SA NCS Mathematics:Glossary
GLOSSARY
affine transformation a transformation (usually in a plane) which takes parallel lines to parallel lines (lengths and angles may change); includes enlargement and reductions, shearing and stretching.
amplitude the maximum difference between the value of a periodic function and its mean.
annuity a fixed sum payable each year or each month either to provide a pre-determined sum at the end of a number of years or months (sometimes referred to as a future value annuity) or a fixed amount paid each year or each month to repay (amortise) a loan (sometimes called a present value annuity).
Formula 1:
- F = future value of the annuity
- x = fixed, regular payments
- i= interest rate per compounding period, as a decimal fraction
- n = number of compounding periods
Formula 2:
- P = present value of the annuity or the amount borrowed which is to be repaid by fixed repayments
- x = fixed repayments
- i= interest rate per compounding period as a decimal fraction
- n = number of compounding periods
association a general term to describe the relationship between two variables. Two variables in bivariate data are associated or dependent if the pattern of frequencies of their bivariate values cannot be explained by only the frequencies of the univariate values. In contrast, two variables are not associated or independent if the frequencies of bivariate values can be determined simply from the frequencies of the values of each variable.
asymptote a straight line to which a curve continuously draws nearer without ever touching it.
bar graph/diagram a diagram that uses horizontal or vertical bars to represent the frequency of classes (or groups or labels) in data consisting of observations of a categorical variable. The height or length of each bar is proportional to the frequency of the corresponding class; the thickness of a bar has no meaning. The bars are not required to touch each other and may be separated. A bar graph is not a histogram.
bivariate data two dimensions (of each object under observation) are recorded as a pair of variables (usually to investigate or describe an association or correlation or relationship between the variables). Numerical bivariate data are often presented visually as a scatter plot on a Cartesian plane where one variable (such as height) is read on the vertical axis and another variable (such as mass) is read on the horizontal axis.
box and whisker diagram a diagram which graphically presents the five-number summary of numerical data values. The diagram shows the range and spread in quartiles around the median (a box indicates the range of the middle two quartiles and whiskers the range of the first and last quartiles).
categorical data data that arises by observing the group or class to which an outcome or an object belongs; often recorded as labels which may be alphabetical or numerical (e.g. gender observed as male or female may be recorded as M or F, and also as 0 or 1).
categorical variable a variable whose values indicate a category to which an observation or an object belongs; may be nominal (e.g. eye colour), ordinal (e.g. low, medium, high), or cyclical (e.g. Monday, Tuesday, ... Sunday).
circumference the (measure of the) perimeter of a circle.
compound event an event consisting of compound outcomes, from a compound experiment. We say the compound event has occurred at one particular repetition of the experiment when any one of its compound outcomes is observed.
compound experiment an experiment that consists of more than one simple component (e.g. throwing a die and also spinning a coin, or three successive spins of a coin). Each repetition of the compound experiment give rise to a single compound outcome, which is recorded as multivariate data (e.g. the outcome may be five and heads, written as (5: H); or heads then tails then heads, written as (H; T; H).
compound outcome any outcome of a compound experiment; generally composed of a sequenced list of the outcome of each of the constituent simple experiments (e.g. throwing a head on spinning a coin and then a six and then a four on successive rolls of a die (H; 6; 4)).
conjecture a tentative solution inferred from collected data.
constraints limiting conditions (usually translated into linear inequalities) in a linear programming problem.
contingency table the elements of a population or group may be classified according to qualitative (categorical) variables. A classification in a two-way table of the elements according to two such qualitative characteristics is called a contingency table. The rows of the table denote the categories of the first variable and the columns the categories of the second variable.
correlation a term meaning association or relationship, often used for numerical data. A measure of correlation for numerical data is derived from a comparison of the regression line and the bivariate data in a scatter plot.
cumulative frequency for data that has been ordered (from minimum to maximum values) the successive values can be assigned frequencies. The cumulative frequency for a value x is the total count of all the data values that are less than or equal in value to x.
data items of information that have been observed and recorded; can be categorical (e.g. gender) or numerical (e.g. age). Data is often arranged in a list or table. Data can be univariate, bivariate or multivariate.
dependent variable the element of the range of a function which depends on the corresponding value(s) of the domain (e.g. in the area of a circle (y) depends on the radius, x. x is the independent variable and y is the dependent variable).
derivative the rate of change of a function with respect to the independent variable.
disjoint groups two groups that have no common member (e.g. male, female); three or more groups of which no pair has any common member (e.g. planes, cars, trains).
effective interest the annual rate which is equivalent to a nominal rate when compounding is effected more often than once a year (e.g. 12% p.a. compounded monthly is equivalent to 12,68% p.a.; the nominal interest rate (i) is 0,12 and the effective interest rate is 0,1268).
enlargement a mapping that increases the distances between parallel lines by the same factor in all directions (in contrast, see reduction).
error margins an understanding of the consequence of rounding or truncating in the course of calculations (the implication is not working with maximum accuracy as far as possible).
event any subset of all the possible outcomes of an experiment. An event occurs at a particular experimental trial if any one of its constituent outcomes is the outcome observed for that trial.
exchange rate the price of the currency of one country in terms of the currency of another.
experiment a statistical experiment is a repeatable activity or process for which each repetition gives rise to exactly one outcome, drawn from the sample space; a simple experiment gives rise to univariate data on the outcome of each trial (e.g. the observed face of a die). The number of trials observed is the sample size n.
extrapolate to estimate an unknown quantity by projecting from the basis of what is already known, but outside the limits of the known data (in contrast, see interpolate).
feasible region the set of all points that satisfy the constraints of a linear programming problem.
five number summary a summary of the observed values of a numerical variable, with sample size n, that consists of five values: the minimum, first quartile, median, third quartile, maximum.
fractals curves or surfaces generated by some repeated process resulting in self-similarity.
frequency a count of the number of times a particular outcome or event was observed in data with a sample size n.
frequency polygon a polygon formed by joining the mid-points of the top of the columns of a histogram.
frequency table a table reporting the groups into which data values were organised, as well as the frequency of each group.
glide reflection a combination of a translation and a reflection.
grouped data data arising from organising n observed values into a smaller number of disjoint groups of values, and then counting the frequency of each group; often presented as a frequency table or visually as a histogram.
histogram a visual representation used for grouped data for a numerical variable. The histogram consists of adjacent rectangles, each standing on a class interval. The area of each rectangle is proportional to the frequency of observations falling in a class interval. Class intervals are often on the horizontal axis and all have the same width; the rectangles are vertical. Frequency density is read on the vertical axis. A bar graph is not a histogram.
independent events the idea that two events do not connect with each other in any observable pattern, and hence that neither event can give any useful information about the other event. In contrast, if two events are not independent, they are said to associated. Numerical variables that are not independent are said to be correlated.
independent variable as used in dealing with functions: the value that determines the value of the dependent variable (e.g. in y the area of a circle (y) depends on the radius (x); x is the independent variable and y the dependent variable).
inflation rate a quantitative measure of consumer goods price increases over time.
inflection (point of) where a curve changes from being concave to convex or vice versa (e.g. the point ( ; 0) is a point of inflection on the graph of ); horizontal points of inflection are called stationary points.
interpolate to estimate an unknown quantity within the limits of what is already known (in contrast, see extrapolate).
isometric transformation a mapping which preserves lengths and angles (in a plane); includes translation, reflection, rotation and glide reflection.
least squares line a line in a scatterplot which is closest to the plotted points in that it will minimise the sum of squared vertical distances from the points to the line.
median a value that splits the sample data of a numerical variable into two parts of equal size, one part consisting of all values less than the median and one part with all values greater than the median; most easily established if the data values are arranged in increasing or decreasing order.
multivariate data two or more dimensions (of each object under observation) are recorded as an ordered string of variables (often to investigate or describe any association or correlation or relationship between the variables. The data is often arranged in a rectangular format of rows and columns, where a specific column is reserved for each variable and a row is allocated to each object that is observed (e.g. one row for the five variables of height, mass, age, eye colour and gender observed for each person, and five columns each with n entries). Categorical variables are summarised by frequency counts. Numerical multivariate data are often presented two variables at a time, with a scatter plot for each pair.
nominal interest rate the quoted (annual) rate per annum. If interest is compounded more often than once a year, the effective rate is greater than the nominal rate (e.g. 12% p.a. compounded monthly is equivalent to 12,68% p.a.; the nominal interest rate (i) is 0,12 and the effective interest rate is 0,1268).
objective function the function that one seeks to optimise in a linear programming problem.
ogive a cumulative frequency curve with cumulative frequency plotted on the vertical axis against increasing values of the observed variate on the horizontal axis.
optimise maximise or minimise (whichever is better in the particular context).
parameter a constant whose value determines in part how other interrelated variables are expressed and through which they may then be regarded as being dependent upon one another.
percentiles values of ranked data separated into one hundred groups of equal size, especially when sample size n is very large.
periodicity the interval after which a periodic (or repeating) function takes the same values.
qualitative data information or data arising from observations which are not numerical; qualitative data can be categorical.
quantitative data data with values are numerical; can be discrete (counted) or continuous (measured).
quartiles the sample quartiles are three values which split the ordered sample values into four groups of equal size; the second quartile is the median.
ranked data numerical data, ordered from say minimum to maximum, and then allocated ranks 1 to n.
ranks positions in an increasing or decreasing ordering, labelled as 1, 2, ... n. The minimum (and maximum) are assigned rank either 1 or n. Ranks may be averaged where values are tied.
reduction a mapping that reduces the distances between parallel lines by the same factor in all directions (in contrast, see enlargement).
reflection a transformation which produces a mirror image of the same shape and size as the original, but reversed; sometimes referred to as a flip.
regression line the straight line which minimises the sum of the squares of deviations of the dependent variable of individual data items from this line.
right circular cone a cone that has its vertex directly above the centre of the circular base.
right cylinder a solid that has one axis of symmetry through the centre of the circular base and a uniform, circular cross-section.
right prism a prism that has lateral sides that are perpendicular to its base.
right pyramid a pyramid that has its vertex directly above the centre of its base.
rigid transformation any transformation which leaves the shape and size unchanged; includes translation, reflection, rotation and glide reflection.
rotation a transformation under which a point or shape is rotated (turned) around a fixed point.
sample size the number of trials or repetitions of an experiment or observation process; often the symbol for the sample size is n (e.g. n=30 implies thirty observations were made and recorded).
sample space the collection of all the possible outcomes in a statistical experiment; may be discrete (consisting of categorised or counted values) or continuous (when a measurement is made on a scale that is continuous, e.g. mass, temperature, height).
scatter plot a graph using a pair of x,y axes to represent bivariate data, each bivariate element being plotted at a position that represents its pair of values; often accompanied by a least squares line fitted to the data.
search line the set of parallel lines used graphically to optimise the objective function O = ax+by, in a linear programming problem, subject to given constraints; the lines are generated by using various values for O with a and b fixed.
shearing angular deformation of a plane figure without a change in area; a type of transformation in which parallel lines remain parallel, but distances and angles are not preserved.
sigma the symbol , denoting the sum.
sinking funds an accounting term for cash set aside for a particular purpose and invested so that the correct amount of money will be available when it is needed; this is an example of a future value annuity.
spherical geometry the study of figures on a spherical surface.
stationary points a point on a graph of where the tangent is horizontal ().
stem and leaf diagrams a way of organising and summarising numerical data values into a vertical array of stems and a horizontal array of leaves on each stem. This method conveys more than a histogram because it also stores the data and allows easy calculation of median and quartiles.
stretch the shape of a graph or object undergoes a vertical or horizontal increase or decrease in scaling.
taxi-cab geometry the study of routes in a rectangular grid including those which provide the shortest distance between two points.
transformation the change of one figure (transformation geometry) or one expression (algebra) to another.
translation a transformation that moves all points the same distance in a common direction.
trial each repetition of a statistical experiment.
turning point a maximum or minimum point on a curve where the y-value changes from increasing to decreasing or vice versa (and the tangent is horizontal). Usually refers to a point on a curve when the slope is equal to zero.
univariate data one dimensional data; any quantity or attribute whose value varies from one observation to another gives rise to univariate data, which may be qualitative or quantitative.
universal set the set of all objects that are under discussion in a particular context.
Venn diagram a diagrammatic method of representing the relationship between subsets of some universal set.