Unit 3: Measurement Scales and Central Tendency

1. Measurement Scales

Measurement in research consists of assigning numbers to empirical events, objects, or properties, or activities in compliance with a set of rules. The type of scale used determines the statistical techniques that can be applied to the data.

The Four Levels of Measurement (NOIR)

A. Nominal Scale

• Definition: The lowest level of measurement. Numbers serve only as labels or tags for identifying and classifying objects.
• Characteristics: No numerical value, order, or distance.
• Examples: Gender (1 = Male, 2 = Female), Student Roll Numbers, Jersey numbers of football players.
• permissible Statistics:
  • Descriptive: Mode, Frequency counts, Percentages.
  • Inferential: Chi-Square.

B. Ordinal Scale

• Definition: A ranking scale in which numbers are assigned to objects to indicate the relative extent to which the objects possess some characteristic.
• Characteristics: Determines order (greater than/less than), but the intervals between points are not necessarily equal.
• Examples: Academic Ranks (1st, 2nd, 3rd), Socioeconomic Status (Low, Medium, High), Customer Satisfaction Rankings.
• Permissible Statistics:
  • Descriptive: Median, Percentile, Quartile.
  • Inferential: Rank-order correlation, Friedman ANOVA.

C. Interval Scale

• Definition: A scale where numerically equal distances on the scale represent equal values in the characteristic being measured.
• Characteristics: Has order and equal intervals, but no true zero point (the zero is arbitrary).
• Examples: Temperature (Celsius or Fahrenheit), Calendar years (2023, 2024), IQ scores.
  • Note: 40°C is not "twice as hot" as 20°C because 0°C does not mean "no heat."
• Permissible Statistics:
  • Descriptive: Arithmetic Mean, Standard Deviation.
  • Inferential: t-tests, ANOVA, Regression.

D. Ratio Scale

• Definition: The highest level of measurement. It possesses all properties of the nominal, ordinal, and interval scales.
• Characteristics: Has an absolute (true) zero point, meaning the absence of the variable. Ratios are meaningful (e.g., 10kg is twice as heavy as 5kg).
• Examples: Height, Weight, Age, Income, Sales figures.
• Permissible Statistics: All statistical techniques (Geometric Mean, Harmonic Mean, Coefficient of Variation).

2. Comparative and Non-Comparative Scales

Scaling techniques are methods of placing respondents on a continuum with respect to their attitude toward the stimulus object.

A. Comparative Scales

In comparative scaling, there is a direct comparison of stimulus objects. Data must be interpreted in relative terms (Ordinal or Rank order data).

1. Paired Comparison Scaling:
   • A respondent is presented with two objects and asked to select one according to some criterion.
   • Example: "Do you prefer Coke or Pepsi?"
2. Rank Order Scaling:
   • Respondents are presented with several objects simultaneously and asked to order or rank them.
   • Example: "Rank the following brands of phones from 1 (Most preferred) to 4 (Least preferred)."
3. Constant Sum Scaling:
   • Respondents allocate a constant sum of units (usually 100 points) among a set of stimulus objects with respect to some criterion.
   • Example: "Distribute 100 points among the following attributes of a laptop based on importance: Battery, Screen, Speed."

B. Non-Comparative Scales

In non-comparative scaling, each object is scaled independently of the others in the set. The resulting data is generally assumed to be Interval or Ratio scaled.

1. Continuous Rating Scales (Graphic Rating Scales):

   • Respondents rate objects by placing a mark at the appropriate position on a line that runs from one extreme of the criterion variable to the other.
   • Example: Reaction to TV Ad: Very Negative _____________________ Very Positive

2. Itemized Rating Scales:

   • Likert Scale: A measurement scale with five (or seven) response categories ranging from "strongly disagree" to "strongly agree."
     • Example: "I enjoy statistics." (1. Strongly Disagree ... 5. Strongly Agree).
   • Semantic Differential Scale: A seven-point rating scale with endpoints associated with bipolar labels (adjectives) that have semantic meaning.
     • Example: Service is: Fast _ _ _ _ _ _ _ Slow
   • Stapel Scale: A unipolar rating scale with ten categories numbered from -5 to +5, without a neutral point (zero). Used to measure the direction and intensity of an attitude.

3. Primary and Secondary Data Sources

A. Secondary Data

Data that have already been collected for purposes other than the problem at hand.

• Internal Sources: Data generated within the organization (Sales invoices, accounting records, CRM data).
• External Sources:
  • Government Sources: Census, tax records.
  • Syndicated Services: Nielsen, Kantar (sell data to many clients).
  • Publications: Academic journals, trade magazines.
• Advantages: Low cost, short time to acquire, easy access.
• Disadvantages: Data may be outdated, units of measurement may not match, accuracy may be questionable.

B. Primary Data

Data originated by a researcher for the specific purpose of addressing the problem at hand.

• Qualitative Methods:
  • Focus Groups: 6-10 people discussing a topic under a moderator.
  • Depth Interviews: One-on-one probing interviews.
  • Projective Techniques: Word association, sentence completion.
• Quantitative Methods:
  • Surveys: Questionnaires (Online, Mail, Telephone).
  • Observation: Watching behavioral patterns (e.g., watching shoppers in a store).
  • Experiments: Manipulating independent variables to measure the effect on dependent variables.
• Advantages: Current, relevant to the specific research question, controllable methodology.
• Disadvantages: High cost, time-consuming, requires specialized skills.

4. Questionnaire and Preparation of Data

A. Questionnaire Design Process

A questionnaire is a structured technique for data collection consisting of a series of questions, written or verbal, that a respondent answers.

1. Specify the Information Needed: Ensure every question relates to the hypothesis/objectives.
2. Specify the Type of Interviewing Method: (Personal, Telephone, Mail, Electronic).
3. Determine Individual Question Content:
   • Is the question necessary?
   • Are several questions needed instead of one?
4. Design Question Structure:
   • Unstructured (Open-ended): "What is your opinion on..."
   • Structured (Closed-ended): Multiple choice, Dichotomous (Yes/No), Scales.
5. Determine Question Wording:
   • Avoid ambiguity (e.g., "frequently").
   • Avoid leading questions.
   • Avoid generalizations.
6. Arrange Question Order:
   • Funnel Approach: General questions first, specific questions later.
   • Sensitive questions (income, age) should be placed at the end.
7. Form and Layout: Professional appearance, easy to read.
8. Pre-testing (Pilot Testing): Testing the questionnaire on a small sample to identify potential problems before the full launch.

B. Data Preparation

Once questionnaires are returned, the data must be converted into a format suitable for analysis.

1. Questionnaire Checking: Checking for completeness and interview quality.
2. Editing: Reviewing questionnaires to increase accuracy and precision. Fixing illegible, incomplete, or inconsistent answers.
3. Coding: Assigning a code (usually a number) to each possible response to each question.
   • Example: Male = 0, Female = 1.
4. Transcribing/Data Entry: Entering data into a computer (Excel, SPSS, R).
5. Data Cleaning:
   • Consistency Checks: Logic checks (e.g., ensuring a respondent isn't 5 years old and married).
   • Treatment of Missing Responses: Substituting a neutral value, mean substitution, or casewise deletion.

5. Measurement of Central Tendency

Measures of Central Tendency provide a single value that describes the "center" or representative value of a data set.

A. Mean (Arithmetic Mean)

• Definition: The sum of all observations divided by the number of observations.
• Formula:
• Advantages: Most popular, uses all data points, capable of further algebraic treatment.
• Disadvantages: Highly affected by extreme values (outliers).
• Application: Used for Interval and Ratio data.

B. Median

• Definition: The middle value of a distribution when the data is arranged in ascending or descending order. It divides the distribution into two equal halves.
• If N is odd: The middle term.
• If N is even: The average of the two middle terms.
• Advantages: Not affected by outliers/extreme values.
• Disadvantages: Does not use all data values; requires sorting.
• Application: Best for Ordinal data or skewed distributions (e.g., Income data).

C. Mode

• Definition: The value that occurs most frequently in a dataset.
• Characteristics: A dataset may have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).
• Advantages: The only average that can be used for Nominal data.
• Disadvantages: Not stable, mathematically less defined.

D. Geometric Mean (GM)

• Definition: The -th root of the product of items.
• Formula:
• Application: Used for finding average rates of change, growth rates, and ratios.

E. Harmonic Mean (HM)

• Definition: The reciprocal of the arithmetic mean of the reciprocals of the individual observations.
• Application: Used for averaging rates and speeds (e.g., average speed of a car traveling different distances).

Relationship between Mean, Median, and Mode

• Symmetrical Distribution (Normal Curve): Mean = Median = Mode.
• Positively Skewed (Tail to the right): Mean > Median > Mode.
• Negatively Skewed (Tail to the left): Mean < Median < Mode.