MDM4U – Grade 12 Data Management Exam Formula Sheet

Key Equations Matrix Operations

Transpose

At =B ,whereb =a m×n n×m ij ji

Multiplication

Scalar Multiplication

kA=C,wherec =ka ij ij

Inverse

Addition

A+D=E,wheree =a +d ij ij ij

Mean:

Variance:
Standard Deviation:

Z-score: Grouped Data:

x= n xw= ∑wi

sX sY

n∑xy − ∑x∑y =

n∑ x 2 − ( ∑ x ) 2 n∑ y 2 − ( ∑ y ) 2 Coefficient of Determination

∑(y −–y)2 r2 = est

σ =

N n−1 ∑(x − μ) s = ∑(x − x)

σ =·

∑ f ( m − μ ) 2 s =· ∑ f ( m − –x ) 2 ii ii

∑ ( y − –y ) 2
Permutations and Organized Counting

Factorial: n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1

n ab1d−b
a f ForH= c d ,H−1= −c a ifad≠bc

A F =G ,whereg =
m×n n×p m×p ij k=1 ik kj ad−bc

Statistics of One Variable

Population

∑x μ = N

∑(x − μ)2 σ2 =

Sample Weighted Mean –∑x –∑wixi

∑ ( x − –x ) 2 s2=

2 – 2 N n−1

x − μ z =

x − –x z =

σs
· ∑fimi – · ∑fimi

n∑x2 −(∑x2)

=
n(n − 1)

μ = ∑f x = ∑f , where mi is midpoint of ith interval ii

Statistics of Two Variables

Correlation Coefficient

N n−1
Least Squares Line of Best Fit

n∑xy − ∑x∑y y=ax+b,wherea= andb=y −ax

sXY r=

n∑x2 − (∑x)2

– –

Permutations

n!
r objects from n different objects: nPr = (n − r)!

n!
n objects with some alike: a!b!c!…

Combinations and the Binomial Theorem

Combinations

n!
r items chosen from n different items: nCr = (n − r)!r!

at least one item chosen from n distinct items: 2n − 1
at least one item chosen from several different sets of identical items: ( p + 1)(q + 1)(r + 1) … −1

Pascal’s Formula: C = C + C n r n−1 r−1 n−1 r

Introduction to Probability

n(A) Equally Likely Outcomes: P(A) = n(S)

Binomial Theorem: (a + b)n = n C an−rbr r=0n r

Discrete Probability Distributions

n Expectation: E(x) = i=1xi P(xi )

1 Discrete Uniform Distribution: P(x) =

n
P(x) = nCx pxqn−x

P(x) = qxp

1 Exponential Distribution: y = ke−kx, where k =

Normal Distribution: y = e 2
Distribution of Sample Means: μx– = μ and σx– = σ

ConfidenceIntervals:–x −zα σ –

Binomial Distribution: Geometric Distribution:

Hypergeometric Distribution:

Continuous Probability Distributions

E(x) = np

Complement of A: P(A′) = 1 − P(A) P(A) hh

Odds: oddsinfavourofA= P(A′) Conditional Probability: P(A | B) =

Independent Events: P(A and B) = P(A) × P(B)
Dependent Events: P(A and B) = P(A) × P(B | A)
Mutually Exclusive Events: P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events: P(A or B) = P(A) + P(B) − P(A and B) Markov Steady State: S(n) = S(n)P

P(A and B) P(B)

IfoddsinfavourofA= k,P(A)= h+k

C× C ax n−ar−x

p ra

P(x) = nCr

E(x) = n

)
Normal Approximation to Binomial Distribution: μ = np and σ = n pq if np > 5 and nq > 5

<μ<x+z 2 n

p−z 2 n

<p<p+z 2 n

σ

α

2 n

n
σ ˆ

α pˆ(1−pˆ ) ˆ

α p ˆ(1−pˆ )

q E(x) =

μ 1 1x−μ2

( σ 2 π