Unit 3: Measurement Scales and Central Tendency - Subjective Questions

Measurement is the systematic process of assigning numbers, symbols, or labels to the characteristics of objects, individuals, or events according to predefined rules.

The four major levels of measurement are:

• Nominal scale: Classifies observations into mutually exclusive categories without indicating order. Examples include gender, religion, and product category.
• Ordinal scale: Classifies and ranks observations, but the differences between ranks are not necessarily equal. Examples include satisfaction levels and class ranks.
• Interval scale: Has ordered categories with equal intervals, but it does not have a true zero. Temperature measured in Celsius is an example.
• Ratio scale: Possesses order, equal intervals, and an absolute zero. Examples include income, weight, age, and sales volume.

The permissible statistical analysis becomes more sophisticated as measurement progresses from nominal to ratio scales.

The four scales differ in terms of classification, order, equal intervals, and absolute zero:

• Nominal data can be counted and analyzed using the mode.
• Ordinal data permit ranking and can be summarized using the median or mode.
• Interval data permit addition and subtraction; the mean can be calculated.
• Ratio data support all arithmetic operations, including meaningful ratios.

For example, an income of is meaningfully twice an income of , whereas is not twice as hot as .

A comparative scale requires respondents to evaluate one object, brand, idea, or stimulus by directly comparing it with another. The resulting data are generally ordinal because they indicate relative preference rather than an absolute magnitude.

Major types include:

• Paired comparison scale: Respondents choose one preferred item from each pair. For objects, the number of comparisons is:
  
• Rank-order scale: Respondents arrange several alternatives from the most preferred to the least preferred.
• Constant-sum scale: Respondents distribute a fixed number of points, usually 100, among alternatives according to their relative importance.
• Q-sort technique: Respondents sort statements or objects into predetermined groups based on similarity, preference, or agreement.

Comparative scales are easy for respondents to understand, but their results are relative to the alternatives included in the study.

In a paired comparison scale, a respondent is shown two alternatives at a time and asked to select one according to a specified criterion, such as preference, quality, or likelihood of purchase.

If there are alternatives, the required number of unique pairs is:

For example, five brands require:

Advantages:

• It is simple and easy to understand.
• It reduces the burden of evaluating many alternatives simultaneously.
• It is useful when differences between alternatives are small.
• It provides direct relative-preference information.

Limitations:

• The number of comparisons increases rapidly as increases.
• It may produce inconsistent or circular preferences.
• Results apply only to the alternatives compared.
• It usually generates ordinal rather than metric data.

The three comparative scaling techniques are:

• Rank-order scaling: Respondents arrange alternatives in order of preference or importance. For example, five brands may be ranked from 1 for the most preferred to 5 for the least preferred. It reveals order but not the magnitude of differences between ranks.
• Constant-sum scaling: Respondents allocate a fixed total, commonly 100 points, among alternatives. More points indicate greater preference or importance. It provides information about both order and relative differences, but respondents may find the allocation difficult.
• Q-sort scaling: Respondents sort a set of statements, cards, or objects into categories ranging from one extreme to another, such as from "most agree" to "most disagree." The number of items placed in each category is often controlled to approximate a normal distribution.

These techniques are useful for comparing multiple alternatives, but the measurements remain dependent on the specific set of alternatives presented.

Non-comparative scales, also called monadic scales, require respondents to evaluate each object independently rather than comparing it directly with another object.

Their two main forms are:

• Continuous rating scale: A respondent marks a position on a continuous line between two extreme descriptions. For example:

  Poor Excellent

  The marked position is converted into a numerical score.

• Itemized rating scale: Respondents select one response from a limited number of ordered categories. Common examples include Likert, semantic differential, and Stapel scales.

Features of non-comparative scales:

• Each object receives an independent score.
• They are useful for measuring attitudes, perceptions, and satisfaction.
• Results can often be compared across respondents and studies.
• Their quality depends on clear labels, balanced categories, and an appropriate number of response options.

These are widely used itemized non-comparative rating scales:

• Likert scale: A statement is followed by responses such as strongly disagree, disagree, neutral, agree, and strongly agree. Scores may range from 1 to 5.
• Semantic differential scale: Respondents rate an object between bipolar adjectives, such as reliable/unreliable or modern/old-fashioned.
• Stapel scale: A single adjective is placed at the center of a scale. Positive scores show that the adjective describes the object accurately, while negative scores show the opposite.

The Likert scale is usually easiest to construct, the semantic differential scale is useful for profile analysis, and the Stapel scale is useful when suitable bipolar adjectives are unavailable.

Comparative scales involve the direct comparison of two or more alternatives, whereas non-comparative scales evaluate each alternative independently.

Comparative scales can detect small differences and are often easy to understand. However, they do not reveal an absolute level of preference. Non-comparative scales provide independent ratings and permit broader statistical analysis, but they may be affected by respondents using the scale differently.

Primary data are original data collected by a researcher specifically for the problem currently under investigation.

Principal collection methods include:

• Surveys: Information is obtained through questionnaires or interviews conducted in person, by telephone, by mail, or online.
• Observation: Behaviors, events, or conditions are systematically watched and recorded without necessarily questioning participants.
• Experiments: One or more independent variables are manipulated to study their effect on a dependent variable under controlled conditions.
• Focus groups: A moderator guides a small group discussion to explore opinions, motives, and reactions.
• In-depth interviews: Detailed, flexible interviews are used to investigate individual experiences and attitudes.
• Panels and diaries: The same respondents provide information repeatedly over time.

Primary data are relevant and current, but their collection can be costly, time-consuming, and operationally difficult.

Secondary data are data that have already been collected for a purpose other than the research problem currently being studied.

Internal sources originate within the organization and include:

• Sales invoices and transaction records
• Customer databases
• Financial statements
• Inventory and production reports
• Previous research reports
• Website and customer-service analytics

External sources originate outside the organization and include:

• Government census and statistical publications
• Reports of international agencies
• Academic journals, books, and theses
• Trade associations and industry reports
• Commercial research databases
• Newspapers, websites, and open-data portals

Secondary data are generally inexpensive and quickly available. However, the researcher must evaluate their relevance, accuracy, definitions, collection method, coverage, and date of publication before using them.

Primary and secondary data can be compared as follows:

A researcher should normally examine secondary sources first because they can clarify the problem and prevent unnecessary data collection. Primary data should be collected when available secondary data are incomplete, outdated, unreliable, or unsuitable for the research objectives.

Secondary data should be evaluated systematically before use. Important criteria include:

• Relevance: The data should correspond to the research question, population, variables, geographic area, and required level of detail.
• Accuracy: The researcher should examine the reputation of the source, sampling method, response rate, and possible errors.
• Currency: The collection and publication dates must be appropriate for the current decision context.
• Methodology: Definitions, measurement procedures, sampling design, and data-processing methods should be transparent.
• Coverage: The data should adequately represent the target population and time period.
• Consistency: Figures should be compared with other credible sources to identify contradictions.
• Objectivity: The purpose and possible bias of the organization that produced the data should be considered.

Data should not be used merely because they are readily available. Their original definitions and collection context must be compatible with the present study.

A questionnaire is a structured set of written or verbal questions designed to obtain information from respondents for a research study.

Major steps in its design are:

1. Define the research objectives and information required.
2. Identify the target respondents and mode of administration.
3. Decide the content of each question.
4. Select open-ended or closed-ended question formats.
5. Choose appropriate wording that is simple, specific, and neutral.
6. Determine suitable measurement scales and response categories.
7. Arrange questions in a logical sequence, beginning with easy and engaging items.
8. Design clear instructions, layout, numbering, and skip patterns.
9. Prepare introductory and classification questions where necessary.
10. Pretest the questionnaire with respondents similar to the target population.
11. Revise, finalize, and document the instrument.

A well-designed questionnaire improves validity, response accuracy, and completion rates.

Open-ended questions allow respondents to answer in their own words, while closed-ended questions require them to select from predefined alternatives.

Open-ended questions:

• Advantages: Provide rich detail, reveal unexpected ideas, and are useful in exploratory research.
• Limitations: Require more time to answer, are difficult to code, and may produce irrelevant or incomplete responses.
• Example: "What improvements would you suggest for this service?"

Closed-ended questions:

• Advantages: Are easy to answer, code, compare, and analyze statistically.
• Limitations: May restrict respondents' views and can introduce bias if important alternatives are omitted.
• Example: "How satisfied are you with this service: very dissatisfied, dissatisfied, neutral, satisfied, or very satisfied?"

Open-ended questions are appropriate when possible answers are not fully known. Closed-ended questions are preferable when categories can be clearly defined and standardized measurement is required.

Effective questionnaire wording should follow these principles:

• Use simple, familiar, and unambiguous language.
• Ask only one issue in each question and avoid double-barrelled questions.
• Avoid leading, loaded, threatening, and emotionally biased wording.
• Specify the relevant reference period, units, and context.
• Avoid unnecessary technical terms, abbreviations, and double negatives.
• Provide response alternatives that are mutually exclusive and collectively exhaustive.
• Include neutral options or a "do not know" choice when justified.

Questions should be sequenced logically:

• Begin with easy, relevant, and interesting questions.
• Move from general questions to specific questions using the funnel approach.
• Group questions on the same topic together.
• Place sensitive, difficult, and demographic questions near the end.
• Use clear transitions and skip instructions.

Good wording and sequencing reduce misunderstanding, order effects, respondent fatigue, and measurement error.

Data preparation converts raw responses into an accurate and analyzable data set. Its main stages are:

• Editing: Reviewing questionnaires for completeness, consistency, legibility, and obvious errors.
• Coding: Assigning numerical or symbolic codes to responses. Open-ended responses may first be grouped into categories.
• Classification: Organizing data into meaningful and homogeneous groups.
• Data entry: Transferring coded information into statistical software or a database.
• Validation: Checking that entered values correspond to the original records and permitted ranges.
• Data cleaning: Detecting duplicates, impossible values, inconsistencies, outliers, and missing observations.
• Treatment of missing data: Choosing an appropriate method such as deletion, imputation, or a separate missing-value code.
• Transformation: Recoding variables, computing composite scores, or standardizing values when required.
• Tabulation: Summarizing observations in frequency tables and cross-tabulations.

Each modification should be documented to preserve transparency and reproducibility.

Central tendency refers to a single representative value that indicates the center or typical value of a data distribution. The principal measures are the arithmetic mean, median, and mode.

A good measure of central tendency should:

• Be clearly and rigidly defined.
• Be easy to understand and calculate.
• Use an adequate number of observations.
• Be representative of the entire data set.
• Not be unduly affected by extreme values.
• Be capable of further algebraic treatment.
• Show stability across different samples drawn from the same population.
• Be appropriate for the scale and shape of the data.

No measure satisfies every requirement in all situations. The mean is algebraically useful, the median is resistant to extreme values, and the mode identifies the most frequent or popular value.

The arithmetic mean is the sum of all observations divided by the number of observations.

For individual observations :

For a frequency distribution in which has frequency :

For grouped continuous data, represents the midpoint of each class:

If observations have different weights , the weighted mean is:

Merits: It uses all observations, is uniquely defined, and supports algebraic analysis.

Limitations: It is affected by extreme values, may be misleading for skewed distributions, and is generally unsuitable for nominal or purely ordinal data.

The median is the middle value of an ordered data set, dividing it into two equal parts.

For individual observations arranged in ascending order:

• If is odd, the median is the value at position .
• If is even, the median is the average of the values at positions and .

For grouped continuous data:

where:

• = lower boundary of the median class
• = total frequency
• = cumulative frequency before the median class
• = frequency of the median class
• = class width

Merits: It is resistant to extreme values, suitable for skewed data, and applicable to ordinal data and open-ended classes.

Limitations: It does not use the exact magnitude of every observation and has limited algebraic usefulness.

The mode is the value or category that occurs most frequently in a data set. For ungrouped data, it is identified by inspecting the frequencies.

For grouped continuous data:

where:

• = lower boundary of the modal class
• = frequency of the modal class
• = frequency of the class preceding the modal class
• = frequency of the class following the modal class
• = class width

For a moderately skewed distribution, the empirical relationship is:

The mode is useful for nominal data and for identifying the most common product size, brand, response, or category. It is easy to understand and unaffected by extreme values. However, it may be absent, may not be unique, and is unstable when sample frequencies change.