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• Should I report the means or the medians of my data?
• Mean or median when one group is normally distributed and the other is not?
• How do I calculate the standard error of the median?
• How do I calculate a 95% confidence interval around a geometric mean?
• What is the best way to describe a small dataset?
• What is the best way to describe my observations on a 5-point Likert scale?
• What should I do if the mean minus the standard deviation lies far outside the range of possible values?
• References
• Additional resources

We looked at the difference in throughput times of the outpatient clinic in two centers. These times were not normally distributed, so we reported the values as median [25th–75th percentile] and performed the Mann-Whitney test to test the differences between the centers. However, a reviewer now asks for the mean throughput times to be reported. Does it make sense to convert the values to means?

With non-normally distributed data, it is indeed common practice to report medians and use non-parametric tests, so you did this well. According to the reviewer, it would be clearer if you reported means and transformed the data. In my view, transforming is not desirable. You would really only do this to be able to perform parametric tests (such as regression or the t-test). The data you obtain after transformation are also difficult to interpret (what does the mean of the logarithm of time even represent?). To meet the reviewer's request, you can report the means alongside the median values.

In my study I compare a patient group with a control group. In my Table 1, I compare the clinical characteristics of the groups. For this, I reported the normally distributed parameters as mean + SD, and the non-normally distributed parameters as median + IQR. Among others, the age of the patient group is not normally distributed, while the age of the control group is normally distributed; so I would have to report one as mean and the other as median. As a result, readers cannot compare the values at a glance. Would it be better in this case to express both as median, or still one as mean and one as median?

I recommend describing both groups with the same descriptive measures. As you write yourself, otherwise you cannot properly compare them. If for one of the groups reporting the mean and SD is not a good representation of where most of the values lie, then I would describe both groups with median and IQR.

If you report data as median (because of a non-normal distribution) and want to add an SE to it, is that the SE of the mean, or should it always be the SE of the median? If it should be the SE of the median, how can you determine that? What steps do I need to take in SPSS to obtain this data (via explore, only the SE of the mean is shown)?

If the data is presented using medians, it would be strange to add an SE of the mean to it. Descriptions of data often use the following notations:

mean +/- SD

median [lower range, upper range]

median [Q1, Q3]

If you prefer to report the SE (and thus are not so much describing the data as quantifying the precision of the estimate of the parameter in question), then in the case of the median you need the SE of the median. SPSS does not provide this, because it is best calculated using bootstrapping. For large samples that are normally distributed, the following approximation can be used: SEmedian = 1.25 * SEmean. But in your case this will not be appropriate, due to the lack of a normal distribution.

Another approach is to report a 95% confidence interval around the median. According to Altman's book (Statistics with confidence), the following formulas can then be used:

$r=\frac{n}{2} - 1.96 * \frac{\sqrt{n}}{2}$

$s=1+\frac{n}{2} + 1.96 * \frac{\sqrt{n}}{2}$

Round r and s to the nearest whole numbers. Now the r-th and s-th observations of the ranked dataset are the bounds of the 95% confidence interval.

So you can either choose one of the alternative descriptions above, or obtain an estimate of the SE of the median using bootstrapping.

I need to calculate the geometric mean with a 95% confidence interval (CI) of a large number of CRP values. I created the variable ln(CRP) and calculated its mean. I understand that I need to take e^(that number) to get the geometric mean. However, I don't know how to obtain a 95% CI for this.

The easiest way is to first construct a 95% CI for the ln(CRP) variable and then back-transform those bounds using the e^(that bound) rule.

I have 5 measurements from spina bifida patients, four from patients with a spinal cord lesion, and 5 from control biopsies. What is the best way to describe the data? I am considering, for example, using the median and range per group, but I'm not sure whether this makes sense.

If you want to describe the data separately per group, you are dealing with small groups (5-4-5). With such small numbers, a standard deviation is usually not very reliable. It therefore does seem appropriate to describe the spread in the data using another measure, such as a range. If the data is reasonably symmetric, you can describe the center of the data with a mean, or, as you suggest, with a median. The median of 4 numbers is somewhat trickier to determine (often the average of the middle two numbers is chosen).

With this approach you use 3 measures (lowest value, mean or median, highest value) to describe 4 or 5 data points. You could also consider describing the data points themselves; this is only 1 or 2 extra numbers, giving the reader all the information. This is usually clearest in a (jitter) plot, i.e. a plot in which the individual points remain visible by plotting them slightly offset from each other.

I used 5-point Likert questions in a questionnaire, and the best way to process these is median + IQR (certainly not the mean). However, when I write this down it doesn't look good (e.g., median is 3, IQR 1, it remains quite abstract), and I am considering simply writing out the percentages and adding a table. Can you give me any further advice on this?

If you have room for it, writing out the percentages per category is certainly a good way to describe the data fully. If you go with median + IQR, instead of just giving the width of the IQR, you can also report the two bounds of the IQR, e.g. median 3, IQR (2-4). Sometimes "range" is used to mean only the distance between Q3 and Q1, and sometimes it refers to a description of Q1 and Q3 themselves; I see the latter version used somewhat more often.

I have a measurement instrument on which outcomes between 0 and 100 are possible. 0 is very poor and 100 is perfect. I have measurements from 58 patients, with a mean of 13 and a standard deviation of 28. The p-value of the Kolmogorov-Smirnov test is greater than 0.05. But that can't really be right, since the instrument cannot measure negative values. Can I still report the mean and standard deviation in an article?

If I understand correctly, you used the Kolmogorov-Smirnov test to test whether it is plausible that your data follow a normal distribution. You assume your data do follow a normal distribution because the p-value of the test is greater than 0.05. Given that the measurements can only fall between 0 and 100, and the values of the mean and standard deviation you give, I suspect your distribution is very skewed. You can check this by making a histogram. The Kolmogorov-Smirnov test has little power to detect deviations from a normal distribution with this number of observations (patients). This means that the distribution can be very different from a normal distribution without the p-value being smaller than 0.05. In this situation, the Anderson-Darling test has more power to detect deviations from the normal distribution (Table 2) [1]. The Anderson-Darling test is, however, not implemented in SPSS.

In practical terms, it is best to report the median and range or interquartile range in your article. These are also appropriate when your observations do not follow a normal distribution.

1. Razali NM, Yap BW. Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. Journal of Statistical Modeling and Analytics Vol 2.1 (2011): 21-33. link

• The difference between the sample mean and the population mean on TheStatsGeek.com A very clear English-language explanation of the difference between a sample mean and a population mean.

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