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 When test scores are more than 30, the abovementioned computational formula is no longer applicable.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