# Variability, range, interquartile range, and standard deviation.

Almost all investigations in clinical research involve groups, eg, groups of study participants or patients. Often groups are compared with regard to 1 feature (parameter) such as levels of blood sugar or cholesterol. Using the mean value of our feature of interest, we represent the central tendency of the groups (see Your Stats Refresher, AMWA Journal 2015;30(1): 31-33). The degree to which the means are a good representation of the groups will, however, depend on the variability of the data. To adequately describe the distribution of data in a group, we need both the central tendency and a measure of variability (Figure 1). This measure of variability quantifies the extent to which the individual data differ from each other in regard to a feature of interest.

Appreciating variability is particularly important when we investigate the effects of a treatment (drug, device, method). The gold standard is to base the assessment of efficacy on a comparison between 2 groups, one that received the treatment (active group) and one that did not (placebo or control group). Only if the 2 groups are very similar in the relevant aspects, can we conclude that the difference we observe is due to the new treatment.

There are many terms for variability, but all of them mean the same: dispersion, variability, spread, heterogeneity, all refer to the observation that individual data differ from each other. The different measures of variability introduced below describe the extent to which the individual observations are scattered or spread, ie, to what extent they differ.

Range

The simplest way to capture the variability of data is to provide the range, ie, the highest and lowest value in our data together with the mean or median. With this we get a first impression of the spread of the data. Suppose we had measured the heart rate (in beats per minute, bpm) in 7 patients. We determined the following values of 55, 60, 45, 62, 63, 110, and 56 bpm. The range of heart rates in our small sample would then be 45 to 110 bpm, and the mean is 64 bpm. Most values are around 60 bpm and close to the mean (64 bpm); the patient who had a heart rate of 110 bpm was clearly an exception. We can see in this little example that providing the range has the downside of giving us only the most extreme values in our data. If one of them is exceptionally high or low, our perception of the variability in our data will be distorted.

Sometimes the range is given as the numerical difference between the highest and the lowest value, ie, it is calculated by subtracting the lowest value from the highest value in the data set. In our little example the range in heart beats could be expressed as 65 (= 110--45). In doing so, the variability in several groups can be compared with each other in a simple way.

Interquartile Range

Although "interquartile" sounds technical and complicated, the concept behind this clearly defined range is straightforward: the interquartile range (IQR) is the range of the central 50% of the data. Together with the median, the interquartile range is particularly suited to provide information on the variability of data that are not normally distributed, ie, that do not produce a symmetric bell shape when plotted.

To arrive at the interquartile range, the data first must be sorted from low to high values (or vice versa). Then the data are divided in quarters, resulting in 4 quartiles. Starting from the lowest number, the first quartile will end at the 25% mark, the second quartile will be the range of 25% to 50% of the data and its upper limit is the median (the median is the data point at which 50% of the data is higher and 50% is lower), the third quartile will extend from the median (50%) to 75% of the data and the fourth quartile will cover 75%-100% of data.

Consider a sample of 16 patients whose heart rates were measured in beats per minute (bpm). The data were ordered from lowest to highest value and then the quartiles were determined. We calculate the median (66.5 bpm) and the interquartile range (60 to 74 bpm). We know that the lower and upper limits of the interquartile range describe the central 50% of the data (Figure 2), ie, in our study 50% of patients have heart rates between 60 bpm and 74 bpm.

The interquartile range summarizes variability in a better way than the range because it disregards extreme values (ie 45 and 110). However, this measure still relies on only 2 values from the dataset; it does not use of all of the data in the data set.

Standard Deviation *

The idea of the standard deviation is to calculate the average distance of each data point from the mean of this data set. The standard deviation can be thought of as the average distance of the individual data from the mean. Unlike the interquartile range, this quantity has the big advantage of considering all data points and we obtain a comprehensive measure of the variability. When the individual data in a data set are clustered around the mean, the standard deviation will be small. When the individual data are spread apart, the standard deviation will be large (Figure 3).

The calculation of the standard deviation is only meaningful if the data are normally distributed, ie, if they produce a symmetrical bell-shaped curve when plotted.

The following description of the standard deviation involves 2 mathematical formulas. I present them in a simplified way; however they are indispensable for a full understanding. The conceptual mechanics of the calculation of standard deviations are fairly simple.

Step 1. As a first step we calculate the mean.

Step 2. We then calculate the difference of each data point from the mean by a simple subtraction x.-mean (individual data point minus the mean). For data points that are smaller than the mean, the subtraction will result in a negative value (for example, using values from our data above, 45--64 = -19).

Step 3. Before we can go on calculating the standard deviation, we must calculate a quantity known as the variance. To do so, we must square the values obtained in Step 2, add them up, then divide by the number of data points (n). ** (We have to square the values from Step 2 before adding them because, if our data are normally distributed, the positive and negative distances from the mean would cancel each other out, resulting in a sum of zero). The variance is usually expressed by the Greek letter of sigma a2. The mathematical description of the calculation of the variance is:

variance = [SIGMA] [(data points [x.sub.i] - mean).sup.2]/n

The variance is not a very helpful quantity as it does not have the right unit. In our example of the study in heart rate (measured in beats per minute), the variance would have the unit of beats per [minute.sup.squared]--a unit without any medical interpretation. To get away from this problem, we have to take the square root of the variance. Thus the formula for the standard deviation is:

standard deviation: = [square root of [(data points [x.sub.i] - mean).sup.2]/n

With this we have arrived at a measure that accurately describes the variability in a group. We can see from the formula that the size of the standard deviation depends on 2 factors, the average distance between the individual data points and the mean and the number of data points. The smaller the average distance, the smaller the standard deviation; also, the greater the number of data points, the smaller the standard deviation.

The standard deviation has another feature that is very useful. For sets of data that are normally distributed (bell shaped), the standard deviation always comprises a fixed proportion of the data. Within 1 standard deviation, that is from -1 SD to +1 SD from the mean there will be 68.3% of all data, within 2 standard deviations (-2 SD to +2 SD) there will be 95.4% of all data and within 3 standard deviations there will be 99.7% of all data (Figure 4).

In summary, range, interquartile range, and standard deviation are measures of variability within a set of data. Knowing the variability is important because it tells us how well the central tendency, that is, the mean or median, represents the data. While the range is the simplest way to describe variability, it has the disadvantage of providing only the extreme values. The interquartile range provides us with the variability of the central 50% of the data. It is not susceptible to extreme values but relies also on only 2 data points. The standard deviation is the most comprehensive measure to capture variability as it considers all data in our data set.

University of Leicester. Student learning development. www.le.ac.uk/succeedinyourstudies. Accessed July 20, 2015.

Altman DG. Practical Statistics for Medical Research. Chapman & Hall; 1999.

Altman DG, Gardner MJ. Presentation of variability [letter]. Lancet. 1986;2(8507):639.

Altman DG, Bland JM. Quartiles, quintiles, centiles, and other quantiles. BMJ. 1994;309:996.

* The term standard deviation is actually a misnomer as there is nothing "standard" about it

** Actually, statisticians always divide by n-1 instead of n. When we divide by n, we obtain the variance of the sample included in our study. Generally, however, we want to use our sample to estimate the variance in the total population. Dividing by n-1 gives us a better estimate of the variance in the total population, although for large samples the difference is clearly negligible.

By Thomas M. Schindler, PhD / Head Medical Writing Europe, Boehringer Ingelheim Pharma GmbH & Co KG, Biberach, Germany @@@MISSING 135-137 MISSING
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