Solutions to Assignments
MCO-03 -
Research Methodology and Statistical Analysis
Question No. 4
Write short notes on the following:
(a) Comparative Scales
Scaling emerged from the social sciences in an attempt to measure or order attributes with respect to quantitative attributes or traits. Scaling provides a mechanism for measuring abstract concepts.
A comparative scale is an ordinal or rank order scale that can also be referred to as a non-metric scale. Respondents evaluate two or more objects at one time and objects are directly compared with one another as part of the measuring process.
For example you could ask someone if they prefer listening to MP3s through a Zune or an iPod. You could take it a step further and add some other MP3 player brands to the comparison. MP3 players would be scaled relative to each other and the scale position of any one player would depend on the the scale position of the remaining players. Because they are being compared differences such as who has the click wheel are effectively forced. Where this is limiting is evident when you find no standard of comparison outside the objects being compared. No generalizations are made outside of these objects. Often used when physical characteristics of objects are being compared.
1. Guttman Scaling
This can also be referred to as a cumulative scoring or scalogram analysis. The intent of this survey is that the respondent will agree to a point and their score is measured to the point where they stop agreeing. For this reason questions are often formatted in dichotomous yes or no responses.
The survey may start out with a question that is easy to agree with and then get increasingly sensitive to the point where the respondent starts to disagree. You may start out with a question that asks if you like music at which point you mark yes. Four questions later it may ask if you like music without a soul and which is produced by shady record labels only out to make money at which point you may say no. If you agreed with the first 5 questions and then started disagreeing you would be rated a 5. The total of questions you agreed to would be added up and your final score would say something about your attitude toward music.
2. Rasch Scaling
This probabilistic model provides a theoretical basis for obtaining interval level measurements based on counts from observations such as total scores on assessments. This analyzes individual differences in response tendencies as well as an item’s discrimination and difficulty. It measures how respondents interact with items and then infers differences between items from responses to obtain scale values. This model is typically used analyze data from assessments and to measure abilities, attitudes, and personality traits.
3. Rank-Order Scaling
This gives the respondent a set of items and then asks the respondent to put those items in some kind of order. The “order” could be something like preference, liking, importance, effectiveness, etc. This can be a simple ordinal structure such as A is higher than B or be done by relative position (give each letter a numerical value as in A is 10 and B is 7). You could present five items and ask the respondent to order each one A-E in order of preference. In Rank-Order scaling only (n-1) decisions need to be made.
4. Constant Sum Scaling
With this ordinal level technique respondents are given a constant sum of units such as points, money, or credits and then asked to allocate them to various items. For example, you could ask a respondent to reflect on the importance of features of a product and then give them 100 points to allocate to each feature of the product based on that. If a feature is not important then the respondent can assign it zero. If one feature is twice as important as another then they can assign it twice as much. When they are done all the points should add up to 100.
5. Paired Comparison Scale
This is an ordinal level technique where a respondent is presented with two items at a time and asked to choose one. This is the most widely used comparison scale technique. If you take n brands then [n (n-1)/2] paired comparisons are required. A classic example of when paired comparison is used is during taste tests. For example you could have a taste test in which you have someone try both Coke and Pepsi and then ask them which one they prefer.
6. Bogardus Social Distance Scale
This is a cumulative score that is a variant of the Guttman scale, agreement with any item implies agreement with the preceding items. This scale is used to measure how close or distant people feel toward other people. Social distance is a concern when it comes to issues related to racial integration or other forms of equality. This is applicable to team formation in the work place for example. Some people accept other people easily and use trustworthiness as the basis of their relationship with other people. Other people do not accept people who are not like them and tend to keep those that are not like them at arms length.
7. Q-Sort Scaling
This is a rank order procedure where respondents are asked to sort a given number of items or statements and classify them into a predetermined number of sets (usually 11) according to some criterion such as preference, attitude, or behavioral intent. Using cards that note an item to be ranked is the most popular and simplest method to use in the sorting process. In order to increase statistical reliability at least 60 cards should be used and no more than 140. This is good for discriminating among a large group of items in a relatively short amount of time.
(b) Purpose of a Report
No research is complete unless the report is written and communicated. It is
necessary for you as the researcher to maintain proper notes on progress made,
e.g., problem statement objectives, justifications for the study, review of
literature, development of instruments for data collection, hypothesis, sample
description and sampling technique, pilot study, problems faced in data
collection, the data and the analysis. These notes help in preparing the research
report.
The main purpose of research report is to let others interested in the subject
know the findings of the research. The researcher himself/ herself may have
definite purpose of writing the research report.
Examples of purposes are listed below:
1) Research is conducted for the partial fulfillment of the degree like M.Sc.,
Ph.D. Therefore writing report is a part sf the academic programme.
2) Research is conducted to find an answer to the problems faced by the
practitioner, teacher or administrator. Here the report is written to
communicate the findings to others in the profession for critiquing,
application of result or future investigation in the area of research.
3) When the research is funded by the government or a research foundation.
They stipulate the requirements of the report.
The reports are usually written as a thesis, monographs or article for
publication in journal or magazine. The content outline can be broadly
divided as introduction, review of literature, methodology, data analysis and
interpretation, summary and conclusions. The introduction section include:
background of the study, need and justification for the study, problem statement,
variables, objectives and hypothesis, scope and limitations, and assumptions.
Methodology includes: justification and explanation of research approach,
sampling technique and size of sample, setting of the study, construction or
selection of instrument for data collection, procedure for data collection and the
plan of data analysis.
Style of writing references and bibliography is recommended by the publishers '
and the research departments of institutions. The writer has to follow them
strictly. Reference are works that are referred in the text whereas bibliography
includes all the relevant literature reviewed irrespective of "referred" or "not
referred" status.
(c) Binomial Distribution
A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.
A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.
Binomial distributions must also meet the following three criteria:
a. The number of observations or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times. This is common sense—if you toss a coin once, your probability of getting a tails is 50%. If you toss a coin a 20 times, your probability of getting a tails is very, very close to 100%.
b. Each observation or trial is independent. In other words, none of your trials have an effect on the probability of the next trial.
c. The probability of success (tails, heads, fail or pass) is exactly the same from one trial to another.
The binomial distribution is closely related to the Bernoulli distribution. According to Washington State University, “If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a binomial Distribution. On the other hand, the Bernoulli distribution is the Binomial distribution with n=1.”
A Bernoulli distribution is a set of Bernoulli trials. Each Bernoulli trial has one possible outcome, chosen from S, success, or F, failure. In each trial, the probability of success, P(S) = p, is the same. The probability of failure is just 1 minus the probability of success: P(F) = 1 – p. (Remember that “1” is the total probability of an event occurring…probability is always between zero and 1). Finally, all Bernoulli trials are independent from each other and the probability of success doesn’t change from trial to trial, even if you have information about the other trials’ outcomes.
Many instances of binomial distributions can be found in real life. For example, if a new drug is introduced to cure a disease, it either cures the disease (it’s successful) or it doesn’t cure the disease (it’s a failure). If you purchase a lottery ticket, you’re either going to win money, or you aren’t. Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.
The binomial distribution formula is:
b(x; n, P) = nCx * Px * (1 – P)n – x
Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials
The binomial distribution formula can calculate the probability of success for binomial distributions. Often you’ll be told to “plug in” the numbers to the formula and calculate. This is easy to say, but not so easy to do—unless you are very careful with order of operations, you won’t get the right answer. If you have a Ti-83 or Ti-89, the calculator can do much of the work for you. If not, here’s how to break down the problem into simple steps so you get the answer right—every time.
(d) Skewness
Skewness is a measure of asymmetry or distortion of symmetric distribution. It measures the deviation of the given distribution of a random variable from a symmetric distribution, such as normal distribution. A normal distribution is without any skewness, as it is symmetrical on both sides. Hence, a curve is regarded as skewed if it is shifted towards the right or the left.
Types of Skewness
1. Positive Skewness
If the given distribution is shifted to the left and with its tail on the right side, it is a positively skewed distribution. It is also called the right-skewed distribution. A tail is referred to as the tapering of the curve differently from the data points on the other side.
As the name suggests, a positively skewed distribution assumes a skewness value of more than zero. Since the skewness of the given distribution is on the right, the mean value is greater than the median and moves towards the right, and the mode occurs at the highest frequency of the distribution.
2. Negative Skewness
If the given distribution is shifted to the right and with its tail on the left side, it is a negatively skewed distribution. It is also called a left-skewed distribution. The skewness value of any distribution showing a negative skew is always less than zero. The skewness of the given distribution is on the left; hence, the mean value is less than the median and moves towards the left, and the mode occurs at the highest frequency of the distribution.
Measuring Skewness
Skewness can be measured using several methods; however, Pearson mode skewness and Pearson median skewness are the two frequently used methods. The Pearson mode skewness is used when a strong mode is exhibited by the sample data. If the data includes multiple modes or a weak mode, Pearson’s median skewness is used.
The formula for Pearson mode skewness:
Where:
X = Mean value
Mo = Mode value
s = Standard deviation of the sample data
The formula for Person median skewness:
Where:
Md = Median value
How to Interpret
- Skewness also includes the extremes of the dataset instead of focusing only on the average. Hence, investors take note of skewness while estimating the distribution of returns on investments. The average of the data set works out if an investor holds a position for the long term. Therefore, extremes need to be looked at when investors seek short-term and medium-term security positions.
- Usually, a standard deviation is used by investors in forecasting returns, and it presumes a normal distribution with zero skewness. However, because of skewness risk, it is better to obtain the performance estimations based on skewness. Moreover, the occurrence of return distributions coming close to normal is low.
- Skewness risk occurs when a symmetric distribution is applied to the skewed data. The financial models seeking to estimate an asset’s future performance consider a normal distribution. However, skewed data will increase the accuracy of the financial model.
- If a return distribution shows a positive skew, investors can expect recurrent small losses and few large returns from investment. Conversely, a negatively skewed distribution implies many small wins and a few large losses on the investment.
- Hence, a positively skewed investment return distribution should be preferred over a negatively skewed return distribution since the huge gains may cover the frequent – but small – losses. However, investors may prefer investments with a negatively skewed return distribution. It may be because they prefer frequent small wins and a few huge losses over frequent small losses and a few large gains.
