# Percentiles and Frequency Distributions

The concept of the median of a one-sample (univariate) data set was introduced in Statistics Defined. Here we seek to provide a deeper understanding for this measure of data location by showing that the median is a special case of the percentiles of a distribution. How does all this relate to probability ? Our aim[…]

# Histograms with Unequal Class Intervals

Recall that a histogram is a powerful method for visualizing a quantitative data set in terms of its frequency distribution. It is a specialized form of bar chart in which there are no gaps between the bars: the base of each bar represents a specific class interval of the data in question. Also, the bars are sometimes referred[…]

# Transformed Trigonometric Functions

Two or more trig. function transformations are dealt with using the general equations:   y2 = asin[b(x  c)] + d  ,   acos[b(x  c)] + d ;   atan[b(x  c)] + d   EXAMPLE 1:      ;   a = 2  &  d = 3,   so  M = ;                                                                                                  Also,  m =   .   EXAMPLE[…]

# Composition of Functions

Given two functions, f(x) and g(x), we can combine them by the operations of addition, subtraction, multiplication, or division: the result will usually be a new function, h(x). If f(x) and g(x) are polynomials, for instance, their product is another polynomial of higher degree. Again if f(x) and g(x) are polynomials, their quotient is a[…]

# Inverse of a Function

Every function, y = f(x), is a binary relation. That is, it consists of a rule which determines a set of ordered pairs, (x, y) ; this set is a subset of the points on the Cartesian plane. The inverse of a function is the set of ordered pairs obtained by interchanging the members of[…]

# Trigonometric Equations

A trigonometric equation is an equation involving one or more trigonometric functions. Therefore the solution to a trigonometric equation is one or more angles. Most trigonometric equations at the secondary level can be analyzed in terms of simpler components.  So we offer the following comments on the most basic form of trig. equation.             The simple (linear[…]

# Piecewise Defined Functions

A function is piecewise-defined if its domain consists of a finite number of pieces (subintervals) each with its own, possibly unique defining equation. The simplest example of a piecewise-defined function is the absolute value function, depicted below.     This function is described by the equation:  y = |x| , where |x| is defined[…]

# General Transformations

Here we consider how the individual transformations studied so far can be combined. One of our important guiding tools will be the equation: y1 = a*f[b(x – h)] + k Given a basic function y = f(x), the above equation allows us to sketch y1 using our knowledge of the transformation effects of the[…]