INEQUALITY

Introduction of Inequality

Equal, greater or less than.

As well as the familiar equals sign (=) it is also very useful to show if something is not equal to (≠) greater than (>) or less than (<)

These are the important signs to know:

Less Than and Greater Than

The "less than" sign and the "greater than" sign look like a "V" on its side, don't they?

To remember which way around the "<" and ">" signs go, just remember:

·         BIG > small

·         small < BIG

The "small" end always points to the smaller number, like this:

Or equal to

Sometimes we know a value is smaller, but may also be equal to!

To show this, we add an extra line at the bottom of the "less than" or "greater than" symbol like this:

All the Symbols

Here is a summary of all the symbols:

MEASURE OF CENTRAL TENDENCY

Introduction of Measure of Central Tendency

A measure of central tendency is a measure that tells us where the middle of a bunch of data lies.

The three most common measures of central tendency are the mean, the median, and the mode.

Mean: is the most common measure of central tendency. It is simply the sum of numbers divided by the number of numbers in a set of data. This is also known as average.

Median: is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of two middles number.

Mode: is the value that occurs most frequently in a set of data.

Examples:

For the data 1, 2, 3, 4, 5, 5, 6, 7, 8 the measures of central tendency are

Mean =  

Median = 5

Mode = 5

PROBABILITY

Introduction of Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Probability

In general :

Probability line

Show probability on a probability line :

GEOMETRIC PROGRESSION

Introduction of Geometric Sequence

Each term is found by multiplying the previous term by a constant.

Sequence is a set of things (usually numbers) that are in order.

In General we write a Geometric Sequence like this:

{a, ar, ar², ar³, ... }

where:

·         a is the first term, and

·         r is the factor between the terms (called the "common ratio")

But be careful, r should not be 0:

·         When r=0, we get the sequence {a,0,0,...} which is not geometric

The Rule

We can also calculate any term using the Rule:

A Geometric Sequence can also have smaller and smaller values:

Why Geometric Sequence?

Because it is like increasing the dimensions in geometry:

Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

ARITHMETIC PROGRESSION

Introduction of Arithmetic Sequences

The difference between one term and the next is a constant.

Sequence is a set of things (usually numbers) that are in order.

In other words, add the same value each time … infinitely.

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

·         a is the first term, and

·         d is the difference between the terms (called the "common difference")

Rule

We can write an Arithmetic Sequence as a rule:

x_n = a + d(n-1)

(We use "n-1" because d is not used in the 1st term).

INDICES

Introduction of Indices

is a useful way of more simply expressing large numbers. It also present us with many useful properties for manipulating them using what are called the Law of Indices.

What are Indices?

- The expression 2⁵ is defined as follows:

We call "2" the base and "5" the index.

Law of Indices

To manipulate expressions, we consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).

Rule 1: 

   a⁰ = 1

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

Example:

Simplify 2⁰: 

   2⁰ = 1

Rule 2:

Example:

Simplify: 2^-2: 

Rule 3: a^m x aⁿ = a^m+n

To multiply expressions with the same base, copy the base and add the indices.

Example:

Simplify: 5 x 5³ : (note: 5 = 5¹)

Rule 4: 

To divide expressions with the same base, copy the base and subtract the indices.

Example:

Simplify

Rule 5: (a^m)ⁿ = a^mn

To raise an expression to the nth index, copy the base and multiply the indices.

Example:

Simplify: (y²)⁶: 

Rule 6:

Example:

Simplify: 125^2/3:

COMBINATION & PERMUTATION : PART 2

Introduction of Permutation

is to count all of the possible ways that a single set of objects can be arranged. For example, consider the letters X, Y and Z. These letters can be arranged in number of different ways (XYZ, XZY, YXZ, etc.) Each of these arrangements is a permutation.

·         A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. This means that XZY is considered a different permutation than ZYX.

·         The number of permutations of n objects taken r at a time is denoted by _nP_r.

Example 1:

How many different ways can you arrange the letters X, Y, and Z?

(Hint: In this problem, order is important; i.e., XYZ is considered a different arrangement than YZX.)

Solution: One way to solve this problem is to list all of the possible permutations of X, Y, and Z. They are: XYZ, XZY, YXZ, YZX, ZXY, and ZYX. Thus, there are 6 possible permutations.

Another approach is to use Rule 2. Rule 2 tells us that the number of permutations is n! / (n - r)!. We have 3 distinct objects so n = 3. And we want to arrange them in groups of 3, so r = 3. Thus, the number of permutations is 3! / (3 - 3)! or 3! / 0!. This is equal to (3)(2)(1)/1 = 6.

Example 2;

In horse racing, a trifecta is a type of bet. To win a trifecta bet, you need to specify the horses that finish in the top three spots in the exact order in which they finish. If eight horses enter the race, how many different ways can they finish in the top three spots?

Solution: Rule 2 tells us that the number of permutations is n! / (n - r)!. We have 8 horses in the race. so n = 8. And we want to arrange them in groups of 3, so r = 3. Thus, the number of permutations is 8! / (8 - 3)! or 8! / 5!. This is equal to (8)(7)(6) = 336 distinct trifecta outcomes. With 336 possible permutations, the trifecta is a difficult bet to win.

COMBINATION & PERMUTATION : PART 1

Introduction of Combinations 

is to count all of the possible ways that a single set of objects can be selected without regard to the order in which they are selected.

·         n objects can be arranged in n(n – 1)(n – 2) … (3)(2)(1) ways. This represented by the symbol n!, which is called n factorial. (By convention, 0! = 1.)

·         A combination is a selection of all or part of a set of objects, without regard to the order in which they were selected. This means that XYZ is considered the same combination as ZYX.

·         The number of combinations of n objects taken r at a time is denoted by _nC_r.

Example 1:

How many different ways can you select 2 letters from the set of letters: X, Y, and Z?

 (Hint: In this problem, order is NOT important; i.e., XY is considered the same selection as YX.)

Solution: One way to solve this problem is to list all of the possible selections of 2 letters from the set of X, Y, and Z. They are: XY, XZ, and YZ. Thus, there are 3 possible combinations.

Another approach is to use Rule 1. Rule 1 tells us that the number of combinations is n! / r!(n - r)!. We have 3 distinct objects so n = 3. And we want to arrange them in groups of 2, so r = 2. Thus, the number of combinations is 3! / 2!(3 - 2)! or 3! /2!1!. This is equal to (3)(2)(1)/(2)(1)(1) = 3.

Example 2:

Five-card stud is a poker game, in which a player is dealt 5 cards from an ordinary deck of 52 playing cards. How many distinct poker hands could be dealt? 

(Hint: In this problem, the order in which cards are dealt is NOT important; For example, if you are dealt the ace, king, queen, jack, ten of spades, that is the same as being dealt the ten, jack, queen, king, ace of spades.)

Solution: For this problem, it would be impractical to list all of the possible poker hands. However, the number of possible poker hands can be easily calculated using Rule 1.

Rule 1 tells us that the number of combinations is n! / r!(n - r)!. We have 52 cards in the deck so n = 52. And we want to arrange them in groups of 5, so r = 5. Thus, the number of combinations is 52! / 5!(52 - 5)! or 52! / 5!47!. This is equal to 2,598,960 distinct poker hands.

MEASURE OF DISPERSION

The mean value for a set of data may given us some information about the set itself, many varying sets can have the same mean value. To determine how the sets are different we need more information. Another way of examining single variable data is to look at how the data is spread out or dispersed about the mean.

Range

The simplest of our methods for measuring dispersion is range. Range is the difference between the largest value and the smallest value in the data set. While being simple to compute, the range is often unreliable as a measure of dispersion since it is based on only two values in the set.

A range of 50 tells us very little about how the values are dispersed.

Are the values all clustered to one end with the low value (12) or the high value (62) being an outlier?

Mean Absolute Deviation (MAD)

The mean absolute deviation is the mean (average) of the absolute value of the difference between the individual values in the data set and the mean. The method tries to measure the average distances between the values in the data set and the mean.

Variance

To find the variance:

• Subtract the mean, , from each of the values in the data set,  .
• Square the result
• Add all of these squares
• And divide by the number of values in the data set.

Standard Deviation

Standard deviation is the square root of the variance. The formulas are:

Mean absolute deviation, variance and standard deviation are ways to describe the difference between the mean and the values in the data set without worrying about the signs of these differences.

These values are usually computed using a calculator.