~cgates/PERSI/papers/freedman79.pdf $\endgroup$ –  Nick Cox Commented Mar 6, 2014 at 15:31
• $\begingroup$ @NickCox That's a nice find! In case that link goes dead, for future readers the reference is "On Rounding Percentages", Persi Diaconis; David Freedman, Journal of the American Statistical Association, Vol. 74, No. 366. (Jun., 1979), pp. 359-364. Despite the rather broad title, it deals with the probability a table of rounded percentages "correctly" sum to 100. The probability declines to around 3/4 with 3 categories, around 2/3 with 4 categories, and $\sqrt{6/\pi c}$ with $c$ categories $\endgroup$ –  Silverfish Commented Mar 30, 2018 at 19:23
• 1 $\begingroup$ I actually don't think it is necessary to ensure that percentages sum exactly to 100, and wouldn't introduce extra (possibly spurious or distracting) decimal places just to ensure it happens. A little footnote that "Percentages may not sum to 100 due to rounding" should suffice. In fact adding decimal places will not always help, eg the simplest case of having three categories with equal frequencies, then neither 33%+33%+33% nor 33.33%+33.33%+33.33% quite solve the "sum to 100" problem. $\endgroup$ –  Silverfish Commented Mar 30, 2018 at 19:34

Different organisations often have conflicting rules for the precision in reporting of results. Ultimately there is a trade-off between when seeing the extra digits is useful, versus cases where unnecessary and excessive precision "can swamp the reader, overcomplicate the story and obscure the message" — a subject explored by Tim Cole (2015) in a piece that I found gave a useful guide to "sensible" precision in reporting, and a comparison of leading style manuals. His advice on percentages was as follows:

Integers, or one decimal place for values under 10%. Values over 90% may need one decimal place if their complement is informative. Use two or more decimal places only if the range of values is less than 0.1% Examples: 0.1%, 5.3%, 27%, 89%, 99.6%

By "complement" he is referring to cases where one might be interested in the "other lot", e.g. if I tell you 98% of patients in a trial got better, you may well be interested in the 2% who did not, and in that case another decimal place to distinguish whether that "2%" really means "2.4% or "1.6%" would actually be useful.

Cole, T. J. (2015). Too many digits: the presentation of numerical data. Archives of disease in childhood , 100 (7), 608-609. http://dx.doi.org/10.1136/archdischild-2014-307149

This is a significant figures issue, and is dependent upon the precision of the numbers underlying the percentages. The technically correct number of significant figures is not dependent upon downstream use or the differences between percentage values.

If you're trying to express a percentage describing 5 items out of 7, it would be absurd to claim that it's 71.4285714285% - you simply don't have the precision to back up all those decimal places. When doing division, your answer should have as many significant figures and the fewest number of sig figs in your starting numbers. Here, you only have 1 significant figure, so the percentage should really just be 70%, not even 71%. If you had another example where you want to express 71428 items out of 100000, then you are justified in using more significant figures, all the way out to 71.428%.

Even if you have great precision, it's often preferable to truncate for human readability. Depending on your domain, adding those two extra decimal places may or may not make a difference. You should never over-report significant figures, but you may be justified in under-reporting them if your statistical precision is greater than what's needed for your application.

• 4 $\begingroup$ Well worth bringing up this issue, but the examples in your second paragraph are rather opaque. I have seven cousins; five are male: in what sense do I lack a precise enough observation to justify reporting this as a percentage to any number of decimal places I please? $\endgroup$ –  Scortchi - Reinstate Monica ♦ Commented Mar 30, 2018 at 22:27
• 2 $\begingroup$ I have to agree with @Scortchi. If you know the value of an integer (e.g. number of electrons orbiting an atom, etc) you know it essentially to infinite precision points. We know the proportion of Scortchi's cousins that are male to infinite precision points. What's pragmatic to report is a different issue. (In this case, I think simply saying "5/7" is the best thing.) $\endgroup$ –  Bridgeburners Commented Nov 8, 2018 at 17:52

The goal is to make it easy for the reader to understand the important differences. Too many digits obscures the meaningful difference between values in a table. Too few leaves out important information. Here's a great discussion: https://newmr.org/blog/how-many-significant-digits-should-you-display-in-your-presentation/ and here's a much more detailed analysis: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4483789/

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Too many digits—the presentation of numerical data

We have all been frustrated reading numbers to too many decimal places, the simplest being digital scales in the outpatient clinic where measurements are probably not accurate to more than 10g although the implication of the weight recorded is that the accuracy is much greater. In an excellent leading article this month Tim Cole takes us back to first principles to discuss this and provide sensible, pragmatic guidelines for the presentation of numerical data. It is interesting and helpful to work through. Remember the difference between decimal places and significant figures. The number of significant figures (digits) is the number of all digits ignoring the decimal point, and ignoring all leading and some trailing zeros. Data should be rounded appropriately—not too much, not too little. Clearly, for example 22.68 (95% confidence interval 7.51–73.67) is more effectively and meaningfully written as 23 (95% confidence interval 7.5–74). The various reporting tools are discussed. Significant figures should be considered rather than just decimal places. The general principle is to use two or three significant digits for effect sizes, and one or two significant digits for measures of variability. There is a helpful summary table included with recommendations given for different scenarios. See page 608 .

Prevalence of severe childhood obesity in England

Bed sharing and sudden infant death.

Bed sharing increases the risk of sudden infant death in infants less than 3 months. The effect is most profound in infants less than 1 month (5 fold increase in risk of SIDS in infants less than 1 month). The mechanism is not clearly defined. Heyman and colleagues review the accidental deaths during sleep (as a cause of sudden infant death in infancy); New Zealand 48 cases, 2002–2009, 0.1 per 1000 live births. Deaths were due to overlay (n=30), or wedging (n=18), with 34 (71%) in a bed sharing situation. Of the overlay group 8 were by a mother while breast feeding, 4 by a sibling and 17 by a parent. In the wedging group 10 were between a sleeping surface and wall or broken cot, 6 between a cushion and couch and 2 between a sleep surface and bedding. The authors conclude these are potentially preventable deaths particularly if bed sharing is avoided, faulty or if inadequately constructed cots are avoided and extra attention is paid to the safety of sleep arrangements particularly if adhoc/temporary. In an accompanying editorial Volpe and colleagues discuss—Infant sleep related deaths: why do parents take risks. The editorial is provocative discussing these issues in the context of other factors, recent guidance from NICE and the need to inform parents about the risks and benefits in order to help them make the best decision for them and their child. See pages 610 and 603 .

Should we discourage daytime napping

Duration and quality of sleep affect child development and health with early childhood being a time in which sleep consolidates into the night and napping ceases. Many factors influence sleep patterns and childhood sleep patterns have the potential to disrupt family functioning and child well being. Thorpe and colleagues report a systematic review of the evidence regarding the effects of napping on child development and health. 26 articles were included—heterogeneous quality, observational study designs. Most of the findings were inconsistent—cognition, behaviour, health impact—probably because of variability in ages and habitual napping status. The most consistent finding was an association between napping and later onset, shorter duration and poorer quality night sleep with evidence strongest in children greater than 2 years. The authors highlight the absolute need for more data before specific advice is given. Lucy Wiggs discusses the findings and their wider implications in an accompanying editorial. It is interesting to reflect on what is normal—how should a nap be identified (quantity, quality, timing), heterogeneity of the individual, influence of the family and environment, and multiple potential outcome measures of impact and therefore difficulty in studying. Certainly napping in young children is universal and the question posed in the title of the editorial—Daytime napping in preschool aged children; is it to be encouraged—is appropriate. Ensuring children receive sufficient amounts of good quality sleep, according to their individual needs, remains the priority. See pages 615 and 604 .

Why do we treat children of Jehovah's witnesses differently from their adult parents

This is a significant, emotive and difficult issue particularly when the clinician is faced with a patient who needs a blood transfusion but refuses it for religious or other reasons. In a thought provoking leading article Robert Wheeler explores these issues, using case law to illustrate and very much highlighting the different issues in children compared to adult and as such is very relevant to us as paediatricians. The decision of a competent adult to refuse blood is legally binding on doctors. This is not the case in a child or young person under age 18 years when the law will no longer defer to a parent's wishes or religious beliefs if such deference will mean that the child is not treated in accordance with his best interests. This clearly needs to be managed carefully and with consideration of alternative options and after social care and legal advice. The issues and some of the practicalities are complex, even more so during adolescence and the article of relevance to how we manage these difficult situations when blood transfusion or other life saving treatment are needed and for complex reasons consent not forthcoming. See page 606 .

Linked Articles

• Leading article Why do we treat the children of Jehovah's Witnesses differently from their adult parents? Robert Wheeler Archives of Disease in Childhood 2015; 100 606-607 Published Online First: 04 Mar 2015. doi: 10.1136/archdischild-2014-307354
• Editorial Infant sleep-related deaths: why do parents take risks? Lane E Volpe Helen L Ball Archives of Disease in Childhood 2015; 100 603-604 Published Online First: 31 Mar 2015. doi: 10.1136/archdischild-2014-307745
• Original article Infant suffocation in place of sleep: New Zealand national data 2002–2009 Rebecca M Hayman Gabrielle McDonald Nick J de C Baker Edwin A Mitchell Stuart R Dalziel Archives of Disease in Childhood 2014; 100 610-614 Published Online First: 25 Nov 2014. doi: 10.1136/archdischild-2014-306961
• Original article Prevalence of severe childhood obesity in England: 2006–2013 Louisa J Ells Caroline Hancock Vicky R Copley Emma Mead Hywell Dinsdale Sanjay Kinra Russell M Viner Harry Rutter Archives of Disease in Childhood 2015; 100 631-636 Published Online First: 27 Jan 2015. doi: 10.1136/archdischild-2014-307036
• Original article Napping, development and health from 0 to 5 years: a systematic review Karen Thorpe Sally Staton Emily Sawyer Cassandra Pattinson Catherine Haden Simon Smith Archives of Disease in Childhood 2015; 100 615-622 Published Online First: 17 Feb 2015. doi: 10.1136/archdischild-2014-307241
• Leading article Too many digits: the presentation of numerical data T J Cole Archives of Disease in Childhood 2015; 100 608-609 Published Online First: 15 Apr 2015. doi: 10.1136/archdischild-2014-307149
• Editorial Daytime napping in preschool-aged children; is it to be encouraged? Luci Wiggs Archives of Disease in Childhood 2015; 100 604-605 Published Online First: 15 Apr 2015. doi: 10.1136/archdischild-2014-307614

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Too many digits: the presentation of numerical data (Q28647834)

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Summarizing Numerical Data

Seeing the forest for the trees.

Once you have your data in front of you, you’ve seen how we can form visual summaries with ggplot2 . But how can we calculate numerical summaries? Furthermore, what if we are concerned about summarizing a portion of our data, like just one species of penguin at a time? We will answer these questions below, and introduce some new functions from the dplyr package (within the tidyverse library) along the way. We’ll also look at how factor() can come in handy while plotting.

If you are playing along in RStudio while reading these notes (which we strongly recommend!), be sure to start off by loading the two packages that are necessary for the tutorial by running the following code.

Calculating Numerical Summaries

One example of a numerical variable we could have examine is the body mass of a particular penguin (measured in grams). Let’s calculate both a measure of center and spread for this variable.

To get an idea of what summaries we should pick, let’s revisit the density plot from earlier.

What we can glean from this figure is that the distribution of body masses across all species of penguin is skewed right. This means that, for instance, a more typical observation lies closer to 4000 grams than 5000 grams.

If we take an average, it is likely to be pulled to the right by the larger, but less typical, observations. The median observation, however, would be more resistant to this pull. Therefore, the median might be a nice choice for a measure of center. Similarly, since the IQR is initially constructed from the median, it will serve well here as a measure of spread.

Now, let’s calculate these values. We should first isolate our variable of interest. We can do this in code by using the dplyr function select() .

As is custom with dplyr functions, the first argument goes to the data frame you are working with. The following arguments are more function specific. In select() ’s case, we tell the computer which column/variable we are interested in.

Now, we can calculate our summaries. When working with a vector, we could use functions like mean() and median() directly, e.g.  median(body_mass_g) . However, body_mass_g is not a standalone vector but is now a column in a data frame called body_mass ! Therefore, we need to access it through a dplyr function called summarise() .

Note that while the first argument goes to the name of the data frame, the following arguments are given to the names of the new columns that summarise() puts in another new data frame (one row by two columns). You can name the columns whatever you would like.

Based on what we’ve found, the median here supports the claim we made above: that a typical penguin has a body mass closer to 4000 grams than to 5000 grams. The middle 50 percent of the penguins have body masses within 1225/2 grams, or roughly 600 grams, of 4050.

Groupwise Operations

Let’s return to the bill length examine of a particular penguin, measured in millimeters. Here is the density plot for all of the data ; for simplicity, earlier we showed you the plot for only the first 16 observations.

This plot is interesting. It appears we have a bimodal shape! While it’s tempting to state that the data is roughly symmetric and calculate an overall mean, we should first see if there are any other variables at play. It stands to reason that different species of penguin might have different anatomical features. Let’s add species to the mix by using the color aesthetic (see if you can code along)!

Aha! We now see that each penguin species has its own shape of distribution when it comes to bill length.

The example above demonstrates a very common scenario: you want to perform some calculations on one particular group of observations in your data set. But what if you want to do that same calculation for every group? For example, what if we’d like to find the average and standard deviation of bill length among each species of penguin separately ?

This task - performing an operation on all groups of a data set one-by-one - is such a common data science task that nearly every software tool has a good solution. In the dplyr package, the solution is the group_by() function. Let’s see it in action.

Like most tidyverse functions, the first argument to group_by() is a data frame. The second argument is the name of the variable that you want to use to delineate groups. In this case, we want to group by species to calculate three separate mean/standard deviation pairs.

Now, assuming we roll with our new grouped_penguins data frame, we can use summarise() like we did before!

From both the visuals and the numbers, we can see that Adelie penguins have much smaller bill lengths on average when compared to Chinstrap and Gentoo penguins. We also see that the Adelie distribution of bill lengths is less variable than the distributions of the other two species.

Plotting with Categorical Variables

Finally, let’s return to the violin plot of bill lengths grouped by species of penguin.

What if I wanted the Adelie violin to show up on the top of the graph? By default, the violin plot puts the level first in the alphabetical order on the bottom of the plot. Therefore, I need to reorder the levels of species to put Adelie at the top. This is where factor() will do the job!

As before, bill_length_mm is not a standalone vector but a column in a data frame! We cannot access it directly, e.g. by factor(species, levels = c("Gentoo", "Chinstrap", "Adelie")) .

Therefore, we use the dplyr function mutate() . A mutation involves changing the properties of an existing column, or adding a new one altogether (which we will explore next week).

The first argument of mutate() is dedicated to our data frame, penguins . The second argument can be the name of an existing column or the name of a new column (next week). We want to change species to be an altered version of itself, hence we name the second argument species . Make sure you understand where each set of parentheses closes and ends.

Now, assuming we roll with our new reordered_penguins data frame, we can use ggplot() like we did before!

A summary of a summaries…this better be brief! Summaries of numerical data - graphical and numerical - often involve choices of what information to include and what information to omit. These choices involve a degree of judgement and knowledge of the criteria that were used to construct the commonly used statistics and graphics.

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