Summarizing test scores
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SUMMARIZING TEST SCORES: MEASURES OF CENTRAL TENDENCY
CHAPTER 12
GLADYS T. AMBUYAT BSED III - ENGLISH
As teachers we need to know how students performed in the examinations we administer. To be able to describe how well or poorly they performed in the examination, there is a need to summarize test scores obtained by our students. This chapter is geared towards orienting prospective elementary and high school teachers on the most commonly used measures of central tendency in summarizing test scores, namely, the mean, median, and more.
THE MEAN
Measures of central tendency provide a single summary figure that best describes the central location of an entire distribution of test scores (May et al, 1990). The mean however, is the most popular among the measures of central tendency. This is oftentimes called the arithmetic average.
THE MEAN
Mean For Ungrouped Test Scores. When test scores are ungrouped, that is N is 30 or less, mean is computed following the formula:
Sx
N M=
Where: M= mean Sx= sum of test scores N= total number of test scores
Let us illustrate the computation of the mean for ungrouped test scores. For instance, the following scores were obtained by Grade VI pupils in a spelling test:
12, 11, 10, 9, 7, 15, 8, 6, 14, 13.
What is the mean score of the pupils in the aforementioned spelling test. To compute the mean, we first have to add the scores (Sx=105) and count the number of scores (N=10). Let us plug in the obtained values into our computation formula.
Sx
NM=
= 105 10
= 10.5
THE MEAN
Mean For Grouped Test Scores. When test scores are more than 30, the abovementioned computational formula is no longer applicable. There are 2 ways of computing the mean for grouped test scores: frequency-class mark method; and the deviation method.
THE MEAN
Frequency-class mark method
Steps:
1. Calculate the class mark or midpoint of each class interval.
2. multiply each class by its corresponding frequency.
3. sum up the cross products of the class mark and frequency of each class.
THE MEAN
Frequency-class mark method
Steps:
4. Count the number of cases or total of number scores.
5. Plug into the computation formula the values obtained in steps 3 and 4. The formula to be applied:
M= Sfcm N
THE MEAN
Where: M= mean
f = frequency of a class
cm = class mark or midpoint of a class
N= total number of test scores
Sfcm = sum of the cross products of the frequency and class mark.
COMPUTATION OF THE MEAN VIA THE FREQUENCY-MIDPOINT METHOD
CLASSES FREQUENCY
(f)
CLASS MARK(cm)
fcm
46-5041-4536-4031-3526-3021-2516-2011-15
579
108644
4843383328231813
2403013423302241387252
N = 53 Sfcm = 1699
It can be seen on the table that the frequency of each class is shown in second column. Class mark is shown in column 3 and is obtained by adding the lower and upper limits of each class and dividing the sum by 2. on the last column are the cross products of each frequency and class mark. The sum of the cross products is 1,699. let us substitute the values into our computational formula to obtain the mean.
M = Sfcm N
1699 53
=
= 32.06
THE MEAN
Deviation method
Steps in calculating the mean using this method are as follows:
1. Select a class from the grouped frequency distribution that shall be your arbitrary origin.
2. assign 0 deviation to the selected class as starting point. Above 0, all deviation
THE MEAN
scores shall be consecutive positive numbers. Below 0 all deviation score shall be consecutive negative numbers.
3. Multiply each deviation score by its corresponding class frequency to obtain fd.
4. Sum up all algebraically the cross products of each class frequency and deviation score the get Sfd.
THE MEAN
5. Determine the assumed mean (AM). The assumed mean is the class mark 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 + Sfd N
i
Where: M = mean
AM = assumed mean
f = frequency of a class
d = class deviation score
Sfd = sum of the cross products of the class frequency and deviation score
i = class size
N = total number of scores
THE MEAN
Let us verify whether the mean obtained in previous table is correct by applying the deviation method in computing the mean for grouped data. The next table will illustrate the procedures in computing the mean through the deviation method.
COMPUTATION OF THE MEAN VIA THE DEVIATION METHOD
CLASSES FREQUENCY(f)
DEVIATION SCORE
(d)
fd
46-5041-4536-4031-3526-3021-2516-2011-15
579108644
76543210
35424540241240
N = 53 Sfd = 202
i = 5AM =
(11+15) 2 =13
THE MEAN
With the obtained values in the previous table, the mean can be computed by plugging them into our computational formula.
THE MEAN