Key Statistical Definitions: Measurement Scales and Data Frequency

Fundamental Statistical Concepts

Estimates and Population Values

Estimates are statistics collected from samples and used to describe population values.

Scales of Measurement

Understanding the different scales of measurement is crucial for data analysis:

• Nominal Scale of Measurement: This is the simplest form of measurement, used when the dependent variable is qualitative. It categorizes data based on the name of some physical or psychological quality or characteristic rather than a numerical score.
• Ordinal Scale of Measurement: This is the simplest quantitative scale used to measure the dependent variable. Data are ranked in some order (such as highest to lowest, biggest to smallest, most important to least important), but it does not measure the extent or magnitude of difference between measures.
• Interval Scale of Measurement: This is a quantitative scale used to measure the dependent variable. Data indicate relative ranks and the degrees of difference between scores, but the zero point on this scale is not meaningful (arbitrary zero).
• Ratio Scale of Measurement: This is the quantitative scale that has all the properties of nominal, ordinal, and interval data, plus a meaningful zero point on the scale. Zero on this scale means the absence of something, which makes ratio comparisons possible.

Data Organization and Frequency Distributions

Basic Data Concepts

Raw Data: These are the scores or numbers that have been collected but have not yet been organized or summarized.

Types of Distributions

• Ranked Distribution: A distribution of numbers in which scores are arranged in order (ranked), typically with the highest number at the top and the lowest number at the bottom of a list.
• Simple Frequency Distributions: Created by listing all possible score values in a distribution and then indicating the frequency (f).

Frequency Measures

Frequency (f): The number of times a score occurs. The sum of the frequencies in a distribution is equal to the total number of scores in the distribution.

Grouped Frequency Distributions and Intervals

In a Grouped Frequency Distribution, the raw data are combined into equal-sized groups called class intervals. The rule of thumb is to create between 10 and 20 class intervals and adjust the size of the intervals accordingly.

Class Intervals: Equal-sized groups of raw data used to summarize data in a grouped frequency distribution.

Limits and Range

• Range: The full extent of scores from the highest to the lowest in the distribution.
• Apparent Limits: The limits of a class interval expressed in the same units as the original data.
• Real Limits: The true extensions of the limits, calculated as the lower apparent limit minus 0.5 unit and the upper apparent limit plus 0.5 unit.
• Midpoint: The average (center) of a class interval.

Cumulative and Relative Frequencies

• Cumulative Frequency (cum f): The total number of scores that fall below the upper real limit of an interval.
• Relative Frequency (rel f): The proportion of scores from the distribution that fall within the real limits of an interval.
• Cumulative Relative Frequency (cum rel f): The total proportion of scores that lie below the real upper limit of the interval.
• Cumulative Percent (cum %): Also known as percentile. It is the percentage of scores that fall below the exact upper limit of the interval. The calculation is: cum % = cum rel f · 100.