Measure Of Dispersion (cont.)

 2.    The Quartile Deviation

      §Quartiles divide a set of ordered data into four groups with equal numbers of values

The three “dividing points” are the first quartile (Q1), median, (sometimes called the second quartile, Q2), and the third quartile (Q3)

The interquartile range is Q1 – Q3, which is the range of the middle of the data.

 The semi-interquartile range is one half of the interquartile range.

 Both these ranges indicate how closely the data are clustered around the median.

The quartile deviation is called the semi-quartile range IT is defined as the amount of dispersion present in the middle of 50% of the values, hence the equation is given by,

Q.D = Q3 – Q1            2

Q.D = the quartile deviation

Q1 = the first quartile

Q2 = the third quartile

                                                

Q1 = L + (N/4 – F1) i              F

Where:

Q = the first quartile

L = the lower limit of the first quartile class (i.e. the class containing the N/4th item)

            Strictly, L is the lower boundaries of the 1st quartile class.

F1 = the cumulative frequency just above the 1st quartile class.

F = the frequency of the 1st quartile class

N = the total frequency

I = the class interval

Class Intervals	Frequency	M	Quartile class	Cf>
45 – 49	3	47		3
50 – 54	4	52		7
55 – 59	6	57	Q1	13
60 – 64	7	62		20
65 – 69	10	67		30
70 – 74	7	72		37
75 – 79	6	77		43
80 – 84	4	82		47
85 – 89	3	87		50

Solution:

Q1 = L + (N/4 – F1) i            F N/4 = 50/4 = 12.5 Q1 = 54.5 (12.7 – 7)5               6 Q1 = 59.08

3rd Quartile:

Formula :

                                   

Q3 = L + (3N/4 – f1) i              F

Where :

F3  =  the cumulative frequency just above the 3rd quartile class.

Example :

Q3 = L + (3N/4 – f1) i                       F 3N/4 = 150/4 = 37.5 Q3 = 74.5 + (37.5 – 37) 5                              6             74.5 + (O.O833) 5             74.5 + O.4165             74.9165             =74.92 Q.D = Q3 – Q1/2 = 74.92 – 59.08/2 = 7.92

  Measure Of Dispersion (Cont.)

 3.    The standard deviation

A deviation is the difference between an individual value in a set of data and the mean for the data.

Standard Deviation averages the square of the distance that each piece of data is from the mean.

The smaller the standard deviation, the more compact the data set.

                                                          Standard Deviation – Population

                                                           

                                                                 Standard Deviation – Sample

Introduction Of Statistical Data (cont.)

Discrete – data can only take on certain individual values.

  Example:       

·         Number of pages in a book is a discrete data.

·         Shoe size is a Discrete data.  E.g.  5, 5½, 6, 6½ etc.  Not in between.

·         Number of people in a race is a discrete data.

   Continuous data – data can take on any value in a certain range.

  Example:

·         Length of a film is a continuous data.

·         Temperature is a continuous data.

·         Time taken to run a race is a continuous data. 

Sample – The subcollection data drawn from the population.

Population – The complete collection of all data to be studied.

Probability (cont.)

TREE DIAGRAMS

§  Independent events and their probabilities can be shown on a tree diagram.  Each event is represented by a branch

§  E.g.  A coin is flipped twice. Draw a tree diagram to show all the possible outcomes

EXAMPLE:

A coin is flipped twice. Draw a tree diagram to show all the possible outcomes.

Permutation and combination

Permutation:  A set of objects in which position (or order) is important. To a permutation, the trio of Brittany, Alan and Greg is DIFFERENT from Greg, Brittany and Alan.  Permutations are persnickety (picky). Combination:  A set of objects in which position (or order) is NOT important. To a combination, the trio of Brittany, Alan and Greg is THE SAME AS Greg, Brittany and Alan.                         Let's look at which is which: Permutation       versus       Combination 1. Picking a team captain, pitcher, and shortstop from a group. 1. Picking three team members from a group. 2.  Picking your favorite two colors, in order, from a color brochure. 2.  Picking two colors from a color brochure. 3.  Picking first, second and third place winners. 3.  Picking three winners.

Formulas:

A permutation is the choice of r things from a set of nthings without replacement and where the order matters.
Special Cases:
A combination is the choice of r things from a set of nthings without replacement and where order does notmatter.  (Notice the two forms of notation.)
Special Cases:

  

Example 1: 
Evaluate  :
                       

Notice how the cancellation occurs, leaving only 2 of the factorial terms  in the numerator.  A pattern is emerging ... when finding a combination  such as the one seen in this problem, the second value (2) will tell you  how many of the factorial terms to use in the numerator, and the  denominator will simply be the factorial of the second value (2).

Set theory

Set – is the collection of distinct objects

example:

SET SYMBOL

 

Probability

This problem asked us to find some probabilities involving a spinner. Let's look at some definitions and examples from the problem above.

Definition	Example
An experiment is a situation involving chance or probability that leads to results called outcomes.	In the problem above, the experiment is spinning the spinner.
An outcome is the result of a single trial of an experiment.	The possible outcomes are landing on yellow, blue, green or red.
An event is one or more outcomes of an experiment.	One event of this experiment is landing on blue.
Probability is the measure of how likely an event is.	The probability of landing on blue is one fourth.

In order to measure probabilities, mathematicians have devised the following formula for finding the probability of an event.

Probability Of An Event

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

Measure Of Dispersion

A measure of dispersion is a method of measuring the degree by which numerical data or values tend to spread from or cluster about central point of average.

The most common measures of dispersion are the following

1.  The Range

The simplest measure of dispersion.

Calculated by finding the difference between the greatest and the least values of the data.

Useful since it is the easiest to understand.

Affected by extreme data.

The range of values 1, 2, 4, 6, 9, 11, 15, 25 is 25 – 1 = 24

Ungrouped:

            R = H – L

= UB – LB

= 89.5 – 44.5

= 45

Grouped Data:

            R = UB – LB = Boundaries

Class Intervals	Frequency	Class Boundaries	Class Marks
45 – 49	3	44.5 + 49.5/2	47
50 – 54	4	49.5 + 54.5/2	52
55 – 59	6	54.5 + 59.5/2	57
60 – 64	7	59.5 + 64.5/2	62
65 – 69	10	64.5 + 69.5/2	67
70 – 74	7	69.5 + 74.5/2	72
75 – 79	6	74.5 + 79.5/2	77
80 – 84	4	79.5 + 84.5/2	82
85 – 89	3	84.5 + 89.5/2	87

Measure Of Central Tendency

Mean of a set of data:   

    1) Add all the values together       

    2) 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.

Examples:

•  Stan 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           

            = 47

Stan’s mean score is 47.

Median – the middle value in a set of data.

   •  First put the numbers in numerical order.          

            •  Example: 

                       Find the median of:         

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

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

                        Median = 5 

            b) 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 only be one median.   

            Median = 6

Mode – is the most common value in a set of data.

Find the mode of: 

   a) red, blue, yellow, red, green        

            Mode = red

           

   b)  4, 5, 6, 5, 7, 8, 5      

            Mode = 5     

It is possible to have 2 modes. 

•  E.g. 4, 7, 7, 8, 8, 5       

           Mode = 7, 8

           

It is possible to have no mode. 

 •  E.g. 4, 7, 8, 5, 6, 2