Interquartile range

The following is an example of a box plot of seizure data by treatment at each of three visits. This first example is created with the traditional PROC GPLOT. This box plot has boxes that span the interquartile range and whiskers that extend to the maximum and minimum values. Also, a connecting line joins the median values for each treatment. 

BOXJTOO MEANS TO CREATE BOX PLOTS WITH BOXES (25TH AND 75TH PERCENTILES - THE INTERQUARTILE RANGE) JOINED (J) AT THE MEDIANS WITH WHISKERS EXTENDING TO THE MINIMUM AND MAXIMUM VALUES (00) AND TOPPED/BOTTOMED (T) WITH A DASH. 

A-allele. Upper extreme of the linear scale has been omitted, (b) Discrimination between genotypes with Invader. In the box plot the black bars represent medians, whiskers interquartile range and circles outliers. 

Quartiles divide the data distribution into four parts corresponding to the 25%, 50%, and 75% percentiles, also called the first (Qi), second (Qt), and third quartile (g3). The second quartile (50% percentile) is equivalent to the median. The interquartile range IQR = Q3 - Qi is the difference between third and first quartile. 

Calculating quartiles and using the interquartile range is useful in order to negate the effect of extreme values in a dataset, which tend to create a less stable statistic. 

Robust Statistics use trimmed data for the calculation of the estimated values. That means, that a part of the data set in the tails is excluded or modified prior to or during the calculation. An easy example is the use of the interquartile range (the range between the first and the third quartile) instead of the whole data set. 

A second, simple measure of variability is the inter-quartile range that is the interval between the upper and lower quartiles. The upper quartile of a set of data is that value that is less than 25% of the data and greater than 75% similarly, the lower quartile is the value that is greater than 25% of the data and less than 75%. For the blood glucose data the lower quartile is 3.6 mmol / Land the upper quartile 4.55 mmol/L, giving an interquartile range of 0.95 mmol/L. 

Fig. 14 ZK-253 effects on tamoxifen-resistant breast cancer xenograft tumours. Estrogen-dependent MCF-7/TAM tumours were implanted on day 0 into one flank of 70 estrogen-and tamoxifen-supplemented nude mice. After tumours had reached approximately 25 mm in size (after about 22 days), mice were randomised into seven groups (10 mice each) three control groups (control tamoxifen, control vehicle or control ovariectomy without estradiol), and the fom treatment groups (ZK-703, ZK-253, raloxifene or fulves-trant) each at 10 mg/kg subcutaneously daily. Treatment was continued either until the end of the experiment or imtil tumoms reached a median of approximately 100 mm (larger tumours were observed in some mice). The tumours were then removed, snap frozen, and used for analysis of ER levels, a Xenograft tumour growth curves. Data are expressed as medians with interquartile ranges, b ERa levels. Data are expressed as mean with upper 95% Cl...

Flavones and flavanones were less frequently consumed during the 4-day collection period. Flavones were not consumed at all by 38 participants, while 29 people did not consume any citrus flavonoids — flavanones. The interquartile range of intake of flavones was relatively limited, ranging from 0.0 to 2.0 mg/day. Flavone consumption was not normally distributed and was negatively skewed toward a lack of consumption of foods rich in flavones such as olives and lettuce. Likewise, flavanone intake was also not normally distributed with a mean flavanone intake of 1 mg/day compared to the median intake of 1.2 mg/day. This is accounted for by the fact that the range of flavanone intakes was very wide (0 to 239 mg/day), 36%i of participants not consuming any flavanone-rich foods. The main dietary... 

Less commonly used is the mode, the most frequently occurring value in the dataset. The mode is the appropriate measure of central tendency for nominal data. Other measures of location (but not of central tendency) are percentiles or quartiles. The percentile of xis the percentage of the total cases that falls at or below xin value. Commonly used percentiles are the 25th and 75th percentiles (or 1st and 3rd quartiles). The median is the 50th percentile. The distance between the first and third quartile is the interquartile range (Figure 21.1). 

Figure 21.1 Cumulative frequency plot illustration 25th, 75th percentiles, and the interquartile range. estimate of the population variance. Population means and variances are by convention denoted by the Greek letters p and o2, respectively, while the corresponding sample parameters are denoted by X and s2.

The robust estimate of the standard deviation can be the MAD defined above, or if there are sufficient data, 1.35 x interquartile range. The interquartile range (IQR) is the range spanning the middle 50% of the data. However, chemistry rarely has the luxury of sufficient data for a meaningful calculation of the IQR. 

Method No. Bond Distances Mean Median Minimum Maxiumum Range Lower Quartile Upper Quartile Interquartile Range Standard Deviation... 

Figure 2.7 Steroid half-lives. The median (second quartile) indicates generally longer elimination half-lives for steroid 1 relative to steroid 2. The interquartile range indicates greater variability for the first steroid...

The fact that the interquartile-range does reflect dispersion can be appreciated if you compare steroid 1 with number 2. With steroid 2 the half-lives are visibly less disperse and quartiles 1 and 3 are closer together. In this case ... 

Just as the median is a robust indicator of central tendency, the interquartile range is a robust indicator of dispersion. Take the longest half-life seen with steroid 1 (15.8 h) and consider what would have happened if that individual had produced a half-life of 100 h (or any other extreme value). The answer is that it would make absolutely no difference to the inter-quartile range. The value of 15.8 h is already something of an outlier, but it had no undue effect on the inter-quartile range. 

The three quartile values indicate the figures that appear 25, 50 and 75 per cent of the way up the list of data when it has been ranked. The second quartile is synonymous with the median and can act as an indicator of central tendency. The interquartile range (difference between first and third quartile) is an indicator of dispersion. The median and interquartile range are robust statistics, which means that they are more resistant to the effects of occasional extreme values than the mean and SD. The robustness of the median can be abused to hide the existence of aberrant data. 

Time until occlusion and patency are expressed as median and the interquartile range/2 (IQR/2). Significant differences (p < 0.05) are calculated by the non-parametric Kruskal-Wallis test. 

The statistical analysis of the data was performed for the determination of outliers by means of their interquartile range ... 

Identify (or eliminate) outliers by a criterion based on the central 50% of the distribution, thus reducing the masking effect of several outliers. Compute the interquartile range (JQR) between the lower and upper quartiles of the distribution (Qi and Q3, respectively) IQR Q3 - Qi. Then identify as outhers data lying outside the two fences Qi - 1.5 IQR and Q3 +... 

With ordinal and quantitative data, the upper and lower quartiles show the range of values within which the middle 50% of individual values fall. This is known as the interquartile range and is the recommended way of describing variability if the median is being used to indicate central tendency. 

Note that the box and whisker plot presents the median, the maximum and minimum values (and hence the range) and the first and third quartiles (and hence the interquartile range). 

For example, the 75th percentile is the value of X below which 75% of the values lie and above which 25% lie. The 50th percentile is synonymous with the median. Likewise the 25th percentile is the value of X below which 25% of the values lie and above which 75% lie. The difference between the 75th and 25th percentiles is called the interquartile range, which can be a useful measure of dispersion when the distribution of the random variable is heavily skewed or asymmetric. 

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