Chapter 4 Analysis Interpretation of Assessment REsults 13B53C PDF

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CHAPTER 4 Analysis and Interpretation of Assessment Results

Introduction Statistics plays a very important role in assessing the performance of students, most especially in interpreting and analyzing their scores through assessment activities. Teachers should know how to utilize these data, particularly in decision-making. Hence, a classroom teacher should have the necessary background in statistical procedures in order for him to give a correct description, and interpretation of student’s performance in a certain test. This lesson is a review of the important tools needed in describing, analyzing and interpreting assessment results. The topics discussed in this module are presentations of data through textual, tabular and graphical, measures of tendency, measures of dispersion, measures of relative positions, other measure and the level of measurement.

Learning Outcomes

At the end of the module, the student should be able to: a. Interpret assessment results accurately and utilize them to help learners improve their performance and achievement; and b. Utilize assessment results to make informed-decisions to improve instruction.

Lesson 1 - Presentation The study of statistics begins with the collection of data or measurements. Data collected should be organized systematically for easier and faster interpretation. Data can be presented in three forms: textual, tabular, and graphical. The tabular and graphical forms are used when more detailed information about the data is to be presented. A table is used when you want to present a data in a systematic and organized manner so that reading and interpretation will be simpler and easier.

A.1 Textual Presentation Ungrouped data can be presented in textual form, as in paragraph form. This involves enumerating the important characteristics, giving emphasis on significant figures and identifying important features of the data.

Example 1. Below are the test scores of 50 students in Statistics: 25

30 43

18 35

17 40

50 9

12

33 37 46 28 19 27

41 10 18 28

21 36 13 31

20

31

35

28

16

42

40 48 13

40 3

39 50

32

32

26

41

Arranging the scores from the lowest to the highest will facilitate the enumeration of important characteristics of the data. The test scores of the 50 students in Statistics arranged from lowest to highest are shown below: 3

13 32

17 35

20 40

27 43

30

9

13 18 21 28 30 33 36 40 46 10 14 18 25 28 31 34 37 40 48 10 15 19 26 28 31 35 With the data now arranged according to magnitude, we can easily see the 38 41 50 important features worth mentioning in the text. One way of describing the data using the 12 16 20 26 29 32 35 textual form is as follows: 39 42 50 The highest score obtained is 50 and the lowest is 3. Ten students got a score of 40 and above, while only 4 got ten and below. Generally, the students performed well in the test with 33 students of 66% getting a score of 25 and above. Arranging a mass of data manually is quite tedious, but using computers for this purpose is so easy. In the absence of a computer, the process is made easy by putting the data in a stem-and-leaf plot. Stem-and-leaf plot is a table which sorts data according to a certain pattern. It involves separating a number into two parts. In a two-digit number, the stem consists of the first digit, and the leaf consists of the second digit. While in a three-digit number, the stem consists of the first two digits, and leaf consists of the last digit. In a one-digit number, the stem is zero.

Using the unarranged test scores in Statistics of 50 students as data, stem-and-leaf plot can be used to arrange from lowest to highest. The stems are as follows: 0, 1, 2, 3, 4, and 5, three being the lowest and 50 the highest. Table 1 Stem-and-Leaf Plot of Unarranged Test Scores in Statistics of 50 Students Stem Leaves 0 3, 9 1 0, 0, 2, 3, 3, 4, 5, 6, 7, 8, 8, 9 2 0, 0, 1, 5, 6, 6, 7, 8, 8, 8, 9 3 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9 4 0, 0, 0, 1, 2, 3, 6, 8 5 0, 0 looking at the stem-and-leaf plot, we can easily rank the data or put them in order. Thus, the ten lowest scores are: 3, 9, 10, 10, 12, 13, 13, 14, 15, and 16, while the ten highest scores are: 40, 40, 40, 41, 42, 43, 46, 48, 50, and 50.

By

A. 2. Tabular Method Sometimes, we cold hardly grasp information from a textual presentation data. Thus, we may present data by using tables. By organizing the data in tables, important features about the data can be readily 42

understood and comparisons can be easily made. Thus, a table shows complete information regarding the data. A table has the following parts:

1. Table number : This is for easy reference to the table. 2. Table title: It briefly explains the content of the table. 3. Column header: It describes the data in each column. 4. Row classifier: It shows the classes or categories. 5. Body: This is the main part of the table. 6. Source note: This is placed below the table when the data written are not original. Below is a table with all its parts indicated: Table Number

Table 2 Distribution of Students in XYZ High School According to Year Level

Row classifier

Table Title

Column Number of Students Header 300 250 Body 285 215 N = 1,050 Source Note Source: XYZ High School Registrar Year Level First Year Second Year Third Year Fourth Year

Another type of tabular presentation is the frequency table also known as a frequency distribution. It is an arrangement of the data that shows the frequency of occurrence of different values of the variables. A frequency distribution table is a table which shows the data arranged into different classes and the number of cases which fall into each class. The frequency distribution table for ungrouped data is simply an arrangement of data from lowest to highest which shows the frequency of occurrence of each value in a set. This is best used when the range of values is not too wide. Example: Table 3 Ungrouped Frequency Distribution for the Ages of 50 Students Enrolled in Statistics Age 14 15 16 17 18 18

Frequency 4 13 25 5 2 1 N = 50

1. Table number is ____. 2. Table title is Ungrouped Frequency Distribution fort he Ages of 50 Students Enrolled in Statistics 3. Column headers are: a) Age b) Frequency 4. Row classifiers: 14, 15, ..., 19 43

Notice that the range of the ages is 5, that is subtracting 14, the lowest age, from 19, the highest age. However, if the range is more than 15, the best way is to group the data into classes using the grouped frequency distribution table. The frequency distribution for grouped data is an arrangement of data into different classes or categories. It involves counting the data which fall into each class. Below are the steps in constructing a frequency distribution table: 1. Find the range of scores: Range = Highest score - Lowest score = 99 - 67 = 32 2. Decide on the number of class interval k Maximum = 20 Minimum = 7 Ideal = 10 - 15

k

Estimate: (rounded to the nearest whole number) where n, is the total number of scores or cases

k  40 6 Thus, for the above scores, or 6.

3. Determine the class size i of the interval.

44

Range 32 i   6 k or i = 5. (An odd value of i is preferred).

4. Determine the lower limit LL and upper limit of the lowest class interval (the class interval containing the lowest score). LL = score or number closest to but less than the lowest score and preferably a multiple of the class size i.

In the given set of scores, the lowest score is 67. The number closest to 67 that is divisible by the class size i = 5 is 65. Thus, LL = 65. UL = LL + (i - 1). Thus, UL = 65 + (5 - 1) = 69.

5. Determine the other class intervals by consecutively adding the class size i to LL and UL until the interval containing the highest score is contained and make a tally. Thus, Illustration: Performance ratings of government employees 76 67 99 82 86

92 85 95 86 93

87 93 79 83 98

Class Interval 95 - 99 90 - 94 85 - 89 80 - 84 75 - 79 70 - 74 65 - 69

78 91 85 87 71

87 85 81 79 81 Tally //// //// - /// //// - //// - // //// - // //// - / // /

88 79 96 92 86

85 92 75 80 80

92 82 88 74 94 f 4 8 12 7 6 2 1

The limits that define the class intervals as indicated above are called apparent limits. To reflect the continuity of scores, the true limits or class boundaries are indicated. These are obtained by adding all upper limits and subtracting all lower limits on-half of the difference between successive adjacent lower and upper limits. For this particular data, the number to be added and subtracted is 0.5 Other information usually included in a frequency distribution are:

45

LL  U 2 X = the class mark or class midpoint of the class interval =

cf = the greater than cumulative frequency = the frequency of the interval plus all frequencies above the interval

46

rf = the relative

f n

47

frequency of the interval =

The same frequency distribution with additional information as cited above is shown below: Table 4 Performance Rating of Government Employees

Class Interval f X rf cf 4 12 24 31 37 39 40

A.3 Graphical Presentation A graph is a diagram which makes a systematic presentation of a class frequency distribution together with comparison and relationship of the classes. As a graph is usually perceptible, it is easily understood. There are two most common methods for graphing frequency distribution: Histogram and the frequency polygon. Histogram represents a pictorial presentation of a frequency distribution. It may be thought of as a series of rectangles and frequencies, respectively. In histogram, the bases is equal to the length of the interval, and the height is equal to the frequency. It resembles a bar graph. Frequency polygon is another method of graphing frequency distribution. It is also pictorial but it is constructed by joining with straight lines a series of points which are the midpoints of the steps as against their corresponding frequencies. It looks like a zigzag line.

Lesson 2 - Quantitative Analysis and Interpretation 2.1 Levels of Measurement Statistics deals mostly with measurements. We define measurement as the assignment of symbols or numerals to objects or events according to some rules. Since different rules are used for the assignment of symbols, then this would yield different scales of measurement. There are four measurement scales, namely, nominal, ordinal, interval, and ratio. 1. Nominal Scale This is the most primitive level of measurement. The nominal level of measurement is used when we want to distinguish one object from another for identification purposes. In this level, we can say that one object is different from another, but the amount of difference between them cannot be determined. We cannot tell that one is better or worse than the other. Gender, nationality, and civil status are of nominal scale. 2. Ordinal Scale In the ordinal level of measurement, data are arranged in some specified order or rank. When objects are measured in this level, we can say that one is better or greater than the other. But we cannot tell how much more or how much less of the characteristic one object has than the other. The ranking of contestants in a beauty contest, of siblings in the family, or of honor students in the class are of ordinal scale. 3. Interval Scale If data are measured in the interval level, we can say not only one object is greater or less than another, but we can also specify the amount of difference. The scores in an examination are of the interval scale of measurement. To illustrate, suppose Maria got 50 in a Math examination while Martha got 40. We can say that Maria got higher than Martha by 10 points. 4. Ratio Scale The ratio level of measurement is like the interval level. The only difference is that the ratio level always starts from an absolute or true zero point. In addition, in the ratio level, there is always the presence of units of measure. If data are measured in this level, we can say that one object is so many times as large or as small as the other. For example, suppose Mrs. Reyes weighs 50 kg, while her daughter weighs 25 kg. We can say that Mrs. Reyes is twice as heavy as her daughter. Thus, weight is an example of data measured in the ratio scale. 2.2 Measures of Central Tendency 2.2.1 Measures of Central Tendency for Ungrouped Data 2.2.1.1 The Mean The mean (also known as the x arithmetic mean) is the most commonly used measure of central position. It is the sum of measures divided by the number of measures in a variable. It is symbolized as (read as x bar). The mean is used to describe a set of data where measures cluster or concentrate at a point. As the measure cluster around each other, a single value appears to represent distinctively the total measures. It is, however, affected by extreme measures, that is, very high or very low measures can easily change the value of the mean. To find the mean of ungrouped data, use the formula where = the summation of x (sum of the measure) N = number of values of x 49

Example: The grades in Chemistry of 10 students are 87, 85, 85, 86, 90, 79, 82, 78, 76. What is the average grade of the 10 students? Solution:

87  84  85  85  86  90  79  82  78  76  x 10 2.2.1.2 The Weighted Arithmetic Mean

xW 859 .  x Occasionally, we want to find the mean of a set of values wherein each value or measurement has a different weight or degree of importance. We call this the weighted mean and the formula for computing it is as follows:

where:

x W

120  123  83  162  80  127 80(1.5)  82(1.5)  83(1)  81(2)  80(1) 85   10 10 55 50

Subject

Units

Grade

2.2.1.3 The Median The median is the middle entry or term in a set of data arranged in either increasing or decreasing order. The median is a positional measure. Thus, the values of the individual measures in a set of data do not affect it. It is affected by the number of measures and not by the size of the extreme values. To find the median of a given set of data, take note of the following: 1. Arrange the data in either increasing or decreasing order. 2. Locate the middle value. If the number of cases is odd, the middle value is the median. If the number of cases is even, take the arithmetic mean of the two middle measures. Example 1: The number of books borrowed in the library from Monday to Friday last week were 58, 60, 54, 35, and 97 respectively. Find the median. Solution: Arrange the number of books borrowed in increasing order. 35, 54, 58, 60, 97 The median is 58. Example 2: Cora’s quizzes for the second quarter are 8, 7,6, 10, 9, 5, 9, 6, 10, and 7. Find the median. Solution: Arrange the scores in increasing order. 5, 6, 6, 7, 7, 8, 9, 9, 10, 10 Since the number of measures is even, then the median is the average of the two middle scores.

 Md  728 2.2.1.4 The Mode

The mode is another measure of position. The mode is the measure or value which occurs most frequently in a set of data. It is the value with the greatest frequency. To find the mode for a set of data 1. select measure that appears most often in the set; 2. if two or more measures appear the same number of items, and the frequency they appear is greater than any of other measures, then each of these values is a mode; 3. if every measure appears the same number of items, then the set of data has no mode.

51

1 4 2

52

Example 1: The shoe size of 10 randomly selected students in a class are 6, 5, 4, 6, , 5, 6, 7, 7 and 6. What is the mode? Answer: The mode is 6 since it is the shoe size that occurred the most number of times. Example 2: The sizes of 9 classes in a certain school are 50, 52, 55, 50, 51, 54, 55, 53 and 54. Answer: The modes are 54 and 55 since the two measures occurred the same number of times. The distribution is bimodal.

2.2.2 Measures of Central Tendency for Grouped Data 2.2.2.1 The Mean of Grouped Data Using the Class marks When the number of items in a set of data is too big, items are grouped for convenience. The manner of computing for the mean of grouped data is given by the formula:

Mean 

(

fX )

where: f is the frequency of each class X is the class mark of class

fX f   53

The Greek symbol (sigma) is the mathematical symbol for summation. This means that all items having this symbol are to be added. Thus, the symbol means the sum of all frequencies, and means the sum of all the products of the frequency and the corresponding class mark.

Class 46 - 50 41 - 45 36 - 40 31 - 35 26 - 30 21 - 25 16 - 20 11 -15

E es:

Class 46 - 50 41 - 45 36 - 40 31 - 35 26 - 30 21 - 25 16 - 20 11 - 15

f 1 5 11 12 11 5 2 1

Frequency 1 5 11 12 11 5 2 1

X 48 43 38 33 28 23 18 13

Compute the mean of the scores of the students in a Mathematics

fX 48 215 418 396 308 115 36 13

xampl

test.

The frequency distribution for the data is given below. The columns X and fX are added.

54

f



4

fX 1,5 55

Mean

   1,5

Mean 

4 56

M

The mean score is 32.27.

2.2.2.2 The Mean of Grouped Data Using the Coded Deviation

32

An alternative formula for computing the mean of grouped data makes use of coded deviation.

 Mean  A.M .     where: A.M. is the assumed mean f is the frequency of each class d is the coded deviation from A.M. i is the class interval

Any class mark can be considered as assumed mean. But it is convenient o choose the class mark with the highest frequency. The class chosen to contain A.M. s given a 0 deviation. Subsequently, consecutive positive integers are assigned to the classes upward and negative integers to the classes downward. This is illustrated in the next examples using the same data in the previous example. Examples: Compute the mean of the scores of the students in Mathematics test. Class 46 - 50 41 - 45

Frequency 1 5

7

36 - 40 31 - 35 26 - 30 21 - 25 16 - 20 11 - 15

11 12 11 5 2 1

The frequency distributor for the data is given below. The columns X, d and fd are added. Class 46 - 50 41 - 45 36 - 40 31 - 35 26 - 30 21 - 25 16 - 20 11 - 15 Solution:

f 1 5 11 12 11 5 2 1

X 48 43 38 33 28 23 18 13

d 3 2 1 0 -1 -2 -3 -4

fd 3 10 11 0 -11 -10 -6 -4

 i 5

fd   A Mf 4 3  ,,,

58

 fd  Mean  A.M .     i   f 

33   5 2,750     30   =

2,750  5 Mean 3,3 The mean gross sale is Php3,300.

2. 2.2.3 The Median of Grouped Data The median is the middle value in a set of quantities. It separates an ordered set of data into two equal parts. Half of the quantities found above the median and the other half is f

where: lbmc is the lower boundary of the median class f is the frequency of each class cf is the cumulative frequency of the lower class next to the median class fmc is the frequency of the median class i is the class interval

The median class is the class that median must be within th...