TOPIC 10: Measure of central tendency

MEASURE OF CENTRAL TENDACY

Measure of central tendency can be the term which defines the centre of data. Measures of central tendency are the mean, mode, median, and standard deviation of a set of data.

There are 3 of measure of central Tendacy .

1. Mean
2. Median
3. Mode

 

Definition Mean:
Mean of data is a set of numerical values is the arithmetic average of the data values in the set. It is found by adding all the values in the data set and dividing the sum by the total number of values in the set.

Definition Median:
For an ordered data set, median is the value in the middle of the data distribution. If there are even number of data values in the set, then there will be two middle values and the median is the average of these two middle values.

Definition Mode Mode is the most frequently occurring value in the data set.

In addition to these three important measures of central tendency, another measure is also defined.

Examples

1) Mean.

3, -7, 5, 13, -2

• The sim of these number is 3 -7 + 5 + 13 - 2 =12
• There are 5 numbers
• The mean is equal to 12 ➗ 5 = 2.4

  The mean of the above numbers is 2.4

here is how to do it one line:

mean = 3 - 7 + 5 + 13 - 2 ➗ 5 = 12 / 5 = 2.4



2) Median

How to find the median value?

Solution: Put them in order:

 3, 5, 12

The middle is 5, so the median is 5.

 

 

3) Mode

 

{19, 8, 29, 35, 19, 28, 15}

 

Arrange them in order: {8, 15, 19, 19, 28, 29, 35}

19 appears twice, all the rest appear only once, so 19 is the mode.

TOPIC9: REPRESENT DATA

REPRESENT DATA

• Data is a collection of facts, such as numbers, words, measurements, observations or even just descriptions of things.
   
• Data can be qualitative or quantitative.

1. Qualitative data is descriptive information (it describes something)
2. Quantitative data, is numerical information (numbers).

 



And Quantitative data can also be Discrete or Continuous.

• Discrete data can only take certain values (like whole numbers)
• Continuous data can take any value (within a range)

*Discrete data is counted, Continuous data is measured*

Example 1:

What do we know about Arrow the dog?

Qualitative:

- He is brown and black

- He has long hair

-He has lots of energy

Quantitative: 

-Discrete:

- He has 4 legs

- He has 2 brothers

-Continuous:

- He weights 25.5 kg

- He is 565 mm tall

Example 2:

A person;s height: could be any value (within the range of human heights), not just certain fixed heights.

Also a dog's weight and the length of a leaf.

Example 3:

The number of students in a class (You can't have half  a student).

The results of rolling 2 dice:

Can only have the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.

TOPIC8: ARITHMETIC PROGRESSION & GEOMETRIC PROGRESSION

ARITHMETIC PROGRESSION

• An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.

   

Arithmetic progression examples

Formula: Tn = a + (n-1) d

                        a = 1^st term

                        n = nth term

                        d = common difference

What is the 10th term?

T10 = 1 + (10-1) 2

        = 1 + (9) (2)

        = 1 + 18

        = 19

What is the first 3 terms?

T2 = 1 + (2-1) 3          T3 = 1 + (3-1) 3

      = 4                              = 7

What is the 16th term?

T16 = 8 (16-1) = 3

        = 8 + (15) (-3)

        = 8 + (-45)

        = -37 

Example1:

-  Write down the first four terms of AP with first term 8 and difference 7.

T2 = 8 + (2-1) 7          T3 = 8 + (3-1) 7          T4 = 8 + (4-1) 7

      = 15                            =22                             = 29

- Write down the first four terms of AP with first term 2 and difference -5.

T2 = 2 + (2-1) -5         T3 = 2 + (3-1) -5         T4 = 2 + (4-1) -5

      = -3                             = -8                             = -13

- Write down the 10th and 19th terms of the AP.

i) 8, 11, 14...

T10 = 8 + (10-1) 3        T19 = 8 + (19-1) 3

        = 8 + (9) (3)                  = 8+ (18) (3)

        = 8 + 27                         = 8 + 54

        = 35                              = 62

ii) 8, 5, 2...

T10 = 8 + (10-1) -3       T19 = 8 + (19-1) -3

        = 8 + (9) (-3)                = 8 + (18) (-3)

        =8 + (-27)                      = 8 + (-54)

        = -19                             = -46

Example2

Write down the first four terms of AP with first term 8 and difference 7.

T2 = 8 + (2-1) 7          T3 = 8 + (3-1) 7          T4 = 8 + (4-1) 7

      = 15                            =22                             = 29

 

Example3

Write down the 10th and 19th terms of the AP.

i) 8, 11, 14...

T10 = 8 + (10-1) 3        T19 = 8 + (19-1) 3

        = 8 + (9) (3)                  = 8+ (18) (3)

        = 8 + 27                         = 8 + 54

        = 35                              = 62

ii) 8, 5, 2...

T10 = 8 + (10-1) -3       T19 = 8 + (19-1) -3

        = 8 + (9) (-3)                = 8 + (18) (-3)

        =8 + (-27)                      = 8 + (-54)

        = -19                             = -46

 

GEOMETRIC PROGRESSION

• A geometric progression is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by "r". The common ratio is obtained by dividing any team by preceding term.

 

 

Formula: Tn = ar ^n-1

                        a = 1^st term

                        r = common ratio

                        n = nth term

2, 6, 18, 54,...     r = 2^nd term/1^st term 

                                = 6/2 

                                = 3

Find the 15th term of the GP?

T15 = 2 x 3 ^15-1

       = 9, 565,938

Example 1:

Find the 10th and 17th term of GP with first term 3 and common ratio 2.

- a) a = 3                 b)   a = 3

       r = 2                         r = 2

      n = 10^th                         n = 17th

T10 = 3 x 2 ^10-1           T10 = 3 x 2 ^17-1

        = 1,536                    = 196, 608

Example 2: 

Find the 7th term of the GP 2, -6, 18....

- r = 2nd term/1st term

     = -8/2

     = -3

 

 

 

TOPIC7: SEQUENCE AND NUMBER PATTERN

SEQUENCE AND NUMBER PATTERN

• In mathematics, a sequence is an ordered list of objects. Like a set, it contains members (also called elements or terms).The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence. 

 

 

 

 

Example 1: 

1, 4, 7, 10, 13, 16......start at 1 and jumps 3

Example 2:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The pattern is continued by adding 5 to the last number each time, like this:

 

 

 

 

Example3:

 

 25,23,21,19,17,15

 

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

 

 

 



TOPIC6: SETS

SETS

Definition

-  a collection of objects that have something in common or follow rule. The objects u the set are called its elements. Set  notation uses curly braces, with elements separated by commas.

Example 1:
What is the set of all fingers?

Solution:
P = {Thumb, Index, Middle, Ring, Little}
- Note that there are others name for these fingers: The index finger is commonly referred to as the pointer finger, the ring finger is also known as the fourth finger, and the little finger is often referred to as the pinky. Thus, we could have listed the set of fingers as:

P = {Thumb, Pointer, Middle, Fourth, Pinky}

Example 2: Let T be the set of all days in a week.

Solution: T = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Question 1: Let G be the set of all whole numbers less then ten?

Question 2: What is the set of all letters in the English alphabet?

Question 3: Which of the following is the set of all oceans on earth?

A. G = {Atlantic, Pacific, Arctic, Indian, Antarctic}
B. E = {Amazon, Nile, Mississippi, Rio Grande, Niagara}
C. F = {Asia, Africa, North America, South America, Antarctic, Europe, Australia}
D. All of the above

TOPIC5: LINEAR PROGRAMMING

LINEAR PROGAMMING

 

Linear Programming is the process of finding the extreme values (maximum and minimum values) of a function for a region defined by inequalities. In real life, linear programming is part of a very important area of mathematics calles "optimization techniques". This field of study are used every day in the organization and allocation of resource. These real life systems can have dozens or hundreds of variable or more. in algebra through you will only work with the simple (and graph able) two variable linear case. 

 

 

 

Example1

Find the maximum value of the function C = 6x + y subject to the constrains
x ≥ 0, y ≥ 0 , 5x + 3y ≤ 15.

 

Step 1: Objective function is C = 6x + y 
Step 2: Constraints are x ≥ 0, y ≥ 0 5x + 3y ≤ 15
Step 3: [Draw the graph.]
The feasible region determined by the given constraints is shown

 

 

 

Step 4: From the graph, the three vertices are (0, 0), (3, 0), and (0, 5).

Step 5: To evaluate the minimum, maximum values of C, we evaluate C = 2x + y at each of the vertices.

Step 6: [Substitute the values.]
At (0, 0) , C = 6(0) + (0) = 0

Step 7: [Substitute the values.]
At (3, 0) , C = 6(3) + (0) = 18

Step 8: [Substitute the values.]
At (0, 5) , C = 6(0) + (5) = 5

Step 9: So, the maximum value of C is 18.Answer is: 18.


Example2
 
If the objective function of the previous exercise had been:



f(x,y) = 20x + 30y


f(0,500) = 20·0 + 30 · 500 = $15,000       Maximum


f(500, 0) = 20·500 + 30·0 = $10,000


f(375, 250) = 20·375 + 30·250 = $15,000     Maximum
In this case, all the pairs with integer solutions of the segment drawn in black would be the maximum.





 

 

Example3

 

A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B.

 

Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.

At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours.

The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximize the combined sum of the units of X and the units of Y in stock at the end of the week.

1. Formulate the problem of deciding how much of each product to make in the current week as a linear program.
2. Solve this linear program graphically.

Solution
Let

• x be the number of units of X produced in the current week
• y be the number of units of Y produced in the current week

then the constraints are: 

50x + 24y <= 40(60) machine A time

The objective is: maximize (x+30-75) + (y+90-95) = (x+y-50)
i.e. to maximize the number of units left in stock at the end of the week
It is plain from the diagram below that the maximum occurs at the intersection of x=45 and 50x + 24y = 2400

30x + 33y <= 35(60) machine B time

x >= 75 - 30

i.e. x >= 45 so production of X >= demand (75) - initial stock (30), which ensures we meet demand

y >= 95 - 90

i.e. y >= 5 so production of Y >= demand (95) - initial stock (90), which ensures we meet demand. 

The objective is: maximize (x+30-75) + (y+90-95) = (x+y-50)
i.e. to maximize the number of units left in stock at the end of the week
It is plain from the diagram below that the maximum occurs at the intersection of x=45 and 50x + 24y = 2400 

 

 



TOPIC 4: LOGARITHM

LOGARITHM

 

 Logarithm is the power to which a number must be raised

in order to get some other number. It is also quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

 

Logarithm Form: 16 = 2⁴                              

                              Log ₂ 16 = 4

                              64 = 8²

                              Log ₈ 64 = 2

 

 

 

Determine the value of logarithm: Log ₃ 9

                                                 9 = 3² 

 

 

Example1

 

the base ten logarithm of 100 is 2, because ten raised to the power of two is 100 :

log 100 = 2 it is because 10² = 100

Example2

What Is log₅(625)?

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s.

Answer: log₅(625) = 4 

 

 

 



 Example3

 

What is log₂(64)?

We are asking "how many 2s need to be multiplied together to get 64?

2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s.

Answer: log₂(64) = 6

 

 

 

 

This are the 3 question for you to try.

 

1. log₁₀(100) 

2. log₃(81)

3. log₄(256)