ORGANIZING TEST SCORES FOR STATISTICAL ANALYSIS

Organizing Test Scores By OrderingOrdering refers to the numerical arrangement

of numerical observations or measurements.

2 Ways by which test scores can be arranged:1. Ascending order – from lowest to highest2. Descending order – from highest to lowest.

Example

The following are scores obtained by 10 students in their quizzes in Math for the first grading period:

110, 130, 90, 140, 85, 115, 125, 95, 135, 100

Organizing Test Scores By Ranking

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

Steps to Rank Test Scores

1. Arrange the scores from highest to lowest.

2. Assign serial numbers for each score. The last serial number has to correspond to the total number of scores arranged in descending order.

3. Assign the rank of 1 to the highest score; and the lowest rank to the lowest score.

4. In case, there are ties, get the average of the serial numbers of the tied scores

R = (SN1 + SN2 + SNk ... )

NTS

Where: R = rank

SN1 = serial number of the first score

SN2 = serial number of the second score

SNk = other serial number

NTS = number of tied scores

Example

Rank the following scores obtained by 20 first year high school students in spelling:

15 14 10 9 8 8 7 62

4 8 7 8 10 9 14 124

4 6

Organizing Test Scores through a Stem-and-Leaf Plot

In this method of organizing scores, a numerical score is separated into two parts, the stem which is the first leading digit of the scores, while the trailing digit is the leaf.

Procedures

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

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

3. For each score in the mass of data, write down the leaf numbers on the line labeled by the appropriate stem number.

Example

Construct a stem-and-leaf plot of the following periodic test results in Biology

30 74 80 57 32 40 55 59

31 77 82 59 90 51 54 62

33 46 65 49 92 69 66 41

42 50 68 48 57 71 68 59

60 53 49 63 53 44 81 68

STEM LEAF

3

4

5

6

7

8

9

0 1 2 3

0 1 2 4 6 8 9 9

0 1 3 3 4 5 7 7 9 9 9

0 2 3 5 6 8 8 8 9

1 4 7

0 1 2

0 2

Organizing Data by Means of a Frequency Distribution

Frequency Distribution is table showing the number of times a score occurs.

2 Types of Frequency Distribution

1. Single Value

2. Grouped Frequency Distribution

Preparing Single Value Frequency Distribution

Steps

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

2. Tally each score in the tally column

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

4. Sum up all the row total tally mark (n = ___)

ExamplePrepare a single values frequency distribution

for the spelling test of First High School students given below.

14 8 8 6 14 6 14

6 6 4 2 6 6 4

9 2 4 3 10 10 6

9 8 2 4 4 5 6

3 8

Preparing Grouped Frequency Distribution

1. Find the lowest and highest score.

2. Compute the range.

3. Determine the class size and class interval. Desired number of classes is from 10 to 15.

4. Determine the score at which the lowest interval should begin. It should be multiple of the class size.

5. Record the limits of all class intervals, with the highest score value at the top.

6. Tally raw score in the appropriate class intervals.

7. Convert each tally to frequency.

Prepare a grouped frequency distribution for the following scores obtained by 50 students in a periodical test in Mathematics

84 80 68 87 86 70 79 90 67 80

82 62 85 86 86 61 86 87 91 78

72 96 89 84 78 88 78 78 82 76

70 86 85 88 70 79 75 89 73 86

72 68 82 89 81 69 77 81 77 83

Setting Class Boundaries and Class Marks

Class boundary is the integral limit of a class.

The apparent limits of a class are comprised of an upper and lower limit.

The real or exact limits of scores in a class extend from one-half of the smallest unit of the measurement below the value of the score to one-half unit above.

Class mark is the midpoint of a class in grouped frequency distribution.

It is used when the potential score is to be represented by one value if other measures are to be calculated.

CM = (LL + UL)

2

Derived Frequencies from Grouped frequency Distribution

3 Types of Frequency Distributions that may derived from a frequency distribution table.

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

RF = (F/N) 100

2. Cumulative frequency distribution indicates the number of scores that lie above or below a class boundary.

2 Types

1. Less than cumulative frequencies are obtained by adding the successive frequencies from the bottom to the top of the distribution.

2. Greater than cumulative frequencies are calculated by adding the successive frequencies from the top to the bottom of the distribution.

Exercises 1. Rank the following test scores. Indicate on the space

opposite each score its appropriate rank.83 87 91 80 64 83 75 90 98 86 72 71 79 80 65 84 83 93 90 68

2. Construct a stem-and-leaf plot for the test scores above.

A group of 24 high students who took the entrance examination test obtained the following scores on numerical ability test:

26 21 29 32 24 17 23 29

17 20 26 23 21 7 28 25

14 23 18 16 18 31 32 27

Prepare a single value frequency distribution of these entrance examination scores

Prepare a grouped frequency distribution for the scores obtained by 40 first year high

school students in their unit test in Math.

68 65 71 64 69 66 66 60 71 67

67 63 66 65 69 74 66 64 62 62

65 63 67 66 72 68 62 70 63 60

67 62 72 64 68 65 61 61 64 75

Measures of Central Tendency

- A measure of central tendency is a typical value or a representative value of a set of data.

Example: Average grade The median score The most popular

Mean

Referred to as the average For a given set of data, the mean M is the

sum of all n values divided by the total frequency.

M = Sx/n

where, M = meanSx = sum of test scoresn = total number of test scores

1. Solve for the mean of 10 , 9, 8, 12, 15

2. Find the mean of the following test scores in Science: 15, 21, 16, 17, 20, 21, 17 and 19.

3. Calculate the mean of the following test scores: 45, 35, 20, 23, 43, 42, 40, 36

Exercise

The teacher gave five tests in Math. Bea got the following scores in the 1st four tests: 82, 76, 79, and 81. What must be her score in the 5th test so that her average is 80?

The test scores of eleven students are shown below. Find their average score.

45, 50, 55, 38, 39, 2, 48, 53, 50, 40, 52

Disadvantage

The mean is easily influenced by extreme values (very high or very low)

What to do?- Set aside or disregard the

extreme value & solve for the mean.

- Do not use the mean. Use other measures

The Weighted Mean

Incorporates into the formula the weight of each term.

Example: Find the average grade of each of two students whose grades in five subjects are as follows:

Subject Grade Units

(1) (2)

A 85 79 2

B 90 88 3

C 95 95 5

D 83 94 3

E 82 89 1

Weight as Percentage

Example:The grading scheme in Statistics is 30%

Quizzes, 10% R, 10% A/SW, 50% M.E. Find the grades of students A & B.

Q R A/SW M.E. A 93 95 96 98 B 90 85 80 95

Weight as Fraction

Example: A teacher gives one quiz and one long exam and would like to determine the weighted average of a student. The grades and the corresponding fractional weights are:

Grade (x) Fractional Weight (W)

Quiz 90 1/3

Exam 98 2/3

The Median

The median (Md) is the middle value in a set of observations arranged from highest to lowest or vice versa.

Example: Test scores of 15 junior students arranged from lowest to highest

14 14 15 15 15 16 16 17 17 17 18 18 19 20 20

Find the median of 8, 12, 5, 6, 13, and 15

Find the median of the set: 25, 28, 22, 20, 18, 23, 30, 24

Find the median score of 11 students in English test

6, 4, 9, 7, 3, 11, 12, 5, 10, 6, 8

The Mode

The mode (Mo) is the observation which occurs most often in a set of values. The value with the highest frequency.

Example: Find the mode of this set:

24 27 32 29 31 35 27 32 24 25 30 24

Find the mode of the following set of data:

6, 4, 9, 7, 3, 11, 12, 5, 10, 6, 8

Find the mode of the test scores of 15

students

15 15 15 14 15 14 19 19 17 16 14 18 20 20 19

Choosing the Measure of Central Tendency

The mean is appropriate for interval and ratio variables.

The mode is preferred for groups of data which do not tend to group around a central point.

The median best measures the central tendency of groups which contain extreme values.

Mean For Grouped Test ScoresFrequency-Class Mark Method1. Calculate the class mark of each class

interval2. Multiply each class mark by its

corresponding frequency3. Sum up the cross products of the class

mark and frequency of each class4. Count the number of cases or total number

of scores5. Plug into computational formula the values

obtained in step 3 and 4.

Using the Deviation Method

M = Sfcm/n

where: M = mean

f = frequency of a class

cm = class mark

n = total number of scores or cases

Sfcm = sum of the cross products of

the frequency and class mark

Finding the Mean for Grouped Test Scores

Score Class Mark Frequency

75-79

70-74

65-69

60-64

55-59

50-54

45-49

40-44

35-39

30-34

77

72

67

62

57

52

47

42

37

32

1

4

2

5

7

9

11

8

4

3

Deviation Method of Solving the Mean

M = AM + (Sfd/n)i

where M = meanAM = assumed meanf = class frequencyd = class deviation scoreSfd = sum of the cross products of class

frequency and deviation scorei = class sizen = total number of scores

Finding the Median from Grouped Data

1. Cumulate the class frequencies from the lowest to the highest interval

2. Compute the locator of the class containing the median by dividing the total number of scores by 2

3. Locate the cumulative frequency (CF) that approximate n/2

4. Locate the median class based on the CF in step 3. The median class is just above it

5. Determine the frequency of the median class (f)

6. Determine the class boundary of the median class

7. Determine the class size (i)

8. Substitute the obtained values into the formula

Md = L + (n/2 – CF)I

f

where: L = lower class boundary of the median class

n/2 = locator of the median class

n = total number of scores

CF = cumulative frequency before the median class

f = frequency of the median class

i = class size

Compute the mean and median of the test scores of second year high school students in Filipino.

Score Frequency

90 – 92

87 – 89

84 – 86

81 – 83

78 – 80

75 – 77

72 – 74

69 – 71

66 – 68

63 – 65

60 – 62

2

7

10

6

8

4

3

4

3

1

2

Mode for Grouped Data

Mo = 3Md – 2M

where: Mo = mode

Md = median

M = mean

The table below gives the score distribution of 90 students in a Statistics test

Score Frequency55 – 59

50 – 54

45 – 49

40 – 44

35 – 39

30 – 34

25 – 29

20 – 24

15 – 19

10 – 14

2

5

10

12

15

16

13

10

4

4