However, it should be borne in mind that this particular data set has only eleven values, and this is too small a number to infer anything definite about any underlying structure. The corresponding lack of symmetry shows up in the boxplot: the right-hand section of the box is longer than the left. These particular data are not symmetric they are right-skew, and in fact the sample skewness is 2.572. Some kind of assessment of symmetry is possible, since symmetric data will produce a boxplot which is symmetric about the median. The unusually large value in this data set is clearly shown and the median gives an assessment of the centre. Thus these aspects of the diagram give an idea of the dispersion of the data set. The length of the box represents the interquartile range and the lengths of the whiskers relative to the length of the box give an idea of how stretched out the rest of the values are. You can see how a boxplot gives a quick visual assessment of the data. The approach adopted here is one of the simplest and is probably the most common. Some textbooks and software always draw the whiskers right out to the minimum and maximum values and do not mark (potential) outliers separately. Some approaches even distinguish between moderate and severe outliers by using different symbols for them. The whiskers may extend as low as one or even up to two interquartile ranges either side of the box. All boxplots show the three quartiles, but the conventions defining the extent of the whiskers vary from text to text and from one computer package to another. It must be stressed that boxplot construction is an area where there are no universally accepted rules. So the interquartile range is needed to construct the whiskers.įigure 1.4 Figure 1.4 Completed boxplot for collapsed runners The lower adjacent value is the furthest observation which is within one and a half iqr (interquartile range) of the lower end of the box and the upper adjacent value is the furthest observation which is within one and a half iqr of the upper end of the box. The whiskers are drawn outwards as far as observations called adjacent values. However, as you will see in the next step, some observations may be classified as potential outliers and in fact the whiskers extend only to cover observations which are not classified as potential outliers. Essentially, each whisker extends outwards from the edge of the box as far as the most extreme observation. These are lines drawn parallel to the scale (so they are horizontal in this course). The vertical line inside the box is located at the median. The ‘box’ is a rectangle with edges defined by the lower and upper quartiles so it indicates where the ‘middle 50%’ of the data can be found. The median of this data set is 110, and the lower and upper quartiles are 79 and 162, respectively. The median and quartiles are used to construct the ‘box’. Since the minimum is 66 and the maximum is 414, a scale from 0 to 500 (say) is suitable in this case. The steps involved in constructing the boxplot in Figure 1.1 for the data set of β endorphin concentrations are as follows.įirst, a convenient scale is drawn covering the extent of the data. The easiest way to understand exactly what a boxplot represents and how it is constructed is to think about how you would draw one by hand. (1987) Beta-endorphin: a factor in 'fun run' collapse? British Medical Journal, 294, 1004.) (Data sourced from Dale, G., Fleetwood, J.A., Weddell, A., Ellis, R.D. Figure 1.1 A boxplot for the collapsed runners Figure 1.1 A boxplot for the collapsed runners
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Julius Caesar replaced each letter of his messages with the letter found three positions later in the alphabetical order so every A became D, every B became E and so on, the last three letters of the alphabet were replaced by the first three.įor example using the Caesar cipher the phrase: si qua occultius perferenda erant, per notas scripsit, id est sic structo litterarum ordine, ut nullum verbum effici posset: quae si qui investigare et persequi velit, quartam elementorum litteram, id est D pro A et perinde reliquas commutet. IIn this case the message was encrypted by transposition and this is the technique that Julius Caesar also adopted and Suetonius in De Vita Caesarum describes it as follows: Julius Caesar surely knew the scital and perhaps also the secret writings of the Old Testament, of which the Atbash can be considered the first true cipher code which consisted in overturning the alphabet so that the first letter became the last and the last the first and, subsequently, all the others. The technique required the use of two twin sticks and a long strip of parchment which, wound on the sticks, had the message engraved lengthwise the correct decryption of the message depended on the diameter of the twin sticks, one of which was needed by the sender to write the message and the other one by the recipient to decrypt it. The same technique includes the use of "nice inks", starting with the well-known lemon juice.Ī different technique, the transposition of the letters of the message, was used by the Spartans who invented the scital or scythe, an instrument with Ephors sent secret messages to the generals engaged in war. Herodotus tells of how the tyrant of Miletus, during a war against Persians, to send a message to a possible ally he had the skull of a messenger shaved, write the message and after the hair had grown back he sent it to the recipient. a technique called steganography was used, ie the text was physically hidden. The need to send messages that had to remain secret to the adversaries has very ancient origins and in Greece already in the VIII-VII century. The cipher we know today is the one that Suetonius describes in his "Life of the Caesars". 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