MEASURES OF TENDENCY

Objectives: 

1. Find Mode

2. Find Median

3. Find Mean 

Introduction...

What is average?

Average is a number expressing the central or typical value in a set of data, in particular the mode, median, or (most commonly) the mean, which is calculated by dividing the sum of the values in the set by their number, OR constituting the result obtained by adding together several amounts and then dividing this total by the number of amounts. 

How to find Average?

The mean is the average of the numbers. so the way how we find it is add up all the numbers, then divide by how many numbers there are, in other words, it is the sum divided by the number of counts.

By the way, there are few terms that you need to know:

1. Mode - The mode is the common value of the data. 

2. Median - The median is the middle value in a set of data. 

3. Mean - The mean is the average of the sum of numbers then divided by the number of counts.

Lets take some example..

Mode 

Find the mode of: 

black, grey, grey, red, white, grey 

mode = grey

why? because grey has 3 possible modes 

black, grey, grey, red, white, grey 

what if:

black, black, grey, red, white, grey 

mode = black, grey

why? because there is possible to have 2 modes 

black, black, grey, red, white, grey

what if: 

black, grey, red, white

mode = no mode

why? yes it is possible to have no mode. 

Median 

To find median: 

First step is to arrange the numbers in numerical order.

for example: 

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

Arrange it = 1, 2, 3, 5, 6, 7, 9

median = 5 

another example:
In a car company that sells car, there are 8 cars altogether with different prices.
$40, 000, $29, 000, $35, 500, $31, 000, $43 ,000, $30, 000, $27, 000, $32, 000

First step ; Arrange the data from least to Greatest. Then find the middle
= $27, 000, $29, 000, $30, 000, $31, 000, $32, 000, $35, 500, $40, 000, $43, 000

= $31,000, $32, 000 (if there is even number of counting in the middle, we take those two numbers and divide it by 2)

= 31000+32000 = 31,500
             2

= Median is $31,500 

Mean 

The mean is the average of the numbers, then dive by how many numbers there are. 

How to calculate? 

Just add all the numbers then divide by how many numbers there are.

for example: 

Find the mean of 2, 4, and 8? 

First Step: Add all the number = 2 + 4 + 8 = 14
Second Step: Divide the sum of all numbers by the number of counts : = 14 = 4.67 

                                                                                                                      3

= The mean is 4.67

LETS TAKE MORE EXAMPLES...

Example 1 : 

The weekly salaries of six employees at Burger King are $140, $220, $90, $180, $140, $200 

Find the Mean, Median and Mode

Mean : 

= $140 + $220 + $90 + $180 + $140 + $200 

                                  6

= $970

     6

= $161.67

Median : 

= $90, $140, $140, $180, $200, $220 

                             2

= $140 + $180 

             2

= $160 

Mode : 

= $140 

Example 2 :

Azwan trapped 10 rabbits, weighed them to the nearest pound, and recorded his results in groups as follows : 

Use the midpoints of the groups to estimate the mean weight of the rabbits Azwan trapped 

Example 3 : 

The populations of crocodiles in 7 national parks is below. Calculate the mean, median and mode.

Data : 14, 13, 8, 19, 13, 24, 15

Mean : 

= 14 + 13 + 8 + 19 + 13 + 24 + 15

                            7

= 106 

     7

= 15.1 Crocodiles 

Median :

= 8, 13, 13, 14, 15, 19, 24 

= 14 Crocodiles 

Mode :

= 13 Crocodiles 

Lets take an exercise...

Exercise 1 :

The number of graffiti convictions for 12 months of monthly data is below. Calculate the mean, the median and the mode. 

Data : 14, 25, 23, 15, 16, 16, 11, 23, 12, 13, 24, 7. 

PROBABILITY

Objectives:

1. To get to know the function of probability
2. The rules of probability
3. Special rule for addition 
4. Special rule of multiplication 
5. The use of tree diagram

What is the meaning of Probability?
Probability is the like hood or chances of an event occurring, or how likely is something is to happen.

For example:
1. Tossing a coin
2. Throwing dice

Tossing coin - When you are tossing a coin, there are two possible outcomes for each throws, of which head or tail is equal likely. Or, we can say that the probability of the coin landing H(head) is 1/2 or 0.5, and the probability of the coin landing T(tail) is 1/2 or 0.5.

Throwing Dice - When somebody throws a dice, there are SIX possible outcomes, which 1, 2, 3, 4, 5, 6. The probability any one of them is 1/6.

**Always remember, probability is always between 0 AND 1.**

Rules of Probability

1.  Probability - is the measure of how likely the event is. Ex; a spinner landing on blue is 1/4 or a coin flipped lands on a head is 1/2.

2. Experiment - is a situation involving chance or probability that leads to results called outcome. Ex; spinning a spinner once or tossing a coin once.

3. Sample point/outcome - is the result of a single trial of an experiment. Ex; possibilities of landing on a blue, green, red or yellow on a spinner or landing on a head or tails in a coin toss.

4. Certain event/sample space - all the possible outcomes of an experiment(s).

5. Two outcomes are mutually exclusive  when one of them cannot happen if the other occurs. Several outcomes are mutually exclusive if the occurrence of one of them excludes all possibilities of any of the others happening. Ex; on any coin toss, either head or tail may turn up but not both.

6. If two or more events occurs at one time - the events are not mutually exclusive. 

7. Independent events are events where the outcome of one of them in no ways affects the outcomes of the others. Ex; whether or not it rains in Australia has no effect on whether or not a football match is cancelled in Brunei.

8. Conditional event is one the outcomes of which is influenced by the outcome of another event. Ex; the probability of a student being able to attend exam would depend on whether they attend the minimum requirement of attendance. 

How to find probability?

In general:

P(A) =	The Number Of Ways Event A Can Occur
	The total number Of Possible Outcomes

Examples:

The rules of Addition in Probability

Addition rule 1 - When two events are mutually exclusive, the probability that X or Y will occur is the sum of probability of each event.

P(X or Y) = P(X) + P(Y)

The rules of Multiplication in Probability 

In the addition rule, we learn how to find a probability of two events at the same time. This lesson will also deal with multiplication rule. The multiplication rules that we learn now will also deals with two events, but in this problem, the events occur as a result of more than one task (rolling one die after another, drawing two cards, etc). 

If the events X and Y are independent, then the probability that events X and Y will both occur is given by, 

P(X and Y) = P(X) x P(Y) 

The Tree Diagram 

The tree diagrams are useful for organizing the different possible outcomes of an event. For each possible outcome of the first event, we draw a line where we must draw a line where we write down the probability of that outcome and the state of the world if that outcome happened. Then, for each possible outcome of the second event, we do the same thing. 

Below is an example of a simple tree diagram, lets take tossing a coin as example: 

Examples of probability:

Example 1 :
A die is thrown once. What is the probability that the score is a factor of 6?

The factor of 6 are 1, 2, 3, 6 = 4 number of ways

There are 6 possible scores when a die is thrown = 6 Total numbers of outcomes

So...

4/6 = 2/3

Answer is 2/3

Example 2 :

Each of the letter of the word MISSISSIPPI are written on separate pieces of paper that are then folded put in a hat, and mixed thoroughly.

One piece of paper is chosen (without looking) from the hat. What is the probability is it an I? 

There are 4 'I' letters in the word MISSISSIPPI = 4 number of ways

There are 11 letters in the word MISSISSIPPI altogether = 11 total number of outcomes

So...

Answer is 4/11

Example 3 : 

There are 10 counters in a bag: 3 are red, 2 are blue and 5 are green.

The contents oft he bag are shaken before Maxine randomly chooses one counter from the bag.

What is the probability that she doesn't pick a red counter? 

There are 7 counters that are not red: 2 Blue and 5 Green = 7 number of ways

There are 10 total number of outcomes = 10 total number of outcomes 

So...

Answer is 7/10 

Lets take an exercise...

Exercise 1 : 

The diagram shows a spinner made up a piece of card in the shape of a regular pentagon, with a toothpick pushed through its center. The five triangles are numbered from 1 to 5.

The spinner is spun until it lands on one of the five edges of the pentagon. What is the probability that the number it lands on is odd?