Analyzing and Analyzing and Using Test Item Using Test Item

DataData

It refers to the process of examining It refers to the process of examining the students response to each item the students response to each item in the test.in the test.

Purposes and Elements of Item Purposes and Elements of Item AnalysisAnalysis

1.1. To select the best available items for the final form of the To select the best available items for the final form of the testtest

2.2. To identify the structural defects in the itemsTo identify the structural defects in the items

3.3. To detect learning difficulties of the class as a whole; andTo detect learning difficulties of the class as a whole; and

4.4. To identify the areas of weaknesses of the students in To identify the areas of weaknesses of the students in need of remediationneed of remediation

Characteristics of an itemCharacteristics of an item

Desirable characteristics can be Desirable characteristics can be retain for subsequent useretain for subsequent use

Undesirable characteristics is either Undesirable characteristics is either to revised or rejectedto revised or rejected

Three main Elements in an Three main Elements in an Item-AnalysisItem-Analysis

1.Difficulty level of the items1.Difficulty level of the items

2. Discrimination power of each item2. Discrimination power of each item

3. Examination of the effectiveness of 3. Examination of the effectiveness of distractersdistracters

Difficulty indexDifficulty index - refers to the - refers to the proportion of the number of students proportion of the number of students in the upper and lower groups who in the upper and lower groups who answered an item correctly.answered an item correctly.

Therefore it can be obtain by Therefore it can be obtain by adding the proportion in the upper adding the proportion in the upper and lower groups who got the item and lower groups who got the item right and divide it by 2.right and divide it by 2.

Index of Discrimination-Index of Discrimination- it is the it is the percentage of high-scoring percentage of high-scoring individuals responding correctly vs. individuals responding correctly vs. the number of low-scoring individuals the number of low-scoring individuals responding correctly to an item.responding correctly to an item.

• Maximum Positive Discriminating Power Maximum Positive Discriminating Power of an itemof an item – it is indicated by an index of 1.00 – it is indicated by an index of 1.00 and is obtain when all the groups answered and is obtain when all the groups answered correctly and no one in the lower group did.correctly and no one in the lower group did.

• Zero Discriminating powerZero Discriminating power – is obtain when – is obtain when an equal number of students in both groups an equal number of students in both groups got the item rightgot the item right

• Negative Discriminating Power of an itemNegative Discriminating Power of an item – it is obtain when more students in the lower – it is obtain when more students in the lower group got the item right than in the upper group got the item right than in the upper group.group.

     

Measures of attractivenessMeasures of attractiveness..

To measure the attractiveness of To measure the attractiveness of the incorrect option in a multiple the incorrect option in a multiple choice test, we count the number of choice test, we count the number of the students who selected the the students who selected the incorrect option in both the upper incorrect option in both the upper and lower groups. The incorrect and lower groups. The incorrect options should attract less of the options should attract less of the upper group than the lower group.upper group than the lower group.

PREPARING DATA FOR ITEM PREPARING DATA FOR ITEM ANALYSISANALYSIS

1.1. Arrange test scores from highest to lowest.Arrange test scores from highest to lowest.2.2. Get one-third of the papers from highest Get one-third of the papers from highest

scores and the other one-third from the scores and the other one-third from the lowest scores.lowest scores.

3.3. Record separately the number of times Record separately the number of times each alternative was chosen by the each alternative was chosen by the students in both groups.students in both groups.

4.4. Add the number of correct answer to each Add the number of correct answer to each item made by the combined upper and item made by the combined upper and lower groups.lower groups.

5. 5. Compute the index of difficulty for each item,Compute the index of difficulty for each item,

index of difficulty = index of difficulty = No. of students responding correctly to an itemNo. of students responding correctly to an item x 100 x 100

Total no. of students in the upper and lower groupsTotal no. of students in the upper and lower groups

6. 6. Compute the index of discriminationCompute the index of discrimination

index of discrimination=index of discrimination=UpperUpperncrncr – – LowerLowerncrncr

No. of students per groupNo. of students per group

Difficulty of a test item can be interpreted with Difficulty of a test item can be interpreted with the use of...the use of...

Range Difficulty Range Difficulty

LevelLevel 20 & below very difficult20 & below very difficult

21-40 difficult21-40 difficult 41-60 average41-60 average

61-80 easy61-80 easy 81-above very easy81-above very easy

Discrimination IndexDiscrimination Index

Range Verbal DescriptionRange Verbal Description

0.40 and above very good item0.40 and above very good item

0.30-0.39 good item 0.30-0.39 good item

0.20-0.29 fair item0.20-0.29 fair item

0.09-0.19 poor item0.09-0.19 poor item

CORRELATING TEST CORRELATING TEST SCORESSCORES

CORRELATION- the relationship CORRELATION- the relationship between two or more paired-factorsbetween two or more paired-factors

or two or more sets of tests scores or two or more sets of tests scores

CORRELATION COEFFICIENT- a CORRELATION COEFFICIENT- a numerical measure of the linear numerical measure of the linear relationship between two factors on relationship between two factors on sets of scoressets of scores

Obtained Correlation Obtained Correlation coefficient can be interpreted coefficient can be interpreted

with the use of….with the use of….

Correlation Coefficient Degree of RelationshipCorrelation Coefficient Degree of Relationship

0.00-0.20 negligible0.00-0.20 negligible

0.21-0.40 low0.21-0.40 low

0.41-0.60 moderate0.41-0.60 moderate

0.61-0.80 substantial0.61-0.80 substantial

0.81-1.00 high to very high0.81-1.00 high to very high

Pearson’s Product-Moment Pearson’s Product-Moment CorrelationCorrelation

1.1. Compute the sum of each set of scores (SX.SY).Compute the sum of each set of scores (SX.SY).2.2. Square each score and sum the squares Square each score and sum the squares

(SX(SX22 ,SY ,SY22 ). ).3.3. Count the number of scores in each group (N).Count the number of scores in each group (N).4.4. Multiply each X score by its corresponding Y Multiply each X score by its corresponding Y

score.score.5.5. Sum the cross product of X and Y (SXY).Sum the cross product of X and Y (SXY).6.6. Calculate the correlation, following the formula:Calculate the correlation, following the formula:

Spearman RhoSpearman Rho

1.1. Rank the scores in distribution X, Rank the scores in distribution X, giving the highest score a rank of 1.giving the highest score a rank of 1.

2.2. Repeat the process for the scores in Repeat the process for the scores in distribution Y.distribution Y.

3.3. Obtain the difference between the Obtain the difference between the two sets of ranks (D).two sets of ranks (D).

4.4. Square each of these differences and Square each of these differences and sum up squared differences (SDsum up squared differences (SD2 2 ) )

5. 5. Solve for Solve for Rho Rho following the formula:following the formula:

RhoRho=1-=1-{{6 SD6 SD22 }} {N{N33 –N} –N}

Where: rho= rank- order correlation coefficientWhere: rho= rank- order correlation coefficient D= difference between paired ranksD= difference between paired ranks

SDSD2 2 = sum of squared differences between = sum of squared differences between paired rankspaired ranks

N= No. of paired ranksN= No. of paired ranks

Organizing Test Scores for Organizing Test Scores for Statistical AnalysisStatistical Analysis

1.Organizing test scores by ordering1.Organizing test scores by ordering2.Organizing test scores by ranking2.Organizing test scores by ranking3.Organizing test scores through a 3.Organizing test scores through a

stem- and leaf plotstem- and leaf plot4.Organizing data by means of a 4.Organizing data by means of a

frequency distributionfrequency distribution

Preparing Single Value Frequency DistributionPreparing Single Value Frequency Distribution 1. Arrange the scores in descending order. List 1. Arrange the scores in descending order. List

them in the X column of the table.them in the X column of the table.2. Tally each score in the tally column.2. Tally each score in the tally column.3. Add the tally marks at the end of each row. 3. Add the tally marks at the end of each row.

Write the sum in the frequency column.Write the sum in the frequency column.4.4. Sum up all the row total tally marks Sum up all the row total tally marks

(N=___).(N=___).

Shapes of the frequency Shapes of the frequency PolygonsPolygons

1.1. Normal- bell- shaped curve.Normal- bell- shaped curve.

2.2. Positive skewed- most scores are below Positive skewed- most scores are below the mean and there are extremely high the mean and there are extremely high scores. (mean is greater than the scores. (mean is greater than the mode)mode)

3.3. Negatively skewed- most scores are Negatively skewed- most scores are above the mean and there are above the mean and there are extremely low scores. (mean is lower extremely low scores. (mean is lower than the mode).than the mode).

4. Leptokurtic- highly peaked and the 4. Leptokurtic- highly peaked and the tails are more elevated above the tails are more elevated above the baseline.baseline.

5. Mesokurtic- moderately peaked 5. Mesokurtic- moderately peaked

6. Platykurtic- flattened peak6. Platykurtic- flattened peak

7. Bimodal curve- curve with two 7. Bimodal curve- curve with two peaks or modepeaks or mode..

8. Polymodal curve- curve with three 8. Polymodal curve- curve with three or more modesor more modes

9. Rectangular Distribution- there is no 9. Rectangular Distribution- there is no modemode

Skewness- degree of symmetry of Skewness- degree of symmetry of the scoresthe scores

kurtosis – degree of peakness or kurtosis – degree of peakness or flatness of the distribution curveflatness of the distribution curve

Sk= Sk= 3( M –Md)3( M –Md)

SDSD

K = K = QQ

(P90 – P10)(P90 – P10)• Normal distribution – 0.263Normal distribution – 0.263• Platykurtic - > 0.263Platykurtic - > 0.263• Leptokurtic - < 0.263Leptokurtic - < 0.263

Organizing Test Scores Organizing Test Scores for Statistical Analysisfor Statistical Analysis

Organizing Test Scores By Organizing Test Scores By OrderingOrdering

Ordering refers to the numerical Ordering refers to the numerical arrangement of numerical arrangement of numerical observations or measurements.\observations or measurements.\

There are two ways of ordering:There are two ways of ordering:

1.1. Ascending OrderAscending Order

2.2. Descending OrderDescending Order

the following are the scores obtained the following are the scores obtained by 10 students in their quizzes in by 10 students in their quizzes in English for the first grading students.English for the first grading students.

A B C D E F G H I J

110 130 90 140 85 87 115 125 95 135

ASCENDING AND DESCENDING ASCENDING AND DESCENDING ORDER respectivelyORDER respectively

85 87 90 95 110 115 125 130 135 140

140 135 130 135 125 110 95 90 87 85

Organizing test scores by Organizing test scores by rankingranking

Ranking is another way by which test Ranking is another way by which test scores can be organized.scores can be organized.

It is process of determining the It is process of determining the relative position of scores, measures relative position of scores, measures of values based on magnitude, of values based on magnitude, worth, quality, or importance,worth, quality, or importance,

Steps in ranking test scores:Steps in ranking test scores:

Arrange the test scores from highest to Arrange the test scores from highest to lowestlowest

Assign serial number for each score.Assign serial number for each score. Assign the rank of 1 to the highest score Assign the rank of 1 to the highest score

and the lowest rank to the lowest score.and the lowest rank to the lowest score. In case there are ties, get the average In case there are ties, get the average

of the serial numbers of the tied scores.of the serial numbers of the tied scores.R= R= ( SN1 + SN2 + SN3 .... SN N)( SN1 + SN2 + SN3 .... SN N)

S TSS TS

Example: Rank the following scores Example: Rank the following scores obtained by 20 ist year high school obtained by 20 ist year high school students in spelling.students in spelling.

Find the rank of 12, 8,7, andFind the rank of 12, 8,7, and15 14 10 9 8

8 7 6 2 4

4 8 7 8 10

9 14 12 4 6

Organizing Test Scores Through A Organizing Test Scores Through A stem and Leaf Plotstem and Leaf Plot

It is a method of graphically sorting and It is a method of graphically sorting and arranging data to reveal its distribution.arranging data to reveal its distribution.

It is a method of organizing a scores, a It is a method of organizing a scores, a numerical score is separated into two numerical score is separated into two parts, usually the first one or two digits parts, usually the first one or two digits and the other digits.and the other digits.

The stem is the first leading digit of the The stem is the first leading digit of the scores while the trailing digit is the leafscores while the trailing digit is the leaf

Scores

SN Rank Scores

SN Rank

15 1 1 8 11 10.5

14 2 2.5 8 12 10.5

14 3 2.5 7 13 13.5

12 4 4 7 14 13.5

10 5 5.5 6 15 15.5

10 6 5.5 6 16 15.5

9 7 7.5 4 17 18

9 8 7.5 4 18 18

8 9 9.5 4 19 18

8 10 9.5 2 20 20

Procedures:Procedures:

Split each numerical score or value into Split each numerical score or value into two sets of digit. The first or leading set two sets of digit. The first or leading set of digits is the stem, and the second or of digits is the stem, and the second or trailing set of digits is the leaftrailing set of digits is the leaf

List all possible stem digits from lowest List all possible stem digits from lowest to highest.to highest.

For each score in the mass of data, For each score in the mass of data, write down the leaf numbers on the line write down the leaf numbers on the line labelled by the appropriate stem labelled by the appropriate stem numbernumber

Illustrate the stem and leaf plot on Illustrate the stem and leaf plot on the following periodical test results the following periodical test results

in biology.in biology.

30 74 80 57 32

31 77 82 59 90

33 46 65 49 92

42 50 68 48 57

Organizing Data by means of Organizing Data by means of frequency distributionfrequency distribution

Preparing Single value Frequency Preparing Single value Frequency DistributionDistribution

1. Arrange the scores in descending order. 1. Arrange the scores in descending order. List them in the x column of the table.List them in the x column of the table.

2. Tally each score in the tally column.2. Tally each score in the tally column.

3. Add the tally marks at the end of each 3. Add the tally marks at the end of each row. Write down the sum in the row. Write down the sum in the frequency column.frequency column.

4. Sum up all the row total tally marks4. Sum up all the row total tally marks

Prepare a single value frequency Prepare a single value frequency distribution for the spelling test distribution for the spelling test

scores of grade 3 pupilsscores of grade 3 pupils

14 2 6 8 8 6 6 9 8 6

4 2 14 9 4 6 2 4 14 4

5 6 3 6 6 10 10 4 3 8

Preparing Group Frequency Preparing Group Frequency DistributionDistribution

StepsSteps Find the lowest and the highest score.Find the lowest and the highest score. Compute the range.Compute the range. Determine the class intervalDetermine the class interval Determine the score at which the Determine the score at which the

lowest interval should begin. lowest interval should begin. Record the limits of all class intervalRecord the limits of all class interval Tally the raw scores in the appropriate Tally the raw scores in the appropriate

class intervalclass interval Convert each tally to frequency.Convert each tally to frequency.

Setting the class boundaries and Setting the class boundaries and class limitsclass limits

Class boundary is the integral limit of Class boundary is the integral limit of a class. These integral limit should be a class. These integral limit should be apparent or real.apparent or real.• The apparent limits of a class are The apparent limits of a class are

comprised of an upper and lower limitcomprised of an upper and lower limit Class mark is the midpoint of a class Class mark is the midpoint of a class

in a grouped frequency distribution.in a grouped frequency distribution.• It is used when the potential score is to It is used when the potential score is to

be represented by one value if other be represented by one value if other measures are to be calculatedmeasures are to be calculated

Derived Frequencies From Derived Frequencies From Grouped Frequency DistributionGrouped Frequency Distribution

Relative frequency distribution Relative frequency distribution indicates what percent of scores falls indicates what percent of scores falls within each of the classes.within each of the classes.

RF = ( F/N) 100RF = ( F/N) 100

..

Computation of relative Computation of relative frequencyfrequency

Class interval

frequency

Relative Frequen

cy

75-77 1 2.5

72-74 3 7.5

69-71 5 27.5

66-68 12 30

63-65 11 25.5

60-62 8 20

40 100

Types:Types:1.1. <cf- are obtained by adding the <cf- are obtained by adding the

successive frequencies from the bottom successive frequencies from the bottom to the top of the distributionto the top of the distribution

2.2. >cf- are obtained by adding the >cf- are obtained by adding the frequencies from top to bottomfrequencies from top to bottom

Cumulative Frequency distribution indicates the

number of scores that lie above or below a class

boundary

Computation of <cf and >cfComputation of <cf and >cf

Class interval

frequency

<cf >cf

75-77 1 40 172-74 3 39 469-71 5 36 966-68 12 31 2163-65 11 19 3260-62 8 8 40

40

Measures of Central Measures of Central TendencyTendency

1. MEAN1. MEAN

It is often called arithmetic average.It is often called arithmetic average.

2. Median2. Median

It is the score that occurs at a point It is the score that occurs at a point on the scale below which 50 % of the on the scale below which 50 % of the scores fall and above which the other scores fall and above which the other 50 % of the scores occur.50 % of the scores occur.

3. Mode3. Mode

It is the most recurring score in a set It is the most recurring score in a set of test scoresof test scores

Measure of DispersionMeasure of Dispersion

To determine the size of the To determine the size of the distribution of the test scores distribution of the test scores

or the portion of it.or the portion of it.

RangeRange

It is the simplest and the easiest It is the simplest and the easiest measure of dispersion.measure of dispersion.

It simply measure how far the It simply measure how far the highest score from the lowest scorehighest score from the lowest score

It is considered as the least It is considered as the least satisfactory measure of dispersionsatisfactory measure of dispersion

For ungrouped data we have:For ungrouped data we have:

R= Hs - LsR= Hs - Ls

ExampleExample

Determine the range of the test Determine the range of the test score of nine students in a score of nine students in a community development course test.community development course test.

Sol: R = 43-19 = 24Sol: R = 43-19 = 24

For Grouped DataFor Grouped Data

R= Hmdpt – LmdptR= Hmdpt – Lmdpt

Compute the range of the following Compute the range of the following frequency distribution of the test frequency distribution of the test

scores in Mathscores in MathClass interval Frequency

60-64 1

55-59 5

50-54 4

45-49 5

40-44 7

35-39 8

30-3425-2920-2415-19

4321

R = 62- 17 R = 62- 17

= 45= 45

Interquartile rangeInterquartile range

It is the range of the score of It is the range of the score of specified group usually the middle specified group usually the middle 50% of the cases lying between Q1 50% of the cases lying between Q1 and Q3and Q3

IQR = Q3-Q1IQR = Q3-Q1

ExampleExample

Determine the interquartile range of Determine the interquartile range of the test score of nine students in a the test score of nine students in a community development course test.community development course test.

Sol: R = 38-23 = 15Sol: R = 38-23 = 15

For Grouped Data:For Grouped Data:

IQR = Q3-Q1IQR = Q3-Q1

Compute the inter quartile range of Compute the inter quartile range of the following frequency distribution the following frequency distribution

of the test scores in Mathof the test scores in MathClass interval Frequency

60-64 1

55-59 5

50-54 4

45-49 5

40-44 7

35-39 8

30-3425-2920-2415-19

4321

IQR = 49.5 – 34.5IQR = 49.5 – 34.5

= 15= 15

The quartile DeviationThe quartile Deviation

It devides the difference of the 3It devides the difference of the 3rdrd and 1and 1stst quartile into two. quartile into two.

It is the average distance from the It is the average distance from the median to the two quartilesmedian to the two quartiles

QD = QD = Q3- Q2Q3- Q2

22

ExampleExample

Determine the quartile deviation of Determine the quartile deviation of the test score of nine students in a the test score of nine students in a community development course test.community development course test.

Sol: 15 / 2 = 7.5Sol: 15 / 2 = 7.5

Compute the quartile deviation of Compute the quartile deviation of the following frequency distribution the following frequency distribution

of the test scores in Mathof the test scores in MathClass interval Frequency

60-64 1

55-59 5

50-54 4

45-49 5

40-44 7

35-39 8

30-3425-2920-2415-19

4321