MODE, MEAN and MEDIAN

Mode

The mode is the most common value in a set of data.

Find the mode of:

a)Blue, red, yellow, blue, orange

Mode = blue

b)  5, 6, 7, 6, 8, 6, 9

Mode = 6 

It is possible to have 2 modes

E.g. 4, 7, 7, 8, 8, 4 

Mode = 7, 8

It is possible to have no mode

E.g. 4, 7, 8, 5, 6, 2 

Median

The median is the middle value in a set of data

First put the numbers in numerical order

Example: find the median of:

7, 6, 2, 3, 2, 9, 9, 5

Ordered: 1, 2, 3, 5, 6, 7, 9

Median = 5

5, 3, 2, 8, 7, 9

Ordered: 2, 3, 5, 7, 8, 9

There are 2 numbers in the middle. The median is in the middle of these. There can be only be one median 

Median = 6  

Mean

To find the mean of a set of data:

a)Add all the values together 

b)Divide by the number of values there are

The mean takes the total of all the values and spreads the total out evenly to get an average

Example:

Qalesh threw ten sets of three darts at a board. His scores were:

34, 45, 20, 41, 60, 83, 70, 30, 26, 61

Find his mean score 

Mean = total score ÷ number of values

     = 470 ÷ 10

Qalesh’s mean score is 47

Questions that you can try:

1.

2.

 

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MATRICES

 

a) A matrix (plural : matrices) is a rectangular arrays of numbers 

b)Example of matrices are: 

c) The numbers in a matrix are called elements

d) A capital letter used to represent a matrix

e) The order of a matrix is no. of rows X no. of column

Example: In a matrix A above, the order is 2x3, i.e 2 rows and 3 column. 

In matrix B, the order is 3x1

In matrix C, the order is 2x2

f)In general, an mxn matrix has m rows and n columns

Special matrices

a)zero matrix : a matrix which has zeros as all of its element. 

b) Row matrix or row vector: a matrix that has only one row.

Example: (1  2  3)  (35   70   60)  (13   25)

c) Column matrix or column vector: a matrix that has only one column.

Example:

d) Square matrix: a matrix that has the same number of rows and columns. 

Example: 

e) Identify matrix: a square matrix that has elements on its main diagonal (from upper left to lower right) equal tp 1 and all other elements equal to 0.

Example:

f)Identify matrix : a square matrix that has elements on its main diagonal (from upper left to lower right) equal to 1 and all other elements equal to 0.

Then, matrices A and B are equal matrices.

Addition and Subtraction of matrices

Example 1: 

Find,

1) P + Q + R 

2) P - Q + R

Answer of 1:

Answer of 2: 

Questions that you can try:

1. 

2.

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LOGARITHM

^{Definition of logarithm?}

logarithm is the exponent that indicates the power 
to which a base number is raised to produce a given number.

Logarithm is invented by John Napier.

Laws of logarithm

qLoga PQ = Loga P + Loga Q

q Loga P/Q = Loga P – Loga Q

q Loga Pn = nLoga P

q Loga 1 = 0

q Loga a = 1

q Loga y = Logb y / Logb a

Exponent and Base

Common logarithm base

Example: the base 10 logarithm of 100 is 2

Example 2: 2^x = 5  ^{(index form)}

• How to find ^x?
• Need to make sure that they are the same base

                    2^x = 5

^x = log₂ 5

Example 3:Find x, Log2 (5x + 7) = 5

5x + 7 = 2⁵

2⁵  is actually 2 x 2 x 2 x 2 x 2

5x + 7 = 32

5x = 32 – 7

5x = 25

X = 25/5

Anwser: X = 5

Questions that you can try:

1.Given: log₈(5) = b. Express log₄(10) in terms of b. 

2. Solve for x the equation 4^{x - 2} = 3^{x + 4}

3. Simplify without calculator: log₆(216) + [ log(42) - log(6) ] / log(49) 

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FACTORISATION

Factorisation

a) By taking out common factors
b) By grouping
c) Of a different of two squares
d) By inspection or trial and error (quadratic) 

Example: Factorise the following expressions 
(i) by taking out common factors

(a) 2a + 4

Anwser =  2 (a + 2)

(b) 3x² y – xy²

Anwser = xy (3x – y)

(ii) by grouping

(a) ac + ad + 3c + 3d 

Anwser = a (c + d) + 3 (c + d)

(b) 3hk + ky – 3hy – kx 

Anwser = k (3h + y) – (3hy – kx)

(c) 3v – 6w + xv – 2xw

[3 x 2 = (6)]

Anwser = 3 (v – 2w) + x (v – 2w)

  

(iii) a difference of two factors 

(a) a² - 16

(16 = 4 x 4)

= a² – 4²

Final Anwser = (a + 4) (a - 4)

(b) 5k² – 125y²

^{(125 = 25 x 25)}

= 5 (x² – 25y²)

(25 = 5 x 5)

= 5 (x² – 5² y²)

Final Anwser = 5 (x + 5y) (x – 5y)

Questions that you can try: 

1. 2y + 6

2. a ² - b ² - c ² + d ² - 2(ad - bc) 

3. 4x ² + 20 x +25

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