Math Paper 2

Measures of central tendency

Mean, Median and Mode

Measures of spread

Range, Quartiles, Variance and Standard Deviation.

Negatively skewed data

Mean < Median (Left)

Positively skewed data

Mean > Median (Right)

Median position

(n+1)/2

IQR

Q3-Q1

Position of Q1

(n+1)/4

Position of Q3

3(n+1)/4

Variance

σ^2 or Σ[(x-mean)^2 / (n or n-1)]

Deviation

x-mean = difference between entry and the mean

Cluster formation

Variance and standard deviation are small.

How to calculate percentile position?

1. Arrange data in ascending order
2. Determine the index (i), the position of the P'th percentile

i = (P/100)(n)

3. If i is not an integer, round off to the top. The P'th percentile is the value in the i'th position
4. If i is an integer, the P'th percentile is the mean of the values in position i and i+1

Semi-IQR

= (1/2)(Q3 -Q1)

Outlier

Any data that lies more than 1,5 IQR to the left of Q1, or to the right of Q3.

Outlier < Q1 - 1,5(IQR) Outlier > Q3 + 1,5(IQR)

5 number summary

Minimum value Q1 Median Q3 Maximum value

How to decide which midpoint to use in grouped data?

THIS NEEDS A DEFINITION

Ogive

Cumulative frequency curve

Checklist for stats graph:

1. Heading
2. Accurate scale
3. Neatness

A graph that has it's bump/peak on the right side is skewed…

…to the left.

(when the boxes are equal size) If the median is skewed toward the upper quartile the data is skewed to…

…the left, the left whisker is longer.

(when the boxes are equal size) If the median is skewed toward the lower quartile the data is skewed to…

…the right, the right whisker is longer.

A graph that has it's bump/peak on the left side is skewed…

…to the right.

A longer right whisker indicates…

…a skew to the right.

A longer left whisker indicates…

…a skew to the left

Types of scatterplots

Linear Quadratic Exponential

Linear regression line standard form

ŷ = a + bx or ŷ = Bx + A

How to get a linear regression information

a+bx mode

Correlation coefficient

r

r = 1

Perfect positive linear correlation

0,8 < r < 1

Strong positive linear correlation

0,5 < r < 0,8

Moderate positive linear correlation

0 < r < 0,5

Weak positive linear correlation

r = 0

No linear correlation

-0,5 < r < 0

weak negative linear correlation

-0,8 < r < -0,5

Moderate negative linear correlation

-1 < r < -0,8

Strong negative linear correlation

r = -1

Perfect negative linear correlation

What is constant difference and what kind of pattern does it apply to?

• T2 - T1 = T3 - T2

• AS (Arithmetic Sequence)

What are the formulae of the sum of an arithmetic sequence?

Sum of a geometric sequence

Formula of sum to infinity

How to calculate arithmetic mean

How to calculate geometric mean

if a;b;c

[LINES] The adjacent angles on a straight line are supplementary.

∠s on a str line

[LINES] If the adjacent angles are supplementary, the outer arms of these angles form a straight line.

adj ∠s supp

[LINES] The adjacent angles in a revolution add up to 360°.

∠s round a pt OR ∠s in a rev

[LINES] Vertically opposite angles are equal.

vert opp ∠s =

[LINES] If AB || CD, then the alternate angles are equal.

alt ∠s; AB || CD

[LINES] If AB || CD, then the corresponding angles are equal.

corresp ∠s; AB || CD

[LINES] If AB || CD, then the co-interior angles are supplementary.

co-int ∠s; AB || CD

[LINES] If the alternate angles between two lines are equal, then the lines are parallel.

alt ∠s =

[LINES] If the corresponding angles between two lines are equal, then the lines are parallel.

corresp ∠s =

[LINES] If the co-interior angles between two lines are supplementary, then the lines are parallel.

coint ∠s supp

[TRIANGLES] The interior angles of a triangle are supplementary.

∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ

[TRIANGLES] The exterior angle of a triangle is equal to the sum of the interior opposite angles.

ext ∠ of Δ

[TRIANGLES] The angles opposite the equal sides in an isosceles triangle are equal. (i.e. Given image - Prove B and C equal)

∠s opp equal sides

[TRIANGLES] The sides opposite the equal angles in an isosceles triangle are equal. (i.e. Given image - prove c and b equal)

sides opp equal ∠s

[TRIANGLES] In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagoras OR Theorem of Pythagoras

[TRIANGLES] If the square of the longest side in a triangle is equal to the sum of the squares of the other two sides then the triangle is right-angled.

Converse Pythagoras OR Converse Theorem of Pythagoras

[TRIANGLES] If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent.

SSS

[TRIANGLES] If two sides and an included angle of one triangle are respectively equal to two sides and an included angle of another triangle, the triangles are congruent.

SAS OR S∠S

[TRIANGLES] If two angles and one side of one triangle are respectively equal to two angles and the corresponding side in another triangle, the triangles are congruent.

AAS OR ∠∠S

[TRIANGLES] If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent.

RHS OR 90°HS

[TRIANGLES] The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. (i.e. Given image - prove DE parallel to BC AND BC = 2DE)

Midpt Theorem

[TRIANGLES] The line drawn from the midpoint of one side of a triangle, parallel to another side, bisects the third side. (i.e. Given image - Given AD = AB and DE is parallel to BC, prove AE = EC)

line through midpt || to 2nd side

[TRIANGLES] A line drawn parallel to one side of a triangle divides the other two sides proportionally. (i.e. Given image - Given that HF is parallel to BC, Prove AH/HB = AF/FC

line || one side of Δ OR prop theorem; name || lines (i.e. - In ΔABC, HF || BC)

[TRIANGLES] If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side. (i.e. Given image - Prove that HF is parallel to BC)

line divides two sides of Δ in prop

[TRIANGLES] If two triangles are equiangular, then the corresponding sides are in proportion (and consequently the triangles are similar).

||| Δs OR equiangular Δs

[TRIANGLES] If the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequently the triangles are similar).

Sides of Δ in prop

[TRIANGLES] If triangles (or parallelograms) are on the same base (or on bases of equal length) and between the same parallel lines, then the triangles (or parallelograms) have equal areas.

same base; same height OR equal bases; equal height

[CIRCLES] The tangent to a circle is perpendicular to the radius/diameter of the circle at the point of contact.

tan ⊥ radius OR tan ⊥ diameter

[CIRCLES] If a line is drawn perpendicular to a radius/diameter at the point where the radius/diameter meets the circle, then the line is a tangent to the circle.

line ⊥ radius OR converse tan ⊥ radius OR converse tan ⊥ diameter

[CIRCLES] The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. (i.e. Given Image - Prove Angle ACE is perpendicular)

line from centre to midpt of chord

[CIRCLES] The line drawn from the centre of a circle perpendicular to a chord bisects the chord. (i.e. Given image - Prove that DC = EC)

line from centre ⊥ to chord

[CIRCLES] The perpendicular bisector of a chord passes through the centre of the circle.

perp bisector of chord

[CIRCLES] The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre).

∠ at centre = 2 × ∠ at circumference

[CIRCLES] The angle subtended by the diameter at the circumference of the circle is 90°.

∠s in semi-circle OR diameter subtends right angle OR ∠ in ½

[CIRCLES] If the angle subtended by a chord at the circumference of the circle is 90°, then the chord is a diameter.

chord subtends 90° OR converse ∠s in semi-circle

[CIRCLES] Angles subtended by a chord of the circle, on the same side of the chord, are equal.

∠s in the same seg

[CIRCLES] If a line segment joining two points subtends equal angles at two points on the same side of the line segment, then the four points are concyclic.

line subtends equal ∠s OR converse ∠s in the same seg

[CIRCLES] Equal chords subtend equal angles at the circumference of the circle.

equal chords; equal ∠s (Angle A = Angle D)

[CIRCLES] Equal chords subtend equal angles at the centre of the circle. (i.e. Given image, prove that angle FAC is = angle EAD)

equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the circumference of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the centre of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] The opposite angles of a cyclic quadrilateral are supplementary.

opp ∠s of cyclic quad

[CIRCLES] If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

opp ∠s quad supp OR converse opp ∠s of cyclic quad

[CIRCLES] The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

ext ∠ of cyclic quad

[CIRCLES] If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.

ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad

[CIRCLES] Two tangents drawn to a circle from the same point outside the circle are equal in length.

Tans from common pt OR Tans from same pt

[CIRCLES] The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.

tan chord theorem

[CIRCLES] If a line is drawn through the end-point of a chord, making with the chord an angle equal to an angle in the alternate segment, then the line is a tangent to the circle.

converse tan chord theorem OR ∠ between line and chord

[QUADRILATERALS] The interior angles of a quadrilateral add up to 360°.

sum of ∠s in quad

[QUADRILATERALS] The opposite sides of a parallelogram are parallel.

opp sides of ||m

[QUADRILATERALS] If the opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

opp sides of quad are ||

[QUADRILATERALS] The opposite sides of a parallelogram are equal in length.

opp sides of ||m

[QUADRILATERALS] If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

opp sides of quad are = OR converse opp sides of a parallelogram

[QUADRILATERALS] The opposite angles of a parallelogram are equal.

opp ∠s of ||m