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Related Concept Videos

What is a Frequency Distribution00:51

What is a Frequency Distribution

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A frequency is the number of times a value of the data occurs. The sum of all the frequency values represents the total number of students included in the sample. It is commonly used to group data of quantitative types. Frequency distributions can be displayed in a table, histogram, line graph, dot plot, or pie chart, just to name a few. A histogram is a graphical representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to...
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Relative Frequency Histogram01:14

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The relative frequency depicts the proportion of data points that have each value. The frequency tells the number of data points that have each value. Like the histogram, a relative frequency histogram also has the same shape with a horizontal scale (the x-axis), but the vertical scale (the y-axis) is marked with relative frequencies (percentages of the whole) instead of actual frequencies. A relative frequency histogram is a graphical representation of a frequency distribution where the...
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Relative Frequency Distribution00:55

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A relative frequency distribution is the proportion or fraction of times a value occurs in a data set. To find the relative frequencies, one can divide each frequency by the total number of data points in the sample. It is very similar to a regular frequency distribution, except that instead of reporting how many data values fall in a class, a relative frequency distribution reports the fraction of data values that fall in a class. These fractions or proportions are called relative frequencies...
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A percentage frequency distribution, in general, is a display of data that indicates the percentage of observations for each data point or grouping of data points. It is a commonly used method for expressing the relative frequency of survey responses and other data. The percentage frequency distributions are often displayed as bar graphs, pie charts, or tables.
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Sometimes, data gathered from an experiment on a large sample or population are organized into concise tables. In such cases, the frequency of the quantitative data set is plotted in the form of a table. Or else, the data values are grouped into the quantity’s intervals, which form classes, and their respective frequencies are known. That is, the data values are distributed over different categories or classes. This is known as frequency distribution.
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Histogram01:05

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The histogram is a graphical representation in the x-y form of data distribution in a data set. The horizontal x-axis is labeled with what the data represents (for instance, distance from your home to school). The vertical y-axis is labeled either frequency or relative frequency (or percent frequency or probability).
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How to Create and Use Binocular Rivalry
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Rank distributions: Frequency vs. magnitude.

Carlos Velarde1, Alberto Robledo2

  • 1Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico.

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Summary

This study reveals that frequency and magnitude ranked data are functional inverses. This relationship is explained through probability distributions and nonlinear maps, offering insights into phenomena like Benford's Law.

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Area of Science:

  • Data analysis
  • Statistical modeling
  • Complex systems

Background:

  • Ranked data can be analyzed by frequency (occurrences) or magnitude (size).
  • Examples include moon craters, earthquakes, and billionaire wealth.
  • Understanding the relationship between these two perspectives is crucial for data interpretation.

Purpose of the Study:

  • To investigate the functional relationship between frequency and magnitude ranked data.
  • To mathematically define this inverse relationship using probability distributions and iterated maps.
  • To explore how this relationship varies across different decay rates.

Main Methods:

  • Analysis of ranked data distributions (frequency vs. magnitude).
  • Formulation of parent probability distributions to model data generation.
  • Development of analog nonlinear iterated maps to reproduce observed distributions.
  • Extension of the framework using thermodynamic and statistical-mechanical concepts.

Main Results:

  • Frequency and magnitude ranked data are functional inverses of each other.
  • The relationship is precisely defined by the underlying probability distribution and a nonlinear iterated map.
  • For hyperbolic decay (Zipf's Law), distributions are identical (power law).
  • Maximum divergence occurs with logarithmic decay and inverse exponential decay (Benford's Law).
  • Generic differences in power-law exponents appear at various ranks for intermediate decay rates.

Conclusions:

  • The inverse relationship between frequency and magnitude ranked data is a fundamental property.
  • This framework unifies diverse phenomena, from Zipf's Law to Benford's Law.
  • The inclusion of thermodynamic concepts deepens the theoretical understanding of ranked data.