Content uploaded by S.N. Chakrabartty
Author content
All content in this area was uploaded by S.N. Chakrabartty on Aug 22, 2019
Content may be subject to copyright.
Journal of Knowledge Management and Information Technology
Scoring and Analysis of Likert Scale: Few Approaches
Satyendra Nath Chakrabartty
Former Director of Kolkata Campus of Indian Maritime University
Abstract
To improve upon summative scoring of Likert Scale, two alternate methods of
scoring have been proposed in this paper. The proposed methods of scoring are
based on weighted sums where weights are taken as empirical probabilities and
enable us to find cardinal scores for individuals as well as items as expected
values satisfying conditions of additivity and linearity. Comparison of the
methods was undertaken with an empirical data. Proposed methods of scoring
Likert Scale increased reliability of the questionnaire in terms of Cronbach’s
alpha, tended to introduce independency among the items, helped in better
exploring and interpreting the factors. Summated Likert scores did not follow
Normal distribution but each of the proposed method of scoring passed test of
normality. Thus, the transformed scores offer platform for undertaking almost all
type of analysis being done for continuous quantitative variable following
Normal distribution.
Key words: Likert-scale, Cardinal scores, Quantitative analysis, Reliability.
1. Introduction
Likert (1932) proposed a summated scale for the assessment of survey
respondent’s attitudes. Likert scaling presumes the existence of an underlying (or
latent or natural) continuous variable whose value characterizes the respondents’
attitudes and opinions. Likert scales are quite popular and are widely used in
different areas like psychology, sociology, health care, marketing, attitude,
preference, customers’ quality perceptions or expectations, and of subjective
well-being in health care, etc.
Individual item in Likert scale usually has odd number of response categories
say 5 or 7. Descriptive response alternatives could be Strongly approved,
Approved, Undecided, Disapproved, and Strongly disapproved. The response
categories are usually assigned numbers like 1, 2, 3, 4, and 5 or by -2, -1, 0, 1 and
2 or any linear transformation of such numbers. However assigning successive
integer values to scale categories has also been criticized for not being realistic.
Major limitations of Likert type data are:
a) Respondents may avoid using extreme response categories( Central tendency
bias) or agree with statements as presented (Acquiescanal bias ) or try to
DOS 4.08.15 DOA 4.02.15 Page | 31, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
portray themselves or their organization in a more favourable light ( Social
desirability bias)
b) One cannot assume that respondents perceive all pairs of adjacent levels as
equidistant. Mid-point or neutral point or zero point is a perception.
c) There is no hard and fast rule to decide on number of response alternatives.
Number of response categories need not be an odd number. Five- or seven-
point formats appear to be the most prevalent.
d) Distance between response alternatives is assumed to be equidistant. If
response categories are a, b, c, d, e the equidistant property can be achieved
by either of the following two approaches.
*− =− viz. 1, 2, 3, 4 5 or 0.1, 0.2, 0.3, 0.4, 0.5
*
=
viz. 1, 2, 4, 8, 16
But verification of such assumptions is usually not done through data.
Assigning successive integer values to scale categories has also been
criticized.
e) Non- Additivity: Naming the response classes by successive numbers does
not mean we can do all further mathematical operations to the data generated
from administration of such a questionnaire. Consider the hypothetical
example where 50% respondents endorsed the response category ‘Most
agree’ and the rest 50% went for “Most disagree”. Then the average score is
neutral, but clearly, they are at two opposite poles. Thus, addition of
responses of Likert items may not be meaningful. If addition is not
meaningful then further statistics like mean, SD, Correlation, analysis like
regression, estimation, testing etc. may not be meaningful. Moreover, there
could be various response distributions even for a fixed number of response
categories where the distributions have equal mean but different SDs.
f) Generated responses: A major problem with Likert items is the generated
responses consisting of a count of responses in each category. The resulting
analysis is inherently limited, primarily to a frequency table, typically with
relative and cumulative relative frequencies. However, assumption is often
made that the Likert item is interval in nature and means and other statistics
are commonly computed. It could be argued that this assumption is incorrect.
The data generated by an instrument based on Likert items simply cannot be
subjected to the more robust, more powerful and more subtle analysis
available with quantitative data. Use of an ordinal scale implies a statement
of “greater than” or “less than” without stating how much greater or lesser
g) Continuous or discrete: Scores of Likert Items sometimes are treated as if
they are not discrete. Few researchers are of the opinion that Likert scale
(from strongly agree to strongly disagree) is ordinal and have to deal with
this scale as a discrete scale. This implies that the mean, variance and
covariance etc. computed from responses to a Likert scale may not be
meaningful.
h) Reliability: Reliability assessment might use the correlation between the
item score and the total score or use a test-retest procedure. In any event, the
Page | 32, Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
items not correlated with the total would be discarded. However, when items
tend to measure various dimensions of the underlying trait, item correlations
may be poor and if this is observed during analysis stage, one may not like to
discard an item.
i)Ordinal or Interval: The level of scaling obtained from Likert procedure is
rather difficult to determine. The scale is clearly at least ordinal. Response
categories tend to be sequential but not linear. Count of each category of
response represents an ordinal variable. In order to achieve an interval scale,
the properties on the scale variable have to correspond to differences in the
trait on the natural variable. In other words, distance between any pair of
response categories must be same. But it seems unlikely that the categories
formed by the misalignment of the five responses will all be equal, the
interval scale assumption seems unlikely.
j) Normality: Assumption of Normality is generally not observed from data
generated from Likert Scale. As a result, statistical analysis in the parametric
set up cannot be meaningfully undertaken from data set generated from
Likert scale.
Thus, need is felt to have new methods of scoring Likert scale so that the new
scores are cardinal and have one to one correspondence with Real number system
or a subset of it and enable us to perform analysis undertaken with quantitative
data in parametric set up.
2. Literature review
Wu (2007) considered example of an injury scale of five categories represented
by none, minor, moderate, severe, and fatal, the degree of injury seriousness
between severe and fatal is more significant than that between none and minor.
Thus, successive integers to the scale categories may not reflect the realistic
differences in injury seriousness between or among scale categories.
While Jacoby et al (1971) suggest only three response categories, Dillman, et al.
(2009) recommend that four or five categories should be used and Fink (1995)
suggested five to seven. Foddy (1994) concludes that a minimum of seven
categories is required to ensure scale validity and reliability. Nine-point format
were used by researchers like Lee & Soutar (2010) and even a 15-point format
was used by Chaiken & Eagly (1983).
Stevens (1946) mentioned that in the strict sense, statistics involving mean and
SD ought not to be used with ordinal scales since mean and SD are in error to the
extent that the successive intervals on the scale are unequal in size or not-
equidistant. However, Muraki (1990) observed that if the data fits the
Polytomous Rasch Model and fulfill the strict formal axioms of the said model, it
may be considered as a basis for obtaining interval level estimates of the
continuum. Chien-Ho Wo (2007) observed that transformation of scale data
based on Snell’s (1964) scaling procedure does not do much to pass the
normality test. From item response theory, it can be seen that even large ordinal
scales can be radically non-linear.
DOS 4.08.15 DOA 4.02.15 Page | 33, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
3.Objective
The paper proposes two new methods of scoring Likert scale which result in
cardinal scores with one to one correspondence with Real number system or a
subset of it and also to discuss advantages of such scoring methods and to
compare among the methods primarily through empirical exploration.
4. Formal description
Suppose there are n – respondents who answered each of the m-items of a Likert
questionnaire where each item has k-numbers of response categories.
Let be a general element of the basic data matrix of order n X m where n-
individuals are in rows and m-items are in columns. represents score of the
i-th individual for the j-th item. Value of ranges between 1 to k i.e. 1 to 5
for a 5-point scale
Note ∑
=1 = Sum of scores of all individuals for the j-th item (Item Score
for the j-th item)
∑
=1 = Sum of scores of all the items for i-th individual i.e. total score of
the i-th individual (Individual score)
∑∑ = Sum of scores of all the individuals on all the items i.e. total test score
In addition, one can have another matrix (( )) of order m X k showing
frequency of i-th item to j-th response category. A row total will indicate
frequency of that item and will be equal to the sample size (n) .Similarly, a
column total will indicate total number of times that response category was
chosen by all the respondents. Grand total will be equal to sample size X number
of items ().
Multinomial model for Likert responses: A k-dimensional random variable X
with components 1,2, … … … , has multinomial distribution or X follows
Multinomial with parameters (n; p1, p2 , ..., pk ) if the probability density function
is given by:
{1=1, … … . =}= !
1!2!….. !1
1 2
2… … .
where pi ≥ 0 and 1+2+⋯… … = 1 and 1+2+⋯. . =
The multinomial distribution Mult (n, 1,2, … … . . , ) is the joint distribution
of the k -random variables Xi. It is therefore a multivariate, discrete distribution
with mean and variance as follows:
[ ]= and ( )=(1 − )
Page | 34, Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
Thus, a Likert item with k- response categories can be viewed as throwing a k-
faced dice, i.e. each throw of the dice may result into 1 or 2 or 3 or 4 or 5 for a 5-
point scale. So responses to a Likert item can be assumed to follow a multinomial
distribution
Maximum likelihood estimate of a Multinomial distribution is =
for j =
1, 2, .k . Thus, estimate of is simply the observed relative frequency of
outcome j i.e. empirical probability. Reproduction theorem says that sum of
independent multinomial random vectors with identical vectors of choice
probabilities follow a multinomial distribution:
5. Methodology: Three approaches were adopted viz.
Approach 1: with total score of an individual as sum of his/her score on each
item i.e. usual summative Likert scores. Here individual score lies between
(for k point scale) and is discrete in nature. Similarly item scores is
also discrete.
Approach 2: assigning uniform weights i.e. w1, w2, w3, w4, and w5 to the
response categories which will remain unchanged for all items. Here, ∑
=1 =
1 where is the empirical probability of the i-th response category[ i = 1, 2,
…..,k] and is calculated as = ∑
=1
∑∑ where number of items are 1, 2, …. m.
In other words, weight for the i-th response category is the ratio of total
frequency of the category (over all items) and grand total of the Item – Response
Categories frequency matrix ().
Here, each item follows a Multinomial distribution with parameters
(.1,2,3,4,5). Mean and variance of the i-the response category =
. and ( 1 − ) respectively. Correlation between i-th and j-th
response category is − �
(1− )(1− ) .
Total score of i-th item is taken as ∑
=1 where denotes frequency of
the j-th response category of the i-th item and denotes weight of the j-th
response category. Similarly, score of the j-th response category is ∑
=1 .
Score of the i-th individual for the j-th item is equal to if he/she
responded to the t-th response category of the j-th item (t= 1,2,3,4,5 for a five
point scale). Thus, both individual scores and item scores are in terms of
probabilities or expected values and hence each provides measurements of
continuous variable.
Approach 3: with different weights to different response categories of different
items, so that sum of weights for each item is equal to one. Here, weight to the i-
DOS 4.08.15 DOA 4.02.15 Page | 35, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
th item and j-th response category is defined as =
∑
=1
i.e. ratio of cell
frequency and total frequency of the item. . Clearly sum of weights for each item
is equal to one and ∑ ∑ =. However, sum of weights for a response
category of all items is different from one. Score of an individual in the i-th item
will be ∑
=1 and score of j-th response category item will
be∑
=1 .
Here, each item will follow multinomial distribution with different values of
parameters. Item scores are in terms of expectations. However, both item scores
and individual scores are continuous variables.
Alternatively, one can find different weights to different item – response category
combinations so that sum of weights (probabilities) of all cells is equal to one by
choosing
=
∑∑ Clearly ∑ ∑ = 1.
Here, transformed cells scores follow multinomial distribution with larger
number of parameters. However, weights in this method are linearly related to
weights proposed in Approach 3. In fact, weights in this approach are equal to
weights from approach 3 divided by number of items.
Weighted scores are in terms of expected values and hence provide
measurements of a continuous variable.
6. Calculation of weights
Data : A test consisting of five Likert type items each with five response
alternatives was administered to 100 respondents where “Strongly agree ” was
assigned 5 and “ Strongly disagree” was assigned 1.
Empirical calculation of weights for various approaches is shown using the Item
–Response Categories frequency matrix derived from the data. Here, number of
items ()= 5 , each item had 5 response categories . = 5 and sample
size =100.
Table – 1
Calculation of weights: Approach – 2
Items
*RC - 1
*RC - 2
*RC - 3
*RC - 4
*RC- 5
TOTAL
1
19
32
35
11
3
100
2
7
33
34
19
7
100
3
14
17
36
27
6
100
4
10
14
38
30
8
100
5
4
31
37
20
8
100
TOTAL
54
127
180
107
32
500
Weights to response
categories
0.108
0.254
0.36
0.214
0.064
1.00
Page | 36, Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
•RC – i denotes i-th Response category
Here 1= 54
500 ,2= 127
500 and so on
Table – 2
Calculation of weights: Approach - 3
Items
*RC- 1
*RC-2
*RC-3
*RC-4
*RC- 5
Total
1
0.19
0.32
0.35
0.11
0.03
1.00
2
0.07
0.33
0.34
0.19
0.07
1.00
3
0.14
0.17
0.36
0.27
0.06
1.00
4
0.10
0.14
0.38
0.30
0.08
1.00
5
0.04
0.31
0.37
0.20
0.08
1.00
Here, weight for the first response category of Item 1 is 11 divided by total
frequency of the item i.e. sample size. Thus, 11 = 19
100 and so on
7. Observations
In approach 2 and 3:
•Weights are taken as probabilities.
•Weights are obtained from data considering the frequencies or probabilities
of Item – Response categories without involving assumptions of continuous
nature or linearity or normality for the observed variables or the underlying
variable being measured.
•It was assumed that there is no item with zero discriminating value i.e. there
is no item with equal frequency for each response category and there is no
item where all individuals recoded their response to only one response
category. The assumption is reasonable since items with zero discriminating
values are excluded as per method of test constructions.
•Item scores and individual scores are obtained as expected values and hence
provides measurement of continues variables satisfying conditions of
linearity since
E (x + y) = E(x) + E(Y)
E (αx) = αE(x)
E (αx +βy) = αE(x) +βE(y)
•Ranking of individuals are invariant under linear transformation for each of
approach 1, 2 and 3
•Distribution of item scores follows multinomial distribution due to
reproduction property of multinomial random variables for approach 2 and 3.
Sum of item scores will follow multinomial distribution if the items are
independent
•For large sample size, individual scores and item score may tend to follow
normal distribution under each method. Primafacy, each of approach 2 and 3
DOS 4.08.15 DOA 4.02.15 Page | 37, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
is likely to tend to normality but the proposition need to be tested
empirically.
•The metric data and linearity of weighted scores by any of the approach 2
and 3 enable us to generate data that is cardinal (quantitative) to permit
calculation of all descriptive statistics and also undertake relevant estimation,
testing of hypothesis, relevant analysis used in multivariate statistics.
8. Analysis
Following analysis was undertaken based on empirical data described above:
8.1 Mean, Variance and Reliability of the questionnaire (Cronbach alpha) for the
three approaches are shown in Table – 3 below:
Table – 3
Mean Variance and Reliability for various Approaches
Description
Approach - 1
Approach - 2
Approach – 3
Test mean
14.36
3.658
3.8464
Test variance
5.9904
0.6357
0.6893
Mean of Item – 1
2.47
0.66484
0.6613
Mean of Item – 2
2.86
0.72744
0.7384
Mean of Item – 3
2.94
0.7406
0.7758
Mean of Item – 4
3.12
0.77472
0.8744
Mean of Item – 5
2.97
0.7582
0.7965
Variance of Item – 1
1.0291
0.1347
0.1085
Variance of Item – 2
1.0604
0.0998
0.0709
Variance of Item – 3
1.2364
0.1209
0.1608
Variance of Item – 4
1.1456
0.1084
0.1861
Variance of Item – 5
0.9891
0.0919
0.0815
Cronbach’s alpha
0.110552
0.157417
0.147983
Observations
*Approach 2 and Approach 3 resulted in reduction of test average and test
variance i.e. the weighted scores made the data more homoscedastic.
*Scores as weighted sum resulted in higher values of alpha. Value of Cronbach’s
Alpha was highest for Approach 2 followed by Approach 3.
8.2 Rank correlation of individual score: Spearman ρ between individual scores
obtained by various approach are shown in Table – 4.
Page | 38,Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
Table - 4
Rank Correlation Matrix (Spearman ρ)
Observations
•All the rank correlations are found to be significant.
•Low value of rank correlation between Approach 1 and other approaches
indicate that ranks of individuals are different for different approaches.
However high value of Spearman between Approach 2 and 3 indicates that
individual ranks are almost unchanged. In other words, ranks of respondents
as obtained from Approach 2 remained more or less same when their ranks
are computed by Approach 3.
8.3 Item Correlation matrix for each approach are shown in Table 5, 6 and 7
Table – 5
Item Correlation Matrix for Approach - 1
*: Significant at 1% level
Table – 6
Item Correlation Matrix for Approach - 2
Items
1
2
3
4
5
1
1.00
0.051
(-)0.007
(-)0.074
0.038
2
1.00
0.009
0.111
0.098
3
1.00
0.052
0.156
4
1.00
(-)0.044
5
1.00
Table – 7
Item Correlation Matrix for Approach - 3
Items
1
2
3
4
5
1
1.0
0.037
0.024
(-) 0.005
(-) 0.055
2
1.0
0.117
0.143
0.101
3
1.0
(-) 0.019
0.189
4
1.0
(-) 0.120
5
1.0
Approach 1
Approach 2
Approach 3
Approach 1
1.0
0.444
0.306
Approach 2
1.0
0.909
Approach 3
1.0
Items
1
2
3
4
5
1
1.0
0.168
0.096
0.003
0.123
2
1.0
(-) 0.007
0.033
0.045
3
1.0
*(-)0.330
(-) 0.11
4
1.0
0.172
5
1.0
DOS 4.08.15 DOA 4.02.15 Page | 39, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
Observations
*Weighted sum used in Approach 2 and Approach 3 resulted in changes of
magnitudes and signs of item-correlations
*Significant correlation between item 3 and 4 as obtained in Approach 1 became
insignificant in Approach 2 and also in Approach 3.
* No significant correlations were found in Approach 2 and Approach 3 which
imply that the weighted scores made the items more or less independent
8.4 Test of Normality
Attempt was made to test whether total score of respondents follow Normal
distribution using Anderson – Darling test for Normality. It is one of most
powerful statistical test for detecting departures from Normality. The underlying
null hypothesis is that the variable under consideration is normally distributed. A
large p-value corresponding to the test statistic (p > 0.05) would indicate
normality. The test statistic is
Table – 8
Values of test statistic and associated p-values
Value of test
statistics
p – values
Remarks
Approach 1
1.1153
0.0061
H0 is rejected
Approach 2
0.4796
0.483315
H0 is accepted
Approach 3
0.6824
0.07256
H0 is accepted
Observations
*It can be inferred that scores of respondents did not follow Normal distribution
for Approach 1. In other words, summated Likert scores did not follow Normal
distribution. This highlights limitation of Likert data. As a result, a number of
statistical analysis, testing and estimation procedures which presume normality of
data cannot be performed with usual summated scores of Likert type data
*Respondents scores followed Normal distribution for Approach 2 and 3 i.e.
non-linear transformations resulted in the desired property of normality. Thus,
the transformed scores offer platform for undertaking almost all type of analysis
being done for continuous quantitative variable following Normal distribution.
* Higher p- value at Approach 2 in comparison to Approach 3 indicates better
normality in case of Approach 2.
( )( ) ( )( )( )
iNi YFYF
N
i
NAD −+
−+
−
−−= 1
1lnln
12
Page | 40, Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
Similar approach was adapted to test whether item scores follow normal
distribution or not. The test results showed that item scores are not normally
distributed under any of the above said three approaches
8.5 Correlation matrix for the approaches: Correlation of individual scores
obtained by various approach are shown in Table – 9.
Table – 9
Correlations between a pair of approaches
Approach 1
Approach 2
Approach 3
Approach 1
1.0
0.445
0.307
Approach 2
1.0
0.927
Approach 3
1.0
Observations
*Maximum correlation was found between Approach 2 and Approach 3.
8.6 Factor structures of the approaches
Factor Analysis with orthogonal vari-max rotation was undertaken with item
correlation matrix under each approach. The results are as follows
Table – 10
Results of Factor Analysis
Factor
Eigen values
Percentage of
Variance explained
Cumulative
percentage of
variance explained
Remarks
APPROACH - 1
1
1.382
27.638
27. 638
Two factors
explaining 52.507%
of variance
2
1.243
24.869
52.507
3
0.958
19.154
71.661
4
0.787
15.741
87.403
5
0.63
12.59
100
APPROACH - 2
1
1.204
24.077
24.077
Three factors
explaining 66.867%
of variance
2
1.105
22.10
46.177
3
1.035
20.69
66.867
4
0.891
17.829
84.696
5
0.765
15.304
100
APPROACH - 3
1
1.277
25.544
25.544
Three factors
explaining 68.755%
of variance
2
1.152
23.042
48.586
3
1.008
20.169
68.755
4
0.825
16.492
85.246
5
0.738
14.754
100
DOS 4.08.15 DOA 4.02.15 Page | 41, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
Observations
*Approach 1 gives two factors, combined effect of which explains only 52.50%
of variance
* Each of Approach 2 and Approach 3 gives three factors explaining
cumulatively 66.87% to 68.76% of variance respectively
*The results appear to be in line with item correlation matrix under each
approach where each correlation was found to be insignificant except one in
Approach 1 and also high correlation observed between Approach 2 and
Approach 3
Thus, the non-linear transformations tended to introduce independency of items
9. Limitations
Application of the proposed scoring of Likert items and Scale should take into
account the following facts:
*The methods take no account of the experiment design behind the data.
*The methods are not applicable for items with zero discriminating value.
*Irregularities in data should be within tolerance.
* Test of Normality may be undertaken before application of the proposed
methods of Scoring since distribution of individual score obtained from
Approach-2 or Approach-3 is yet to be established.
10. Conclusions
Weighted scores where weights are data driven and proportional to probabilities
helps to find total score of a respondent and also total score of an item as
expected values and enable us to perform usual analysis for a continuous
quantitative variable. Computation of weights considered the frequencies or
probabilities of Item – Response categories without involving assumptions of
continuous nature or linearity or normality for the observed variables or the
underlying variable being measured.
It was assumed that there is no item with zero discriminating value i.e. there is no
item with equal frequency for each response category and there is no item where
all individuals recoded their response to only one response category. The
assumption is reasonable since items with zero discriminating values are
excluded as per method of test constructions.
Scores in terms of expected values resulted in higher reliability of the
questionnaire. Approach 2 registered highest value of Cronbach’s alpha among
the three approaches. Such scores tended to introduce independency among the
items. The Approach 2 and 3 resulted in a situation where the five items were
almost independent and accordingly gave higher number of independent factors.
Page | 42, Vol -1, Issue -2
Journal of Knowledge Management and Information Technology
Thus, scores as per the proposed methods helped in better exploring and
interpreting the factors.
Usual Likert type score did not follow normal distribution. However, weighted
scores as per Approach 2 and also for Approach 3 resulted in the desired property
of normality and are suitable for use in methods of analysis requiring assumption
of normality. Thus, the proposed scores offer platform for undertaking almost all
type of analysis being done for continuous quantitative variable following
Normal distribution. For example, individual scores obtained through Approach
2 or Approach 3 tended to satisfy assumptions of AVOVA, regression analysis,
t-test for testing equality of means, F-test for testing equality of variances,
Discriminant analysis, etc. Proposed methods of Scoring conform better to
Normality.
Thus, scoring of Likert Scale as per Approach 2 or Approach 3 has many
desirable properties and avoids some of the major limitations of usual summative
scores. However, high correlation (over 0.9) was found between Approach – 2
and Approach – 3. This may imply possible use of Approach 2 instead of
Approach 3 primarily because of easiness to computation. Thus, Scoring method
as proposed in Approach -2 is recommended for Likert-type data for clear
theoretical advantages and easiness in calculations with minimum processing
hour.
References
[1]. Chaiken, S., & Eagly, A. H. (1983) "Communication Modality as a
Determinant of Persuasion: The Role of Communicator Salience". Journal of
Personality and Social Psychology, Vol 45, No. 2, pp 241-256.
[2]. Chien-Ho Wu (2007) An Empirical Study on the Transformation of Likert-
scale Data to Numerical Scores, Applied Mathematical Sciences, Vol. 1,
2007, no. 58, 2851 – 2862
[3]. Dillman, D. A., Smyth, J. D. & Christian, L. M. (2009) Internet, mail and
mixed-mode surveys: The tailored design method, John Wiley & Sons Inc.,
Hoboken, N.J.
[4]. Fink, A. (1995) How to ask survey questions, Sage Publications, Thousand
Oaks
[5]. Foddy, W. (1994) Constructing questions for interviews and questionnaires:
Theory and practice in social research, Cambridge University Press,
Cambridge.
[6]. Jacoby Jacob, Matell Michael S (1971) Three-point likert scales are good
enough, Journal of Marketing Research; Nov 1971; Vol 8 pg. 495-500
[7]. Lee, J. A. & Soutar, G. (2010) "Is Schwartz's Value Survey an Interval Scale,
and Does It Really Matter?" Journal of Cross-Cultural Psychology,Vol 41,
No 1, pp 76-86.
DOS 4.08.15 DOA 4.02.15 Page | 43, Vol – 1, Issue - 2
Scoring and Analysis of Likert Scale: Few Approaches
[8]. Likert R (1932). A Technique for the Measurement of Attitudes. Archives of
Psychology; p. 140.
[9]. Muraki, E (1990) – Fitting a Polytomous Item Response Model to Likert –
Type Data. Applied Psychological Measurement, Vol. 14, No. 1, March, pp
59 – 71
[10]. Snell, E. (1964) A Scaling Procedure for Ordered Categorical Data,
Biometrics,
20(3), 592-607.
[11]. Stevens, S. S. (1951). Mathematics, measurement and Psychophysics. In
Handbook of Experimental Psychology. S. S. Stevens (ed.), New York: John
Wiley & Sons pp. 1–49.
[12]. Wu, Chien-Ho (2007) An Empirical Study on the Transformation of Likert-
scale Data to Numerical Scores. Applied Mathematical Sciences, Vol. 1,
2007, no. 58, 2851 - 2862
Author Profile
Prof. Satyendra Nath Chakrabartty is an M. Stat. (Specialisation - Psychometry)
from Indian Statistical Institute and was Director, Kolkata Campus of Indian
Maritime University. His current research interests include multi-dimensional
measurements and their properties to assess overall progress or overall distance
from the set of goals along with identification of critical areas. He also works on
estimation of true scores, true score variance, reliability of a battery of tests under
classical theory approach, Likert type tests and Non-parametric Reliability and
introducing linearity among non-linear relationships etc.
Page | 44, Vol -1, Issue -2
