Statistics

class	class width	freq	mid values	freq density	cumulative freq
50 - 59	10	5	54.5	2.5	5
60 - 64	5	2	62	2	7
65 - 68	4	4	66.5	5	11

Lower class limit: 50, 60, 65, upper class limit: 59, 64, 68

True class limits/ class boundaries = ±0.5 + class limit
Class width = upper class boundary - lower class boundary
Freq density = freq class width
Freq polygon: used mid-values
Cumulative freq: use upper class limit to plot pts
Mean = sum of fX total freq (f = freq for a group, X = midpt for a group-grouped data / class)
lower quartile: class corr. to of cum. freq, upper quartile: class corr. to of cum. freq
inter-quartile range: upper - lower quartile, percentile: class corr. to x% of cum. freq
(25th percentile = lower quartile, 75th percentile = upper quartile, 50th percentile = median)

Variation

y is proportional to x	y is inversely proportional to x
y ∝ x	y ∝ 1/x
y = kx, k = constant	y = k/x, k = constant

Transformation
P is mapped onto P₁ under a transformation U U(P) = P₁

Reflection
P is mapped onto P₁ under a reflection in the line, l


Rotation
P is mapped onto P₁ under a clockwise rotation of x about C


Translation
P is mapped onto P₁ under a translation of translation vector


Enlargement
AB is mapped onto A₁B₁ under an enlargement of scale factor, k w/ a centre of enlargement, C, k = length of (image/object) or length of (AC/A₁C) (k: -ve > image & object on opp sides of C)


Shear
ABCD is mapped onto A₁B₁CD under a shear w/ shear factor, k and l as the invariant line, k = dist bet (A and A₁/A and invariant line) (AA₁ // l) (k: +ve > up and right, k: -ve > left and down)


Stretch
ABC is mapped onto ABC₁ under a stretch w/ stretch factor, k and invariant line, l, k = dist bet (C & l/ C₁ & l)


Combination of transformations
Transformation XY = transformation (Y > X)

Matrix representation of transformations

Translation, T: object + translation vector = image



Reflection, rotation, enlargement, shear, stretch: pre-multiply


Vectors

x = horizontal dist, y = vertical dist
right/up: +ve, left/down: -ve
equivalent vectors: same magnitude & direction
-ve sign reverses direction of vector

= c = c


zero vector (a pt) = 0 = 0 =


Position vector of P (x, y), where O is origin


ka = vector parallel to a, same direction, k times magnitude of a
-ka = vector parallel to a, opposite direction, k times magnitude of a


magnitude of a = |a| =


A, B, C are collinear

Probability
Probability of (E), P(E) = n(E) / n(S)
n(E) = # of favourable outcomes, n(S) = # of possible outcomes (sample space)
0 P(E) 1, P(E) + P(E') = 1
Possibility diagram

P(both heads) = ¼
Tree diagram

P(both heads) = ½ × ½ = ¼

A & B: mutually exclusive events: A & B cannot occur at once
p(A or B occurring) = p (A B) = p(A) + p(B)
To find probability of 2 or more events happening together using a tree diagram
-determine if probability of outcome is affected by previous event
-calculate respective probabilities along branches of tree diagram
-multiply probabilities along relevant branches

Loci
2-D loci
Locus of a pt x cm from given pt A = a circle w/ radius of x cm & A as the centre

Locus of a pt x cm from given line, l = parallel lines on both sides of l and each x cm from l (& same length of l)

Locus of a pt equidistant from 2 given pts, A & B = perpendicular bisector of line segment AB

Locus of a pt equidistant from 2 given intersecting lines, m & l = a pair of perpendicular lines bisecting the angle's bet l & m

Locus of a pt equidistant from 2 given touching lines, l & m = bisector of the angles(s) bet l & m


To draw an incircle/ inscribed circle of triangle ABC
-draw 2 loci of pts equidistant from 2 different pairs of 2 sides of the triangle
-draw perpendicular from a line passing through the pt of intersection, perpendicular = radius of circle


To draw a circumscribed/ circumcircle of triangle ABC
-draw perpendicular bisectors of 2 sides
-draw the circle w/ pt of intersection of loci as centre of circle


3-D loci
locus of a pt d cm from given pt A = SA of a sphere w/ A as the centre & a radius of d cm

locus of a pt d cm from given line l = curved surface or a cylinder w/ l as its axis & a radius of d cm

locus of a pt equidistant from 2 given pts, A & B = a plane perpendicular to & bisecting line segment AB


Emaths1

Back to 'O' level notes index

Back to notes index