MEASURES OF VARIATION

Measures of variation – is a single value that is used to describe the spread of the scores in a distribution.

Types of Absolute MEASURES of VARIATION:

4 Kinds of Absolute Variation:

1.   Range – is the difference between the highest score and the lowest score in a distribution.

a.   Range for Ungrouped Data

Formula: R= HS-LS

Where: R – range value

        HS – highest score

        LS – lowest score

Example:Find the range of the two groups of score distribution

                   

Group A	Group B
10(LS) 12 15 17 25 26 28 20 35(HS)	15(LS) 16 16 17 17 23 25 26 30(HS)

R= HS-LS

R=35-10

R= 25

b.   Range for Grouped Data

Formula: R= HSUB – LSLB

         
 Where: R- range value

    HSUB – upper boundary of the highest score

    LSLB– lower boundary of the lowest score

  Example:Find the value of range of the scores of 50 students in Mathematics achievement test.

X	F
25-32	3
33-40	7
41-48	5
49-56	4
57-64	12
65-72	6
73-80	8
81-88	3
89-97	2
	N= 50

                  

Solution:

LL of the LS = 25
LSLB – 24.5

UL of the HS – 97

HSUB = 97.5

R= HSUB – LSLB

R= 97.5-24.5

R= 73

Properties of Range:

1.   It is quick and easy to understand.

2.   It is a rough estimation of variation.

3.   It is easily affected by the extreme scores.

Interpretation of the range Value:

          When the range value is large, the scores in the distribution are more dispersed, widespread or heterogeneous. On the other hand, when the range value is small the scores in the distribution are less dispersed, less scattered or homogeneous.

2.   Inter-quartile range (IQR) and Quartile Deviation (QD)

Inter-quartile Range – is the difference between the third quartile and the first quartile.

            IQR=Q3-Q1

Properties of Inter-quartile Range

1.   Reduces the influence of extreme values.

2.   Not as easy as to calculate as the range.

3.   Only considers the middle 50% of the scores in the distribution.

4.   The point of dispersion is the median value.

Quartile Deviation – indicates the distance we need to go above the median to include the middle 50% of the scores.

Formula: QD= Q3-Q1

                   2

Steps in Solving Quartile Range:

·         Solve for the value of Q1.

·         Solve for the value ofQ3.

·         Solve for the value of QD using the formulaQD= Q3-Q1

                                                                     2

A.   Quartile Deviation for Ungrouped Data

          Example:

         

x(scores) 6 8 10 12 12 14 15 16 20

Solve for the Q1:                    Solve for Q3:

N=9                                       Q₃       = [ ¾ n + (1- ¾) ]

Q1      = [¼ n + (1 – ¼)]                           = [ ¾ (9) +(1- ¾) ]

          = [¼ (9) + (1 – ¼)]                        = [ 27/4 + ¼ ]

          = [ 9/4 + ¾]                              = [28/4]

          = [12/4]                                  = 7^th score

          = 3^rd score                               = 15

          = 10

  IQR = Q₃-Q1                            QD = Q3-Q1

       = 15-10                                          2

       = 5                                          = 15-10

                                                 2

                                            = 5

                                              2

                                       QD = 2.5

 B.   QUARTILE DEVIATION OF GROUPED DATA

QD= Q3-Q1

                                   2

Example:

x	f	Cf<
25-32	3	3
33-40	7	10
41-48	5	15
49-56	4	19
57-64	12	31
65-72	6	37
73-80	8	45
81-88	3	48
89-97	2	50
	N=50

Solve for the value of Q₁.

             n/4 = 50/4 =12.5

            Q₁C = 41 – 48

            LL   = 41

            L_B   = 40.5                     
              Cfp = 10

            Fq   = 5

            c.i   = 8

Interpretation of IQR and QD

          The larger the value of the IQR or QD, the more dispersed the scores at the middle 50% of the distribution. On the other hand, if the IQR or QD is small, the scores are less dispersed at the middle 50% of the distribution. The point of dispersion is the median value.

3.   Mean deviation – measures the average deviation of the values from the arithmetic mean.

a.   MEAN DEVIATION FOR UNGROUPED DATA

Where:

          MD = mean deviation value

            X =  individual score

            x  = sample mean

            n = number of class

Steps in solving Mean Deviation for ungrouped Data:

1.      Solve for the mean value.

2.     Subtract the mean value from each score.

3.     Take the absolute value of the difference in step2.

4.     Solve for the mean deviation using the formula.

a.   MEAN DEVIATION FOR GROUPED DATA

Where:

          MD = mean deviation value

                  f = class frequency

                x_m = class mark or midpoint

                x   = mean value

                n = number of class

Steps in solving Mean Deviation for Grouped Data:

1.      Solve for the value of mean.

2.     Subtract the mean value from each midpoint or class mark.

3.     Take the absolute value of each difference.

4.     Multiply the absolute value and the corresponding class frequency.

5.     Find the sum of the results in step 4.

6.     Solve for the mean deviation using the formula.

     

Analysis:

          The mean deviation of the 40 scores of students is 10.63. this means that on the average, the value deviated from the mean of 33.63 is 10.63.

 4. Variance and the Standard Deviation

Variance - one of the most important measures in variation.

Steps in solving Variance for Ungrouped Data:

1.   Solve for the mean value.

2.   Subtract the mean value from each score.

3.   Square the difference between the mean and each score.

4.   Find the sum of the results in step3.

5.   Solve for the population variance or sample variance using the formula of ungrouped data.

a.   Variance for Ungrouped Data

b.   Variance of Grouped Data

 

Note: If the standard deviation is already solved, square the value of the standard deviation to get the variance.

Steps in solving Variance of Grouped Data:

1.   Solve for the mean value.

2.   Subtract the mean value from each midpoint or class mark.

3.   Square the difference between the mean value and midpoint or class mark.

4.   Multiply the squared difference and the corresponding class frequency.

5.   Find the sum of the step 4.

6.   Solve for the population variance or sample variance using the formula.

Standard Deviation – is the most important measures in variation. It is also known as the square root of the variance. It is the average distance of all the scores that deviates from the mean value.

Reflection

          Variation is a way to show how data is dispersed, or spread out. Several measures of variation are used in describing the variation scores: absolute measures of variation and relative measures of variation.

           Every learner has distinct characteristics that differs from any one. Some students are fast learner but some are not. To determine the gap between the learning ability of the children, measures of variance can be used. The result of your solving will determine the students who are lacking and the students who were good. It helps the teacher to identify to whom he/she must give more effort. 

         Handling diverse children is surely a headache. Its a real crap knowing that there are students who cannot keep up with the lessons and their peers but it is the job of the teachers in keeping the on the right track. Measures of variation is really a big help to the teachers to determine how close or how far is the distance of the scores of students in a certain test. This will give information that a teacher can use in his teaching process, on who to be given extra support and what to do ito ensure that no one is left behind.