Quartiles calculation

The probabihty-density function for the normal distribution cui ve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile Xm is used in conjunction with the lower quartile or 25 percentile Xl or with the upper quartile or 75 percentile X[j. These statistics are frequently quoted, and determination of values of a, b, and c by using Xm with Xl and with Xu is an indication of the symmetry of the cui ve. When the agreement is reasonable, the mean v ues of o so determined should be used to calculate the corresponding value of a. 

Semiquartile Distance. When all the data in a group are ranked, a quartile of the data contains one ordered quarter of the values. Typically, we are most interested in the borders of the middle two quartiles Qx and Q3, which together represent the semiquartile distance and which contain the median as their center. Given that there are N values in an ordered group of data, the upper limit of the jih quartile ( >-) may be computed as being equal to the [(jN — l)/4th] value. Once we have used this formula to calculate the upper limits of Qx and Q3, we can then compute the semiquartile distance (which is also called the quartile deviation, and as such is abbreviated as QD) with the formula QD = (Q3 — Q )/2. 

In order to tear a piece of paper into four equally wide strips, three tears must be made. One to tear the original paper in half and the other two to tear those halves in half again. A quartile is the mathematical equivalent of this to a range of ordered data. You should realize that the middle quartile (Q2 ) is, in effect, the median for the range. Similarly, the first quartile (Qi) is effectively the median of the lower half of the dataset and the third quartile (Q3) the median of the upper half. In the same way as for the median calculation, a quartile should be represented as the mean of two data points if it lies between them. 

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. 

FIGURE 6 Summary of measurements of in situ rates of bacterial DOC consumption, calculated as the sum of bacterial production and respiration, and in vitro rates of DOC consumption, calculated from declines in DOC during batch incubations. Box-and-whisker plots show median, and upper/lower quartiles (box), and the range of values (bars). Extreme outliers are denoted by open circles. The in situ consumption rates are from the global dataset of simultaneous measurements of bacterial production and respiration collected by del Giorgio and Cole (1998), with the addition of unpublished data (see text). Because the in situ rates have been measured under a range of temperature, for this comparison the bioassay rates have not been corrected for temperature. 

Detailed instructions are provided for the calculation of the mean, median and SD (but not quartiles) using Microsoft Excel. Readers are referred to the accompanying web site for detailed instructions on generating all these descriptive statistics (including quartiles) using Minitab or SPSS. Generalized instructions that should be relevant to most statistical packages are provided in the book. 

The early chapters (1-5) are fairly basic. They cover data description (mean, median, mode, standard deviation and quartile values) and introduce the problem of describing uncertainty due to sampling error (SEM and 95 per cent confidence interval for the mean). In theory, much of this should be familiar from secondary education, but in the author s experience, the reality is that many new students cannot (for example) calculate the median for a small data set. These chapters are therefore relevant to level 1 students, for either teaching or revision purposes. 

Figure 6 Relationship between median, upper, and lower quartiles of Al, As, Be, Cd, Cu, Mn, Pb, and Zn, and pH in Czech Republic brooks in the late 1980s. Values were calculated after sorting the water samples n = 12,988) into groups with 0.2 pH unit ranges. Volume-weighted average concentrations in bulk precipitation (o) and throughfall ( ) in 1991 samples from the Bohemian Forest are shown (after Vesely and Majer, 1996, 1998). The declines of Be, Cd, Mn, and Zn at very low pH are likely caused by depletion of exchangeable trace metals from the watershed soils.

IQR=Q3-Q , where Q3 is the third quartile and Qj is the first quar-tile, when the median divides the experimental sample into two parts. Outliers were found in experiments No 1, 3 and 6 and may be rejected but we did not find enough good reason for that and they were used in calculations of the average corrosion rate value and its standard deviation. 

The first step of Croux and Ruiz-Gazen making PCA more robust is centering the data with a robust criterion, the LI-median, that is, the point which minimizes the sum of Euclidean distances to all points of the data. In a next step, directions in the data space, which are not influenced by outliers, are determined by maximizing a robust parameter, the estimator. To calculate this estimator, first all objects are projected onto normalized vectors passing through each point and the LI-median center. Then for each projection, the Qn, that is, the first quartile of all pairwise differences, is calculated as follows ... 

Calculate the interquartile range (IQR) =(QUARTILE(range, 3) - QUARTILE(range, 1))... 

Fig. 3 Box-plot of MTBE concentrations found in the vicinity of an airport (A) at different water bodies and (B) detailed for coastal water samples ( = 8) through seven sampling campaigns. For each variable, the box has lines at the lower quartile (25%), median (50%), and upper quartile (75%) values. The whiskers are the lines extending from each end of the box to show the extent of the data up to 1.5 times the interquartile range (IQR). The mean value is marked with (a) and outliers with (x) symbols. Each sample ( ) was analyzed in triplicate, and the average value was considered for calculations. Non-detected levels were expressed as half of instrumental limit of detection (5 X 10 rgL )...

On the basis of the data obtained during studies conducted over the last few years in many areas, our research team has been able to make a fairly reliable data analysis for the purpose of determining reference levels for fresh honey and bees. The minimum and maximum thresholds have been dehned by calculating a quartile so as to derive two median values for a group of data the low quartile and the high quartile (Table 11.7). From the two sets of data - drawn from the literature and experimentally - it was possible to derive the approximate reference values shown in Table 11.8. 

All dietary and whole body tissue concentrations were expressed as dry weights. In some cases, typically fish, residues had to be calculated from wet weight concentrations using measured or assumed (75%) moisture contents. The moisture contents of whole body fish appeared to be relatively consistent, with reported values of 71,75, and 77% (n=2,051), corresponding to 25th, 50th, and 75th quartiles, respectively (Seiler and Skorupa 2001). 

Positive dimensional scores are presented for 83 acute hospitals participating in the second Belgian comparative research exercise. Positive dimensional scores (percentages of positive response) were calculated at hospital level by dividing the number of positive answers by the total number of answers for each dimension. Positive dimensional scores are displayed using box plots, which provide an indication of the dispersal between hospitals, possible skewing of data and outliers (hospital level). The box plot includes the smallest observation (sample minimum), lower quartile (Ql), median (Q2), upper quartile (Q3) and largest observation (sample maximmn). 

Usually, the survey group is divided into a tiered performance structure such as first, second, third, and fourth quartiles. Based on the energy performance index calculated above, you can find out which performance quartile your process unit belongs to. This indicates where your process unit stands among your peers. 

In non-parametric statistics the usual measure of dispersion (replacing the standard deviation) is the interquartile range. As we have seen, the median divides the sample of measurements into two equal halves if each of these halves is further divided into two the points of division are called the upper and lower quartiles. Several different conventions are used in making this calculation, and the interested reader should again consult the bibliography. The interquartile range is not widely used in analytical work, but various statistical tests can be performed on it. 

Thus, epidemiological cohort studies can provide associations between vitamin intake either from food or from fortified food or supplements and a specific disease, and the RCTs can provide a proof whether this association is causal or not. The major differences are that the cohort studies usually include healthy people at baseline, while the RCTs usually include patients who suffer from the disease (secondary prevention). Observational studies usually have longer follow-up periods and they assess food intake, from which vitamin intake is calculated from. It may also be mentioned that an observational study may either find an increased disease risk at low intake or low plasma levels of a nutrient (usually in the lowest quartile or quintile of the cohort), which is opposite to the finding of a reduced disease risk at high intake or high plasma levels. RCTs, however, aim to find a reduced disease risk at high intake levels that is achieved through the nutrient supplement used. This difference is discussed in more detail below. 

The histogram of function values at 20000 uniformly distributed random points in the same subregion is shown in Figure Id. Twice as many function evaluations have been used because the interval finiction evaluation uses approximately two times more calculations than real function evaluations. The ranges of the horizontal axis are the standard interval. Most function values are in one quartile of the standard interval. 

This uses only the y-values of the selected samples. The median, quartiles, and interquartile range of these values are calculated, and samples with y-values more than delta times the interquartile range beyond the quartiles are rejected. This is a fairly standard statistical approach to the identification of outliers, with delta =1.5 being the usual default. 

For all the 11 questions, the following statistical parameters have been calculated mean, median, standard deviation and inter-quartile range. 

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