CMA Foundation Fundamentals of Business Mathematics and Statistics Syllabus

What is the discriminant of a quadratic, and what does its sign tell about the roots?

Discriminant D = b^2 - 4ac. If D > 0 roots are real and distinct; if D = 0 roots are real and equal; if D < 0 roots are imaginary (complex).

For ax^2 + bx + c = 0, what are the sum and product of the roots?

Sum of roots = -b/a; Product of roots = c/a.

What are the formulas for the area and perimeter of a rectangle and a square?

Rectangle: Area = length x breadth; Perimeter = 2(length + breadth). Square: Area = side^2; Perimeter = 4 x side.

State the formulas for the area of a triangle (base-height) and the circumference and area of a circle.

Triangle area = (1/2) x base x height. Circle: Circumference = 2(pi)r; Area = (pi)r^2, where r is the radius.

What is Heron's formula for the area of a triangle with sides a, b, c?

Area = sqrt[s(s-a)(s-b)(s-c)], where s = (a + b + c)/2 is the semi-perimeter.

State the volume and total surface area of a cube and a cuboid.

Cube (side a): Volume = a^3; Total surface area = 6a^2. Cuboid (l, b, h): Volume = lbh; Total surface area = 2(lb + bh + hl).

State the volume and curved surface area of a right circular cylinder and a sphere.

Cylinder (radius r, height h): Volume = (pi)r^2 h; Curved surface area = 2(pi)rh. Sphere (radius r): Volume = (4/3)(pi)r^3; Surface area = 4(pi)r^2.

State the volume and curved surface area of a right circular cone.

Cone (radius r, height h, slant height l): Volume = (1/3)(pi)r^2 h; Curved surface area = (pi)r l, where l = sqrt(r^2 + h^2).

Define statistics and distinguish between primary and secondary data.

Statistics is the science of collecting, organising, presenting, analysing, and interpreting numerical data. Primary data is collected first-hand by the investigator for a specific purpose; secondary data is already collected by someone else and reused.

Distinguish between a census and a sample survey.

A census collects data from every unit of the population. A sample survey collects data from a representative subset (sample) of the population and infers about the whole.

Differentiate between discrete and continuous variables.

A discrete variable takes only specific, countable values (e.g., number of children). A continuous variable can take any value within a range (e.g., height, weight).

What is the difference between a frequency and a class frequency in a frequency distribution?

Frequency is the number of times a value occurs. Class frequency is the number of observations falling within a particular class interval in a grouped distribution.

Name the common methods of presenting statistical data.

Textual presentation, tabular presentation (tables), and diagrammatic/graphical presentation (bar diagrams, pie charts, histograms, frequency polygons, ogives).

What is an ogive, and what are its two types?

An ogive is a cumulative frequency curve. The two types are the 'less than' ogive (using upper class limits) and the 'more than' ogive (using lower class limits); their intersection gives the median.

Name the three main measures of central tendency.

Arithmetic Mean, Median, and Mode.

How is the arithmetic mean calculated for ungrouped and grouped data?

Ungrouped: Mean = (sum of x) / n. Grouped: Mean = (sum of f x) / (sum of f), where f are frequencies and x are values (or class mid-points).

Define the median and how to locate it for grouped data.

The median is the middle value that divides an ordered data set into two equal halves. For grouped data: Median = L + [(N/2 - cf)/f] x h, where L = lower limit of median class, N = total frequency, cf = cumulative frequency before the class, f = frequency of class, h = class width.

Define the mode and state the empirical relationship between mean, median, and mode.

The mode is the value occurring most frequently. Empirical relation: Mode = 3 Median - 2 Mean.

What are partition values: quartiles, deciles, and percentiles?

Quartiles divide data into 4 equal parts (Q1, Q2, Q3). Deciles divide it into 10 equal parts (D1...D9). Percentiles divide it into 100 equal parts (P1...P99).

What is dispersion, and name the common measures of dispersion.

Dispersion is the extent to which data values are spread out from a central value. Common measures: range, quartile deviation, mean deviation, standard deviation, and variance (with coefficient of variation as a relative measure).

Define range and quartile deviation.

Range = Largest value - Smallest value. Quartile Deviation (semi-interquartile range) = (Q3 - Q1)/2.

What is standard deviation and how does it relate to variance?

Standard deviation (sigma) is the positive square root of the mean of squared deviations from the mean; it measures absolute dispersion. Variance is the square of the standard deviation (sigma^2).

What is the coefficient of variation and what is it used for?

Coefficient of Variation (CV) = (Standard Deviation / Mean) x 100. It is a relative measure used to compare the consistency or variability of two or more data sets; a lower CV means more consistency.

Define correlation and state the range of the correlation coefficient.

Correlation measures the degree and direction of the linear relationship between two variables. The correlation coefficient (r) ranges from -1 to +1; +1 = perfect positive, -1 = perfect negative, 0 = no linear correlation.

What is Karl Pearson's coefficient of correlation formula (covariance form)?

r = Cov(x,y) / (sigma_x x sigma_y) = [sum of (x - mean x)(y - mean y)] / sqrt[sum(x - mean x)^2 x sum(y - mean y)^2].

What is Spearman's rank correlation coefficient formula?

R = 1 - [6 x sum(d^2)] / [n(n^2 - 1)], where d is the difference between ranks of paired items and n is the number of pairs.

What is regression, and what is the relationship between the two regression coefficients and r?

Regression estimates the value of one variable from another using a best-fit line. The correlation coefficient is the geometric mean of the two regression coefficients: r = plus or minus sqrt(b_yx x b_xy), taking the sign of the regression coefficients.

What is an index number, and what does it measure?

An index number is a statistical measure (expressed as a percentage) showing relative change in a variable or group of variables over time, place, or other characteristic, relative to a base period (taken as 100).

Distinguish between Laspeyres and Paasche price index numbers.

Laspeyres uses base-year quantities as weights: P01 = (sum p1 q0 / sum p0 q0) x 100. Paasche uses current-year quantities as weights: P01 = (sum p1 q1 / sum p0 q1) x 100. Fisher's index is the geometric mean of the two.

Define probability of an event and state its range and the classical formula.

Probability measures the likelihood of an event occurring; P(E) = (number of favourable outcomes)/(total number of equally likely outcomes). It ranges from 0 (impossible) to 1 (certain).

State the addition theorem of probability for two events and the condition for mutually exclusive events.

General: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events (cannot occur together), P(A and B) = 0, so P(A or B) = P(A) + P(B).

State the multiplication theorem of probability and the condition for independent events.

General: P(A and B) = P(A) x P(B|A). For independent events (one does not affect the other), P(A and B) = P(A) x P(B).

What is the mean and variance of a Binomial distribution with parameters n and p?

For a Binomial distribution B(n, p): Mean = np and Variance = npq, where q = 1 - p. The probability of r successes is P(r) = nCr p^r q^(n-r).

State the mean, variance, and key property of the Poisson distribution.

For a Poisson distribution with parameter m (= lambda): Mean = Variance = m. P(r) = (e^-m x m^r)/r!. It models rare events; it is the limiting case of the binomial when n is large and p is small.

State the main properties of the Normal distribution.

It is a symmetric, bell-shaped continuous distribution where mean = median = mode. It is defined by mean (mu) and standard deviation (sigma); the total area under the curve is 1; about 68% of values lie within 1 sigma, 95% within 2 sigma, and 99.7% within 3 sigma of the mean.