The possible methods of distribution regarding numerical data PDF

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Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. ... Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense....

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Numerical Distributions Dot Plots A collection of dots where each dot represents a discrete data value plotted above a number line. They are useful for displaying small data sets. Example VCAA 2009 Exam 2 Question 1 The dot plot below displays the distribution of the number of rainy days for the 12 months of 2008. For the year 2008, the median number of rainy days per month is    in the ⎯⎯⎯⎯= 6.5th position which is ⎯⎯⎯⎯⎯= 15.5 days. 



Stem Plots (Stem-and-Leaf Plots) A special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (the last digit). They are useful for displaying small to medium number of discrete or continuous data. The leaves are written in ascending order outwards from the stem. Each stem can be split if the data is heavily clustered in one stem. A key must be provided to show how the data in the stem plot is to be read. For example 1|3 = 13 means that the number 3 in the row 1 is read as 13. 1|3 = 1.3 means the same number 3 in the row 1 is read as 1.3. Example VCAA 2011 Exam 2 Question 1 The stemplot shows the distribution of the average age, in years, at which women first marry in 17 countries. For these countries, the median average age of women at   9th position, which is 28.2 first marriage is in the ⎯⎯⎯⎯=  years. Histograms A visual representation of the frequency of discrete or continuous data. Similar to a bar chart, the height represents the frequency or percentage for that value. However, the bars are placed directly next to each other to show the continuity of the numerical values which allows us to analyse the shape. Continuous data is often grouped as the exact values can vary so much. The ends of the group are written at the edge of each bar, whereas the discrete values are written in the middle. Example VCAA 2009 Exam 1 Question 5 The percentage histogram below shows the distribution of the fertility rates (in average births per woman) for 173 countries in 1975 In 1975, for these 173 countries, fertility rates were most frequently between 6.5 and 7.5.

Example VCAA 2001 Exam 1 Question 1 & 3 The systolic blood pressures of 79 middle -aged people were measured. The results are summarised graphically in the histogram below. For this group of people, the distribution of their systolic blood pressure is best described as positively skewed. From the histogram, the median systolic blood pressure is  in the ⎯⎯⎯⎯= 40th position, which is somewhere between 

150 and 160.

Description of Numerical Distributions Numerical distributions are commented on about their shape, centre, spread, and outliers. • The shape of the distribution is [symmetric/positively skewed/negatively skewed/bi-modal].  The distribution has a [mean/median] of [value].  The distribution has a [range/𝐼𝑄𝑅/standard deviation] of [𝑚𝑎𝑥 − 𝑚𝑖𝑛/𝑄 − 𝑄 /𝑠].  The distribution [has/has no] outliers. Shape of a Numerical Distribution The shape of dot plots, stem -plots, histograms, and boxplots can be interpreted in similar ways. Symmetric Data is clustered around the middle. Narrows out on both ends evenly. Mean and median are roughly the same value.

Bi-Modal Data has two peaks of roughly the same height.

Negatively Skewed Data is clustered around the positive/right end. Narrows out to the negative/left end. Generally, the median is further right than the mean.

Positively Skewed Data is clustered around the negative/left end. Narrows out to the positive/right end. Generally, the median is further left than the mean.

Example VCAA 2008 Exam 1 Question 2 The box plot below shows the distribution of the time, in seconds, that 79 customers spent moving along a particular aisle in a large supermarket. The shape of the distribution is best described as positively skewed with outliers. Example VCAA 2006 Exam 1 Question 3 The back-to-back ordered stemplot below shows the distribution of maximum temperatures (in °Celsius) of two towns, Beachside and Flattown, over 21 days in January. The distribution of maximum temperatures for Flattown is best described as approximately symmetric with outliers....