Standardized interquartile range

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... 

Interquartile range The middle 50% of a set of data arranged in ascending order. The normalized interquartile range serves as a robust estimator of the standard deviation. (Section 2.6.2)... 

Distribution plot of 421 samples. Data were analyzed in a statistical program and showed the following mean = 13.18 mode = 1 1.68 median = 12.00 lower quartile =10 upper quartile = 16 interquartile range = 6 standard error of mean (SEM) - 0.28 variance = 33.20 SD = 5.76 CV = 0.44. 

Descriptive statistics—mean, median, trimmed means, standard deviation and standard error, variance, minimum, maximum, range, interquartile range, skewness, kurtosis Frequency statistics—outlier identification boxplots, stem-and-leaf plots, and histograms Frequency statistics—description percentiles, probability plots, robust estimates or M-estimators, Kolmogorov-Smirnov and Shapiro-Wilk normality tests Variance homogeneity—Levene s test for equality of variance... 

Because the use of a sample mean, and particularly a sample standard deviation, relies on the assumption of normality of distribution of the data, outliers in the data must be suitably dealt with. Alternatively, robust z-scores may be calculated from the median and the normalized interquartile range (norm IQR). 

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. 

A simple robust estimate of the standard deviation is provided by the interquartile range (IQR, see Section 6.2). For a normal error distribution, the IQR is ca. 1.35standard deviation estimate that is not affected by any value taken by the largest or smallest measurements. Unfortunately, the IQR is not a very meaningful concept for very small data sets. Moreover, and somewhat surprisingly, there are several different conventions for its calculation. For large samples the convention chosen makes little difference, but for small samples the differences in the calculated IQR values are large, so the IQR has little application in analytical chemistry. 

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. 

Dispersion n The variation in the value of a variable within a set of observations. A small dispersion indicates closely spaced values whereas a large dispersion indicates widely spaced values. There are several measures of dispersion which define quantitative values related to the spread of the values in a set of observations. These include range, interquartile range, standard deviation, and variance. A measure of dispersion along with a measure of central tendency give a simple quantitative description of shape of the distribution of values within the set of observations. 

Calculate the following descriptive statistics for the data on water hardness (mmoll ) given as follows arithmetic mean, median, standard deviation, variance, standard error, confidence interval at a significance level of 0.01, range, and the interquartile distance - 8.02 7.84 7.98 7.95 8.01 8.07 7.89. 

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