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 Dispersion (Cont.)  2.    The Quartile Deviation


Measure Of Dispersion (Cont.)  3.    The Standard Deviation

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

Introduction Of Statistical Data

n  Primary data – data you collect

–         Example :

·         Surveys

·         Focus groups

·         Questionnaires

·         Personal interviews

·         Experiments and observational study

n  Secondary data – data someone else has collected

–         Example :

·         County health departments

·         Vital Statistics – birth, death certificates

·         Hospital, clinic, school nurse records

·         Private and foundation databases

·         City and county governments

n  Qualitative Data - Can be separated into different categories that are distinguished by some nonnumeric characteristics

•       Deals with descriptions.

•       Data can be observed but not measured.

•       Colors, textures, smells, tastes, appearance, beauty, etc.

•       Qualitative → Quality

  n  Quantitative Data - Numbers representing counts or measurements

•       Deals with numbers.

•       Data which can be measured.

•       Length, height, area, volume, weight, speed, time, temperature, humidity, sound levels, cost, members, ages, etc.

•       Quantitative → Quantity

Example 1:  Oil Painting Qualitative data: ·         red/green color, gold frame ·         smells old and musty    ·         texture shows brush strokes of oil paint ·         peaceful scene of the country ·         masterful brush strokes

Example 1:  Oil Painting Quantitative data: ·         picture is 10" by 14” ·         with frame 14" by 18” ·         weighs 8.5 pounds     ·         surface area of painting is 140 sq. in. ·         cost $300

Example 2:  Latte Qualitative data: *robust aroma *frothy appearance * strong taste *glass cup

Example 2:  Latte Quantitative data: *12 ounces of latte *serving temperature 1500 F. *serving cup 7 inches in height *cost $4.95

Example 3:  Freshman Class Qualitative data: *friendly demeanors *civic minded *environmentalists *positive school spirit

Example 3:  Freshman Class Quantitative data: *672 students *394 girls, 278 boys *68% on honor roll *150 students accelerated in mathematics

Introduction of statistical data (Cont.)

Topics

Introduction of statistical data

-          Give examples to distinguish:

          o   Primary and secondary data

    o   Qualitative and Quantitative data

    o  Discrete and continuous data

-          Distinguish between sample and population

-          Identify various methods of choosing a sample from a population

-          Identify various methods of collecting data

Interpretation of statistical representations

-          Interpret pictograms, bar charts, line graphs, component bar charts, pie
      charts, histograms and tables

-          Use computer software where possible to interpret statistical data

-          Apply techniques of interpreting statistical data to solve work related problems

Measure of central tendency

-          Introduce the arithmetic mean, median and mode

-          Identify situations where each becomes an appropriate measure of central

       tendency

-          Determine from ungrouped data the arithmetic mean, the mode and the median

-          Determine from grouped data:
   o   The arithmetic mean

   o   The approximate value of the mode

   o   The approximate value of the median

-          Use computer software where possible to solve problems involving measures
      of central tendency

-          Use appropriate measures of central tendency to solve work related problems

Measure of dispersion

-          Understand and use difference measure of dispersion and variation (range, 

        interquartile range, standard deviation) in comparing and contrasting 
       sets of data

-          Determine the median value, the quartiles and the interquartile range of a set 
      of data

-          Use a box plot to represent the dispersion of data

-          Calculate the standard deviation from a set of ungrouped and grouped data

-          Use computer software where possible to solve work related problem
       involving measures of dispersion

-          Use appropriate measures of dispersion to solve work related problems

Probability

-          Understand the terms associated with probability

-          Calculate the probability of single event

-          Calculate the probability of combined events, including the use of possibility

       diagrams and tree diagrams

-          Distinguish between dependent and independent events

-          Calculate probabilities of mutually exclusive and independent events

-          Use of computer software to solve problem involving probability

-          Apply probability to solve work related problem

Permutation and combination

-          Understand the principles of permutation

-          Find the total number of different ways in which an arrangement can be done

-          Understand the principles of combination

-          Find the number of combinations (or selections) of n unlike objects taken r 
      at a time