Summary measures for univariate data

To summarize univariate data such as the heights in the previous table, the following quantities are of paramount interest:

• Sample size is the number of objects making up the sample. It is also the number of rows n in the data matrix. It strongly determines the significance of inferences deduced from the sample about the original population from which the sample was taken. For example, sample statistics based on a sample with only 11 objects are not likely to be very representative of statistics for the whole population of meteorologists at the University of Reading. Many results in statistics are only really valid in the asymptotic limit when the sample size $n\rightarrow\infty$.

• Central Location is the typical average value about which the sampled variable is located. In other words, a typical size for the variable based on the sample. It can be measured in many different ways, but one of the most obvious and simplest is the arithmetic sample mean:

  
  

  $$\overline{x}=\frac{1}{n}\sum_{i=1}^{n} x_i$$ (2.1)

  
  

  For the example of height in the previous table, the sample mean is equal to 174.3cm which gives an idea of the typical height of meteorologists in Reading.

• Scale is a measure of the spread of the sampled values about the central location. The simplest measure of the spread is the range, R=max(x)-min(x), equal to the difference between the largest value in the sample and the smallest value in the sample. This quantity, however, is based on only the two most extreme objects in the sample and ignores information from the other n-2 objects in the sample. A more democratic measure of the spread is given by the standard deviation

  
  

  $$s=\sqrt{\frac{1}{n}\sum_{i=1}^{n} (x_i-\overline{x})^2}$$ (2.2)

  
  

  which is the square root of the sample variance. ^2.2

• Shape of the distribution can be summarized by calculating higher moments about the mean such as

  
  

  $$b_1=\frac{1}{n}\sum_{i=1}^{n} \left(\frac{x_i-\overline{x}}{s}\right)^3$$ (2.3)

  $$b_2=\frac{1}{n}\sum_{i=1}^{n} \left(\frac{x_i-\overline{x}}{s}\right)^3$$ (2.4)

  
  

  b₁ is called the moment measure of skewness and measures the asymmetry of the distribution. b₂-3 is the moment measure of kurtosis and measures the flatness of the distribution.