DESCRIBING RELATIONSHIP

Correlation – another statistical method used in analyzing test results.  This is the tool that we are going to utilize if we want to determine the relationship or the association between scores of students in two different subjects. It refers to the extent to which the distribution are linearly related or associated between two variables.

Coefficient Correlation – extent of correlation is indicated numerically. The Coefficient Correlation (r_xy) also known as the Pearson Product Moment Correlation Coefficient in honor to Karl Pearson who developed the said formula.

          2 ways in identifying correlation between variables:

1.   Using the formula;

2.   Using the scatter point or scattergram.

Kinds of Correlation:

1.   Positive Correlation – high scores in distribution x are associated with high scores in distribution y. This means that as the value of x increases the value of y too increases or as the value of x decreases, the y values will also decrease.

2.   Negative correlation –high cores in distribution x are associated with low scores in distribution y. Low scores in high distribution x are associated with high scores in in distribution y. This means that as the value of x increases, the value of y decrease or when the values of x decrease, the value of y increase.

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3.   Zero Correlation – no association between scores in distribution x and in scores in distribution y. No single can be drawn that best fitted to all points as shown in the scattergram of zero correlation.

Pearson Product Moment Correlation:

R_xy=     (n) (Σxy) –(Σx)(Σy) 

     [(n)( Σx²) - (Σx)²] [(n)(Σy²) – (Σy)²]

Graph:

Scattergram of Correlation

Scattergram of Positive Correlation

 

 Scattergram of Negative Correlation

Scattergram of Zero Correlation

     Computation of Correlation Coeffecient Using Pearson r:

Analysis:

          The value of the correlation coefficient is r_xy = 0.93, which means that there is a very high positive correlation between the scores of 10 students in mathematics and in science. This means that students are good in mathematics are also good in science.

Spearman Rho Coefficient

          Another way of finding the correlation between two variables is the Spearman rho Correlation coefficient and is denoted by a Greek letter rho (ρ). The Spearman rho correlation coefficient (ρ) is a measure of correlation when the given sets of data are expressed in ordinal level of measurement rather than raw scores as in Pearson r.

Formula:

                    

                        ρ =1 -6 ∑D2

                             N(N2-1)

Where:

                   ρ = Spearman rho Correlation Coefficient value

                   D = difference between a pair of ranks

                   N = number of students/cases

Steps in Solving Spearman rho Correlation Coefficient:

   1.   Rank the scores in the distribution if raw scores are given.

   2.   Find the difference between each pair.

   3.   Square the difference.

   4.   Find the summation of the squared difference.

   5.   Solve the value of the Spearman rho Correlation Coefficient using its formula.

          REFLECTION:

                   There are many ways on how can we measure the relationship of different scores, test items and so on.

                  One way to measure the relationship between two variables such as score of students and different subjects is the correlation, Through this technique, teachers are able to recognize what are the possible reason about high performance in a test or low performance in a test. It can also measure the relationship between two groups of score.

MEASURES OF SKEWNESS

Measure of Skewness– describe the degree of departure of the scores from symmetry.

 Formula:

Skewness can be classified according to the skewness coefficient.

Ø  If the SK < 0, it is called POSITIVELY SKEWED distribution.

Ø  If the SK > 0, it is NEGATIVELY SKEWED distribution.

Ø  If the SK = 0, the scores are NORMALLY distributed.

POSITIVELY SKEWED or the skewed to the right is the distribution where the thin end tail of the graph goes to the right part of the curve.

NEGATIVELY SKEWED or skewed to the left is a distribution where the thin end tail of the graph goes to the left part of the curve.

NORMAL DISTRIBUTION – is a special kind of symmetric distribution and it represents some properties in mathematics. It is very important when comparing between scores and making statistical decisions.

Properties of Normal distribution:

1.   The curve has a single peak, meaning the distribution is unimodal.

2.   It is a bell-shape curve.

3.   It is symmetrical to the mean.

4.   The end tail of the curve can be extended indefinitely in both sides and the asymptotic to the horizontal line.

5.   The shaped of the curve will depend on the value of the mean and standard deviation.

6.   The total area under the curve is 1.0. Hence, the area of the curve in each side of the mean is 0.5.

7.   The probability between two given points in the curve is equal to the area between the two points.

4 types of Standard Scores:

1.   Z-scores – is used to convert a row score to standard score to determine how far a raw score lies from the mean in standard deviation units.

Where:

Analysis:

          The score of Ritz Glenn in Business Calculus is one unit standard deviation below the mean. His score, in Production Management is two units standard deviation above the mean. Therefore, we can conclude that Ritz Glenn performed better in Production Management than Business Calculus

 2.   T-scores – another type of standard score where the mean is 50 and the standard deviation is 10.

Analysis:

          Z-score 0f -1 is equivalent to a T-score of 40, and z-score of +2 is equivalent to a T-score of 70.  The negative value is eliminated in the T-score equivalent. Therefore, Ritz Glenn performed better in Production Management than in the Business Calculus due to higher value of T-score which is equal to 70.

3.  Standard nine (STANINE) – is a nine-point grading scale ranging from 1-9, 1 being the lowest and 9 the highest. Stanine grading is easier to understand than the other standard model.

4.   Percentile Ranks – indicates the percentages of scores that lies below a given scores. If the scores are normally distributed, percentile rank can be inferred from the standard score.

Formula:

Steps in Solving Percentile Ranks:

      1.   Arrange the test scores from highest to lowest.

      2.   Make a frequency distribution of each score and the number of students obtaining each score.

      3.   Find the cumulative frequency by adding the frequency in each score from the bottom upward.

      4.   Find the percentile rank in each score using the formula and the result as indicated in column 

          REFLECTION:

                 Skewness is really useful if we want to measure how many of our students did great in their exam and how many failed. On the other hand, normal distribution helps in comparing between scores and making statistical decisions.

MEASURES OF VARIATION

              Measures of variation is a single value that is used to describe the spread of the scores in a distribution. The term variation is also known as variability or depression. There are several ways of describing the variation of scores: absolute measures of variation and relative measures of variation.

TYPES OF ABSOLUTE MEASURES OF 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 1

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 2 :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= 5

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 formula QD= Q3-Q1

                                                                                                                     

 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.

Population Standard Deviation

Sample Standard Deviation

RELATIVE MEASURE of VARIATION:

          Coefficient of Variation – shows variation relative to the mean. It is used to compare two or more groups of distribution of scores.

Formula:

REFLECTION:

       Not all distribution of scores can be measured by just using the central tendency because central tendency does not actually describe distribution. But, with the help of variation, spread of scores are easily described.