Dispersion

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance or the standard deviation.
== Measures ==
A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse.
Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in Pro Kit cards, so is the measure of dispersion.
Examples of dispersion measures include
* standard deviation,
* range (displayed on the Dispersion per box tab on pages of random containers like Gift Boxes or Pro Kit Boxes)
* and several more.
Another measure of dispersion is the variance (the square of the standard deviation). The unit of the variance for Pro Kit Boxes would be cards2 which is less intuitive to understand.
== Examples and visualizations ==
=== Bar chart by range ===
Dispersion measured by range can be displayed in a simple bar chart. All infoboxes for random containers on Asphalt Wiki use this chart type in the Dispersion per box tab to give a quick overview of the minimum and maximum together with the mean value. The longer the bar, the greater the dispersion.
The example to the right shows the dispersion of different card types in Asphalt 8 Daily Kit Boxes. While the number of common, rare and legendary cards can vary, the number of Blueprints, Engines, Tech and Parts is not dispersed at all () which results in a bar of zero length. Mean values are marked by a red vertical line.
=== Binomial distribution ===
The disadvantage of the above-mentioned horizontal bar charts is that they do not show if every outcome within the range occurs with the same frequency (and is thus equally probable). If the statistical data is displayed in an area or line chart with the possible numbers of cards on the x-axis and the number of boxes that granted those numbers on the y-axis, one can also see how many boxes granted cards. Counting the occurence of cards of a certain type is a Bernoulli process (getting 1 card or not, getting 2 cards or not and so on), so the bigger the sample size gets, the more the results will approach the bell-shaped binomial distribution. This means that results near the mean value will occur more often while results with great deviations will be rarer. The more dispersed a card type is, the more the chart will be horizontally stretched (more deviations from the mean), and vice versa. This is the case for all random containers in Asphalt games where it is observed if an item is granted or not (or, more generally, if an event occurs or not).
=== Uniform distribution ===
Some random containers always grant a certain amount of an item. For example, all Gift Boxes in Asphalt Streetstorm grant a certain number of , so here it is of interest how many Subscribers are granted, not if they are granted or not.
Applying the same chart type as for the Daily Kit Boxes above results in the graph to the right that does not seem to follow any bell-shaped line. Also, it is interrupted twice, once in an interval around 50 and once around 100. Within the resulting three intervals, values seem to be completely random and not concentrated around a maximum at some mean value. This is an indicator that the underlying random process follows a discrete uniform distribution which means that within one of the three intervals, all values are equally likely to occur. A simple example of a discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each score has the same probability of 1/6. The more often the die is rolled, the more the values for each number of pips will approach the same value (see law of large numbers). This will also be the case for the values shown in the graph as the sample size increases.
Another form of visualization for uniformly distributed outcomes is the scatter chart. The example to the right uses the same Subscriber data from Gift Boxes as above, but has the number of performed trials on the x-axis and the number of granted Subscribers on the y-axis, with each data point represented by a dot.
While the generally higher occurence of values within the lowest interval appears as higher peaks in the area chart, it appears as a higher dot density in the scatter chart.
The scatter chart also displays dispersion in a more intuitive way as the dots appear to the eye as more or less "evenly dispersed". 
== See also ==
* Arithmetic mean
* Law of large numbers
Category:Statistics