A Written Report on Measurement and Evaluation

8/2/2019 A Written Report on Measurement and Evaluation

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A Written Report on

Measurement andEvaluation

Measures of Central Tendency:

Their Procedures Use andInterpretation

Submitted by:

Apoloan, Karen

Paray, Jenibel

III- BSPT

Submitted to:

Prof. Rafael Panganiban



I. ContentIntroduction

Educational data, such as test results, can be organized, analyzed andinterpreted through the use of simple statistical measures —measures thatcan easily reveal the quality of students’ performance. As Santrock (2004)puts it, “a measure of central tenden cy is a number that provides informationabout the average of typical score on a set of data”. It has been anestablished fact, that when a group of students is tested, the distribution ofscores portrays a picture so that only very few students get high or lowscores; most of the scores have the tendency to cluster at the middle or at thecenter. The point where the score cluster in the distribution is known as the measure of central tendency . This is where the density of scores of theexaminees lies or the point of central location of the group.

The tendency of students’ scores to cluster at a certain point in a distributionis valuable to teachers. This tendency can reveal the quality and quantity of agroup’s performance. Teachers can interpret individua l score as below, aboveor within the central tendency. Their interpretation depends upon their needs.

A. The MeanThe mean is nothing more than the average of a group of scores. It isprocedurally defined as “the sum of a set of values divided by the number ofvalues in the set”. It is the most commonly used measure of central tendency.

The mean has several characteristics that make it the measure of centraltendency most frequently used. One of these characteristics is stability. Sinceeach score in the distribution enters into the computation of the mean, it is morestable over time than other measures of central tendency, which consider onlyone or two scores.

Another characteristic of the mean is that it is affected by extreme scores. Thismeans that a few very high scores in a distribution composed of low scores (apositively skewed distribution) or a very few low scores in a distributioncomposed of primarily high scores (a negatively skewed distribution) will “pull”the value of the mean down or up toward the extreme scores. Since mean isaffected by extreme scores, it is usually not the measure of choice when dealingwith skewed distribution. In such case, a measure that is more resistant toextreme scores is required.



Other characteristics of the mean are as follows:

1. The mean is the arithmetic average of the measurements.

2. It lies between the largest and smallest measurements of a set of test scores.

3. There is only one mean in a set of test scores.

Mean for Ungrouped Test Scores . When the test scores are ungrouped, that is N is30 or less mean is computed following the formula :

Formula Plain English Version

Where:

M = mean

∑X = sum of the test scores

N = total number of test scores

Example

The following scores were obtained by ten Fourth Year High School Students in a

Physics quiz: 12, 11, 10, 9, 7, 15, 8, 6, 14, and 13. What is the mean score of the pupilsin the aforementioned Physics quiz?

Solution:

=

=

M = 10.5

Mean for Grouped Test Scores. When test scores are more than 30, the abovementioned computational formula is no longer applicable. There are two ways ofcomputing the mean for grouped test scores: frequency-class mark method , and thedeviation method .



To compute the mean using the frequency-class mark method , the following stepshave to be observed:

4. Calculate the class mark or midpoint of each class interval.

5. Multiply each class mark by its corresponding frequency.6. Sun up the cross products of the class mark and frequency of each class.

7. Count the number of cases or total number of scores.

8. Plug into the computational formula the values obtained in steps 3 and 4. Theformula to be applied is given below:

M = ∑fcm / N

Where : M = the mean

f = frequency of a class

cm = class mark or midpoint of a class

N = total number of scores or cases

∑fcm = sum of the cross products of the frequency and classmark.

Example:

The table below shows how the mean for grouped data is composed using thefrequency-class mark method..

Computation of the Mean Via the Frequency-Class Mark Method

Classes Frequency (f) Class mark (cm) Fcm46-5041-4536-4031-35

26-3021-2516-2011-15

579

10

8644

N= 53

48433833

28261813

240301342330

2241387252

∑fcm



Solution:

M = ∑fcm/N

= 1699/53

M = 32.06

Another method of computing the mean for grouped data is the deviation method. The steps in calculating the mean using this method are as follows:

1. Select a class from the grouped frequency distribution that shall be yourarbitrary origin.

2. Assign 0 deviations to the selected class as starting point. Above 0, all deviationscores shall be consecutive positive numbers. Below 0 all deviation scores shallbe consecutive negative numbers.

3. Multiply each deviation score by its corresponding class frequency to obtain fd .

4. Sum up algebraically the cross products of each class frequency and deviationscore to get ∑fd.

5. Determine the assumed mean (AM). The assumed mean is the class mark ofthe class with 0 deviation.

6. Count N or the total number of scores and determine class size ( i ).

7. Substitute the values into the following computational formula to get the mean:

M = AM + (∑fd/N) i

Where: M = mean

AM = assumed mean

f = class frequency

d = class deviation score

∑fd = sum of the class products of the class frequency anddeviation score

i = class size

N = total number of scores



Example 1 :

The table below shows how the mean for grouped data is composed using thedeviation method .

Classes Frequency (f) Deviation Score fd46-5041-4536-4031-3526-3021-2516-2011-15

579

108644

N= 53

76543210

35424540241240

∑fd= 202

M = AM + ( Sfd / N) i

= 13 + (202/53) 5

= 13 + (1010/53)

= 3 + 19.06

M = 32.06

Example 2:

Computation of the Mean via the Deviation Method with the Highest Class Interval asthe Point Of Origin

Classes Frequency (f) Deviation Score fd46-5041-4536-4031-35

26-3021-2516-2011-15

579

10

8644

N= 53

0-1-2-3

-4-5-6-7

0-7

-18-30

-32-30-24-28

∑fd= -169



Solution

M = AM + (∑fd / N) i

= 48 + (-169/53) 5

= 48 + (-845/53)

= 48 + (- 15.94)

M = 32.06

The foregoing clearly shows that the same answer can be obtained whether the startingpoint is the lowest, middlemost, or highest class interval.

B. The Median

Definition:The second most frequently encountered measure of central tendency.

The median is the score that splits a distribution in half: 50% of the scores lieabove the median and 50% of the score lie below the median. Thus the medianis also known as the 50 th percentile. It is the midpoint of a distribution of scores.

Characteristics: 1. The median is the central value; 50% of the test scores lie above it and 50%

fall below it. 2. It lies between the largest and smallest measurements of a set of test scores. 3. Most stable measure of central tendency. It is not influenced by test scores. 4. There is only one median for a set of test scores.

Median for Ungrouped Data

Case 1: When the Number of test scores (N) is odd:

Example: Find the median for the scores:

40, 32, 37, 30, 28, 33, 45

Arrange the scores either ascending or descending.

45, 40, 37, 33, 32, 3, 0

Then get the middlemost score.



45, 40, 37, 33 , 32, 3, 0 → Therefore median is 33.

Case 2: When the Number of test scores (N) is even:

Example: Find the median for the scores:

40, 32, 37, 30, 28, 33, 45

Arrange the scores either ascending or descending.

15, 25, 16, 31, 36, 44, 40, 28

Then get the average of the two middlemost scores.

44, 40, 36, 31, 28, 25, 16, 15 → and the median is (31+28)/2= 29.5

Median for Grouped Data

Example: Find the median for the distribution given below:

Class Number of Students1-5 2

5-10 511-15 1216-20 621-25 3

1. Construct a cumulative frequency table of less than type for the above data.

Here 2 students got the marks between 1 and 5 which means 2 students have marksless than 5. Now 5 students got marks between 5 - 10. So the students who got marksless than 10 are (2 + 5) i.e. 7 students. Proceeding in the similar way, we get thefollowing cumulative frequency table.

Class Number of

StudentsLess than 5 2

Less than 10 2 + 5= 7Less than 15 7 + 12= 19Less than 20 19 + 6= 25Less than 25 25 + 3= 28



2. Compute the locator of the class containing the median by dividing the totalnumber of scores (N) by 2. → N/2= 28/2= 143. Locate the cumulative frequency (CF) before the median class. → CF= 74. Determine the frequency of the median class . → f= 125. Determine the lower limit of the median class. → L= 11- 0.5= 10.5 6. Determine the class size (i). → i= 57. Plug- in the needed values in the formula of getting the median:

( ) )

Where: L = Lower limit of the class that contains the median = 10.5 N = Number of numbers = 28CF = Number of numbers before the class containing the median = 7

f = number of numbers in the class containing the median = 12i = class interval (size) = 5

C. Mode

Definition:

The mode or modal score, in a distribution is the score that occurs mostfrequently. However, a distribution may have one score that occurs mostfrequently (unimodal), two scores that occur with equal frequency and morefrequently than any other scores (bimodal), or three or more scores that occur

with equal frequency and more frequently than any other score (multimodal). Ifeach score in a distribution occurs with equal frequency the distribution is calleda rectangular distribution and it has no mode.There are two kinds of mode. The crude mode in an ungrouped test scores is themost frequently occurring score in an array. Consider the following test scores:

33 34 32 49 33 32 34 48 44

Crude mode = 33

True mode of both grouped and ungrouped test scores is obtained byusing the following formula:

Mo= 3Md= 2M where: Mo= mode

Md= medianM = mean



If Md= 37.35 and M= 38.45, what is the true mode? By substituting thegiven values to our computational formula, the obtained mode is 35.15 as shownbelow.

Mo= 3(37.35)- 2(38.45)

= 112- 76.9= 35.15

Characteristics:

1. The mode is the most frequent score in array.2. It is not influenced by extreme scores.3. There can be more than one mode for a set of scores. If there are two modes,

the set of scores is bimodal, for three or more, the set is multimodal.

Mode for Ungrouped Data

Get the most frequently occurring score in an array.33 34 32 49 33 31 37 48 44 → Mo= 33

Mode for Grouped Data

Class Frequency

100- 104 395- 99 4

90- 94 8

85- 89 580-84 2

A Comparison of the Mean, Median, and Mode The mean, median, and mode are affected by what is called skewness in the data.

• negative skew : The left tail is longer; the mass of the distribution is concentratedon the right of the figure. It has relatively few low values. The distribution is saidto be left-skewed or "skewed to the left ".

• positive skew : The right tail is longer; the mass of the distribution is concentratedon the left of the figure. It has relatively few high values. The distribution is said tobe right-skewed or "skewed to the right ".

In the given frequenciesthere are maximum (8)

frequencies in the classinterval 90- 94. So themidpoint is 92, the mode.



Here is Figure, which showed a normal curve, a negatively skewed curve, and apositively skewed curve:

Look at the above figure and note that when a variable is normally distributed,the mean, median, and mode are the same number.

When the variable is skewed to the left (i.e., negatively skewed), the mean shiftsto the left the most, the median shifts to the left the second most, and the modethe least affected by the presence of skew in the data.

Therefore, when the data are negatively skewed, this happens:mean < median < mode.

When the variable is skewed to the right (i.e., positively skewed), the mean isshifted to the right the most, the median is shifted to the right the second most,and the mode the least affected.

Therefore, when the data are positively skewed, this happens:mean > median > mode.

If you go to the end of the curve, to where it is pulled out the most, you will seethat the order goes mean, median, and mode as you “walk up the curve” for negatively and positively skewed curves.

You can use the following two rules to provide some information about skewness evenwhen you cannot see a line graph of the data (i.e., all you need is the mean and themedian):

1. Rule One. If the mean is less than the median, the data are skewed to the left .2. Rule Two. If the mean is greater than the median, the data are skewed to the

right



II. Reflections

Being a teacher goes beyond classroom discussion. We must assure

that we have imparted new things to our students before we leave the

classroom. We must be confident enough to say that we are forming ageneration fully equipped with the knowledge and wisdom that they need as

they fulfill they destiny.

To ensure that our students really learn from us, measurement and

evaluation is needed. One way of evaluating a group’s performance is to

measure the central tendency of a distribution of scores. In a distribution of

scores, there is a part where the scores of the students tend to cluster. This is

the measure of central tendency. The measure of central tendency does not

only reflect the performance of the group but also reveals the effectiveness of

the teacher. If the scores of the students tend to cluster at the range of the

low scores, they might be wrong with the strategies of the teacher. Then this

will serve as an alarm to improve or change the teaching style of the

educator. This will also help the teacher to decide if there is a need to teach

the lesson again.

As teachers, we must not neglect low performance of the students. If

they will learn nothing from us, they will suffer more when they go to the next

level of schooling.

- Apoloan, Karen

- (Reporter of the Introduction of Measuring Central Tendency and Mean)



Measuring Central Tendency has great application in educational

evaluation. Back when I was in High School I could not understand why the

every end of an exam especially the periodic exam we always have to tally

our score. Back then I thought it’s just an additional work to be done in a

public school and they just wanted to show everyone who got the highest or

lowest score for class competition purposes. But now as a teacher in the

making myself have I come to realize the importance of the score tally. The

mean, median and mode can show how the students fare in the exam. It will

show a teacher like me if my students did learn from my lectures, depending

on their scores. Further more if the examination is too easy, too hard or just

right for them. In which in return I could do some revisions in my exams and

also improve my strategies of teaching. During the National AchievementTest, the whole school performance on the exam is the one being evaluated.

The use of median would have been better for this situation since the

students partaking are mixed from fast learners to slow learners meaning it

could result to extreme scores. As you can see the use of the mean, median

and mode must depend upon the situation and I as a teacher evaluating my

students must learn which of which is best to use.

- Paray, Jenibel

- (Reporter of the Median and Mode)

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A Written Report on Measurement and Evaluation