Statistics quartiles

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. 

Box plots, also known as box and whisker plots, are commonly used to display univariate statistics for a given variable across another variable. The statistics typically displayed in a box plot are the minimum, first quartile, median, third quartile, and maximum values. Mean values are often included in box plots as well. The following is a sample box plot of a clinical response measure showing how three different drug therapies compare to one another. 

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. 

The one thing at which Excel does not excel is statistical analysis. The (very basic) level of material covered in this chapter is about at (if not beyond) the limit of its capabilities. It can be used to generate means, medians, SDs and quartiles, but while the first three are OK, the quartile values generated are somewhat unconventional and will not be pursued further. The mean, median and SD of a data set can be generated by using either worksheet functions or the Data Analysis tool. 

The number of observations (n) and the statistics referred to in this chapter should be fairly obvious. The second quartile (Q2) is generally not shown as it is the same as the median. The SEM may not be familiar at the moment, but is an important statistic that will be described in Chapter 4. 

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. 

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. 

Other statistical measurements used in geology for particle size distribution characterization (moment, quartile and others) have been defined [135,136]. 

The heights of the bars or columns usually represent the mean values for the various groups, and the T-shaped extension denotes the standard deviation (SD), or more commonly, the standard error of the mean (discussed in more detail in Section 7.3.2.3). Especially if the standard error of the mean is presented, this type of graph tells us very litde about the data - the only descriptive statistic is the mean. In contrast, consider the box and whisker plot (Figure 7.2) which was first presented in Tukey s book Exploratory Data Analysis. The ends of the whiskers are the maximum and minimum values. The horizontal line within the central box is the median, fhe value above and below which 50% of the individual values lie. The upper limit of the box is the upper or third quartile, the value above which 25% and below which 75% of fhe individual values lie. Finally, the lower limit of the box is the lower or first quartile, the values above which 75% and below which 25% of individual values lie. For descriptive purposes this graphical presentation is very informative in giving information about the distribution of the data. 

The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical normal distribution the mean (d) will be approximately equal to the median (w) the lower and upper quartiles will be... 

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. 

Graphically, the most important details of descriptive statistics of data can be represented in a box-and-whisker plot or, for short, box plot (Figure 2.5). Along the variable axis, here the ordinate, a box is drawn, with the lower and upper quartile being the bottom and top of the box, respectively. The width of the box has no meaning. 

Number of Potential Outliers (0 to n) To count outliers in a distribution, we used the 1.5 x IQR [interquartile range the difference between the first quartile (Qi) and the third quartile (03)] criterion that is the basis of a rule of thumb in statistics for identifying suspected outliers (Moore and McCabe, 1999). An item of value d is considered as a suspected (mild) outlier if... 

For more general applications, you may want to know a bit more about statistical metrics. As a brief summary, the basic statistics include mean, standard deviation, quartiles, standard error, and confidence intervals. Let us explain the physical... 

Step 2 The panel s responses are analyzed by the facilitator, who then prepares a statistical summary of the forecast values, for example, the median value and the two quartiles (75th and 25th percentile values). 

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. 

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