Unit 3: Measurement Scales and Central Tendency

I. Orientation

Research methodology depends on converting abstract concepts such as satisfaction, income, intelligence, or productivity into observable and analysable information. Measurement assigns numbers or labels to characteristics according to defined rules, while scaling places observations along a continuum or category system. The quality of later statistical analysis depends on the scale used, the source and preparation of data, and the summary measure selected.

• Objectivity: Measurement rules should be consistent so that different researchers obtain comparable results.
• Validity: The instrument should measure the intended characteristic; a satisfaction scale should not mainly measure respondent mood or reading ability.
• Reliability: Repeated measurement under similar conditions should produce reasonably stable results.
• Level of measurement: Nominal, ordinal, interval, and ratio scales permit progressively stronger mathematical operations.
• Data quality: Accurate collection, editing, coding, classification, and tabulation are necessary before analysis.
• Representative summary: Central tendency identifies a typical or central value, but the appropriate measure depends on the data distribution and measurement scale.

II. Measurement Scales — Rules for Assigning Meaning to Data

Measurement scales provide the structure through which numerical values or categories represent attributes. Their level determines which comparisons and statistical calculations are meaningful.

A. Measurement Scales

Measurement scales are systems for classifying observations according to identity, order, equal distance, and meaningful origin.

• Nominal scale: It classifies observations into mutually exclusive categories without ranking.
  • Example: department codes such as 1 = Finance, 2 = Marketing, and 3 = Human Resources.
  • Valid operations include counting, percentages, and identifying the mode; calculating an average of department codes has no meaning.
• Ordinal scale: It ranks observations, but the distances between ranks are not necessarily equal.
  • Example: service ratings of Poor, Fair, Good, and Excellent.
  • A respondent choosing rank 4 is above rank 3, but the difference between “Fair” and “Good” may not equal the difference between “Good” and “Excellent.”
• Interval scale: It has ordered values with equal intervals, but zero does not represent complete absence.
  • Example: temperature in Celsius; the difference between 20°C and 30°C equals the difference between 30°C and 40°C, but 40°C is not twice as hot as 20°C.
  • Mean and standard deviation are generally appropriate.
• Ratio scale: It has order, equal intervals, and a true zero indicating absence of the measured quantity.
  • Example: income, age, distance, weight, and number of purchases.
  • Statements such as “20 kg is twice 10 kg” are meaningful.
• Scale selection: Nominal data require categorical analysis, whereas interval and ratio data can support more extensive arithmetic and statistical procedures.

III. Comparative and Non-Comparative Scales — Measuring Judgements

Scaling techniques are commonly divided according to whether respondents evaluate alternatives against one another or assess each alternative independently.

A. Comparative and Non-Comparative Scales

Comparative scales require a relative judgement between objects; non-comparative scales require an independent evaluation of each object.

• Comparative scales: The respondent compares two or more objects directly.
  • Paired comparison: One item is compared with another, such as choosing between Brand A and Brand B.
  • If n objects are compared in every possible pair, the number of comparisons is:
    
    TEXT

    Number of pairs = n(n - 1) / 2

  • Here, n is the number of objects. For four brands, the researcher obtains 4(3)/2 = 6 pairs.
  • Rank-order scaling: Respondents arrange objects from most preferred to least preferred, such as ranking five banks from 1 to 5.
  • Constant-sum scaling: A fixed total, commonly 100 points, is distributed among attributes. Giving 50 points to price and 30 to quality leaves 20 points for other attributes.
• Non-comparative scales: Each object is rated independently, allowing several objects to receive the same score.
  • Likert scale: Respondents indicate agreement, commonly from 1 = Strongly disagree to 5 = Strongly agree.
  • Semantic differential: An object is rated between bipolar adjectives, such as Unreliable 1 2 3 4 5 Reliable.
  • Stapel scale: A single adjective is rated on a positive and negative numerical range, such as +5 to -5 for “modern.”
• Choice of technique: Comparative scales are useful for preference ordering and market choice, while non-comparative scales are useful for measuring attitudes, perceptions, and intensity.
• Limitations: Comparative results may depend on the alternatives shown; non-comparative results may suffer from acquiescence, central tendency, or extreme-response bias.

IV. Primary and Secondary Data Sources — Origins of Evidence

Research data may be collected directly for the current investigation or obtained from information previously collected for another purpose. The distinction affects relevance, cost, and control over quality.

A. Primary and Secondary Data Sources

Primary data are first-hand observations gathered for a specific research problem, whereas secondary data already exist in published or unpublished form.

• Primary data: The researcher collects information directly from respondents, events, or measurements.
  • Methods: Surveys, interviews, observation, experiments, focus groups, and direct physical measurement.
  • Strength: The researcher controls the population, variables, wording, timing, and collection procedure.
  • Example: Interviewing 300 employees in 2025 to measure current job satisfaction.
  • Limitation: Collection may require substantial time, money, training, and field supervision.
• Secondary data: Information collected earlier by another person, institution, or agency.
  • Internal sources: Sales records, payroll files, customer complaints, accounting statements, and previous research reports.
  • External sources: Government census reports, statistical abstracts, academic journals, trade associations, company reports, and credible databases.
  • Strength: It is usually faster and less expensive than conducting a new survey.
  • Limitation: Definitions, sampling methods, dates, units, and accuracy may not match the present study.
• Evaluation criteria: Researchers should examine authority of the source, purpose of collection, method used, coverage, date, consistency, and possible bias.
• Combined use: Secondary data can define the problem and sampling frame, while primary data can fill gaps or provide current evidence.

V. Questionnaire and Preparation of Data — From Responses to an Analysis File

A questionnaire converts research objectives into standardised questions. Preparation of data then transforms completed instruments into a clean, structured dataset suitable for analysis.

A. Questionnaire

A questionnaire is a systematically arranged set of questions used to obtain comparable information from respondents.

• Purpose and alignment: Every question should serve an objective or variable in the research design; unnecessary questions increase fatigue and non-response.
• Question wording: Questions should be simple, specific, neutral, and limited to one idea.
  • Poor: “Are the staff helpful and efficient?”
  • Better: Separate questions on helpfulness and efficiency because they are different attributes.
• Question types: Open-ended questions permit unrestricted answers; closed-ended questions provide alternatives such as yes/no, categories, ratings, or rankings.
• Response options: Categories should be mutually exclusive and collectively exhaustive. Age categories such as 18–25, 26–35, and 36–45 should not overlap.
• Question order: Begin with easy, relevant questions; place sensitive or demographic questions later unless they are needed for screening.
• Instructions and layout: Clear directions, numbering, adequate spacing, and logical skip instructions reduce interviewer and respondent error.
• Pilot testing: A small pre-test reveals ambiguity, missing response choices, excessive length, and technical problems before the main survey.
• Ethics: Respondents should receive appropriate information about purpose, voluntary participation, confidentiality, and the use of their responses.

B. Preparation of Data

Preparation of data is the process of checking, transforming, and organising raw responses before statistical analysis.

• Editing: Completed questionnaires are checked for completeness, consistency, legibility, and eligibility.
  • Example: A questionnaire from an ineligible respondent should be identified before coding.
• Coding: Verbal answers are assigned symbols or numbers for entry.
  • Example: Yes = 1, No = 0; numerical codes are labels unless the scale genuinely supports arithmetic.
• Classification: Similar responses are grouped into meaningful categories, such as grouping occupations into professional, technical, administrative, and service work.
• Data entry: Responses are entered into a data file where rows commonly represent cases and columns represent variables.
• Cleaning: The dataset is checked for duplicate records, impossible values, missing observations, and inconsistent responses.
  • Example: A reported age of -4 is an invalid value requiring verification or treatment as missing.
• Tabulation: Frequencies, percentages, cross-tabulations, and summary tables organise the data for interpretation.
• Missing-data treatment: The researcher may retrieve the answer, leave it missing, use a justified statistical treatment, or exclude the case; the decision should be documented.
• Data file structure: A codebook should identify each variable, label, unit, permitted values, and missing-value code.

VI. Measurement of Central Tendency — Identifying a Typical Value

Measures of central tendency condense a distribution into one representative or central value. The mean, median, and mode are the principal measures, and each has different conditions of suitability.

A. Measurement of Central Tendency

Central tendency describes the location around which observations cluster, but no single measure is best for every dataset.

• Arithmetic mean: The mean is the sum of all observations divided by their number.
  
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  x̄ = Σx / n

  • x̄ is the sample mean, Σx is the sum of observations, and n is the number of observations.
  • For 4, 6, 8, 12, the mean is (4 + 6 + 8 + 12)/4 = 7.5.
  • It uses every value and is useful for interval or ratio data, but extreme values can distort it.
• Weighted mean: Values receive different importance or frequencies.
  
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  x̄w = Σ(wx) / Σw

  • w is the weight and x is the value. A course grade based on 30% assignment and 70% examination uses these percentages as weights.
• Median: The median is the middle observation after values are arranged in ascending or descending order.
  • For an odd number of observations, it is the central value; for an even number, it is the average of the two central values.
  • It is appropriate for ordinal data and skewed income or property-price distributions because it is less affected by extremes.
• Mode: The mode is the most frequently occurring value or category.
  • In 2, 3, 3, 4, 5, the mode is 3; for nominal data such as most-selected transport type, it may be the only suitable central measure.
  • A distribution may be bimodal or multimodal, or may have no mode if every value occurs equally often.
• Geometric mean: The geometric mean is useful for growth rates and multiplicative changes.
  
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  GM = (x₁ × x₂ × ... × xₙ)^(1/n)

  • x₁ ... xₙ are positive observations and n is their number. It should not be used with zero or negative values.
• Harmonic mean: The harmonic mean is suited to averaging rates when the same distance, quantity, or base is involved.
  
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  HM = n / Σ(1/x)

  • x represents each positive rate or value. It is especially sensitive to small observations.
• Selection principle: Use the mode for nominal categories, the median for ordinal or highly skewed data, and the mean for reasonably symmetric interval or ratio data.
• Interpretation: Central tendency should be considered with dispersion and distribution shape; two datasets can share a mean while having very different variability.