Descriptive statistics provide simple summaries about the sample and about the observations that have been made. Such summaries may be either
quantitative, i.e.
summary statistics, or visual, i.e. simple-to-understand graphs. These summaries may either form the basis of the initial description of the data as part of a more extensive statistical analysis, or they may be sufficient in and of themselves for a particular investigation. For example, the shooting
percentage in
basketball is a descriptive statistic that summarizes the performance of a player or a team. This number is the number of shots made divided by the number of shots taken. For example, a player who shoots 33% is making approximately one shot in every three. The percentage summarizes or describes multiple discrete events. Consider also the
grade point average. This single number describes the general performance of a student across the range of their course experiences. The use of descriptive and summary statistics has an extensive history and, indeed, the simple tabulation of populations and of economic data was the first way the topic of
statistics appeared. More recently, a collection of summarisation techniques has been formulated under the heading of
exploratory data analysis: an example of such a technique is the
box plot. In the business world, descriptive statistics provides a useful summary of many types of data. For example, investors and brokers may use a historical account of return behaviour by performing empirical and analytical analyses on their investments in order to make better investing decisions in the future.
Univariate analysis Univariate analysis involves describing the
distribution of a single variable, including its central tendency (including the
mean,
median, and
mode) and dispersion (including the
range and
quartiles of the data-set, and measures of spread such as the
variance and
standard deviation). The shape of the distribution may also be described via indices such as
skewness and
kurtosis. Characteristics of a variable's distribution may also be depicted in graphical or tabular format, including
histograms and
stem-and-leaf display.
Bivariate and multivariate analysis When a sample consists of more than one variable, descriptive statistics may be used to describe the relationship between pairs of variables. In this case, descriptive statistics include: •
Cross-tabulations and
contingency tables • Graphical representation via
scatterplots • Quantitative measures of
dependence • Descriptions of
conditional distributions The main reason for differentiating univariate and bivariate analysis is that bivariate analysis is not only a simple descriptive analysis, but also it describes the relationship between two different variables. Quantitative measures of dependence include correlation (such as
Pearson's r when both variables are continuous, or
Spearman's rho if one or both are not) and
covariance (which reflects the scale variables are measured on). The slope, in regression analysis, also reflects the relationship between variables. The unstandardised slope indicates the unit change in the criterion variable for a one unit change in the
predictor. The standardised slope indicates this change in standardised (
z-score) units. Highly skewed data are often transformed by taking logarithms. The use of logarithms makes graphs more symmetrical and look more similar to the
normal distribution, making them easier to interpret intuitively. ==References==