Chapter 2: Frequency Distributions Notes

Frequency Distributions: Key Concepts

  • Frequency distribution: an organization of all scores (or categories) showing how often each score occurs in a dataset.
  • Purpose: summarize large data by counting occurrences, facilitating understanding of data shape, central tendency, and variability.
  • Raw scores: the scores measured in a study (the individual data points).
  • Frequency (f): the number of times a particular score occurs.
  • N: the total number of scores in the dataset (sample size).
  • A frequency distribution table lists each score (or category) and its frequency f; N is the sum of all f values in the table.

New Symbols and Terminology

  • Raw scores: the scores we initially measure in a study.
  • Frequency of a score: how many times a given score occurs in the data (denoted by f).
  • N: the sample size or total number of scores in the data.
  • Summary relation: a frequency distribution organizes scores based on the frequency of occurrence.

How to Create a Frequency Distribution

  • Identify all possible scores or categories (the X-axis values).
  • Tally how many times each score occurs (compute f for each category).
  • Compute total N as the sum of all frequencies: N =
    \sum f.
  • Optional: compute relative frequencies and percentages:
    • Relative frequency: RF=fN\text{RF} = \frac{f}{N}
    • Percentile of a score (see the Percentile section).

Practice: Raw Scores to f Table (Concept)

  • Example context: A practice item converting raw scores to a frequency table, where N is the sum of reported frequencies.
  • Typical steps shown: list X values (e.g., days per week, item responses), count f for each X, and confirm that (\sum f = N).
  • Note: The transcript includes a practice exercise with a small dataset (N = 14 in one example) to illustrate building the frequency table from raw scores.

Graphing Frequency Distributions

  • A frequency distribution graph always has scores on the X axis and frequency on the Y axis.
  • The type of graph depends on the measurement scale:
    • Nominal/Ordinal data: use a Bar Graph.
    • Interval/Ratio data: use a Histogram or a Frequency Polygon.
  • The choice of graph affects interpretation and readability, not the underlying data.

Relative Frequency and Percentiles

  • Relative frequency is the proportion of the time a score occurs in a sample:
    • Standard formula: RF=fN\text{RF} = \frac{f}{N}
    • Percent relative frequency: RF%=fN×100%\text{RF}_{\%} = \frac{f}{N} \times 100\%
  • Note on a transcript inconsistency: one slide lists Relative frequency as RF=Nf\text{RF} = N f (which is not the standard definition). The typical, accepted definition is RF=fN.\text{RF} = \frac{f}{N}.
  • Caution: Always verify which version your course uses; many texts present RF as a proportion and RF% as a percentage.

Finding Relative Frequency Using the Normal Curve

  • The proportion of the total area under the normal curve occupied by a group of scores corresponds to that group’s relative frequency.
  • In practice: approximate RF by the area under the normal curve corresponding to that score range.
  • Conceptual link: RF aligns with the probability under the normal distribution curve for the interval containing the scores.

Graphing Frequency Distributions: Graph Types and Uses

  • Bar graphs (nominal/ordinal data): show category frequencies; example: preferred exercise type (Machines, Spinning, Aerobics).
  • Histograms (interval/ratio data): show distribution of a continuous variable (e.g., days per week you buy overpriced coffee).
  • Frequency polygons: another way to display interval/ratio data; lines connect the midpoints of histogram bins.
  • Key idea: All three graphs present the same data; the axes differ and affect interpretation.

Frequency Distribution Shapes

  • The shape of the frequency distribution is important for choosing statistical procedures and for interpretation.
  • Common shapes include Normal, Skewed (negative or positive), and Bimodal.

The Normal Distribution

  • Shape: bell-shaped, symmetric, with most scores near the center and fewer scores as you move away from the center.
  • Tails: far-left and far-right portions contain relatively low-frequency scores (extreme scores).

Skewed Distributions

  • Definition: not symmetrical; has one pronounced tail.
  • Negative skew: tail extends toward the left (toward lower scores).
  • Positive skew: tail extends toward the right (toward higher scores).
  • Direction of the distinctive tail indicates the type of skew.

Bimodal Distributions

  • Definition: two distinct peaks (modes) in the distribution.
  • Mode: the most common value in the dataset.
  • Interpretation: often indicates two different groups within the data (e.g., two subpopulations).
  • Example: grade distributions where many students score A and many score C, suggesting two groups.

Cumulative Frequency

  • Definition: the cumulative frequency is the number of scores in the data that are at or below a particular score.
  • Notation: for a score x, the cumulative frequency CF(x) = number of scores ≤ x.
  • Calculation: CF(x) is obtained by summing f values up to and including x.

Percentile and Cumulative Frequency (Concepts and Calculation)

  • Percentile: the percent of all scores that fall at or below a particular score x.
  • Operational method: determine percentile by finding the area under the normal curve to the left of the score, or by using cumulative frequency.
  • Common relation: percentile of x ≈ 100 × (CF(x) / N).
  • Practical note: Percentiles convert an absolute score into a relative standing within the distribution.

Worked Example: Thermostat Data (Cumulative and Relative Frequency)

  • Data set (temperatures X in degrees):
    • 68, 69, 69, 69, 69, 70, 70, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 73, 73, 73
  • N = 25
  • Frequency f by temperature:
    • 68: f=1f=1
    • 69: f=4f=4
    • 70: f=7f=7
    • 71: f=5f=5
    • 72: f=5f=5
    • 73: f=3f=3
  • Cumulative Frequency CF(x) (≤ x):
    • CF(68) = 1
    • CF(69) = 1+4 = 5
    • CF(70) = 5+7 = 12
    • CF(71) = 12+5 = 17
    • CF(72) = 17+5 = 22
    • CF(73) = 22+3 = 25
  • Relative Frequency RF (proportion):
    • RF(68) = (\frac{1}{25}) = 0.04 (4%)
    • RF(69) = (\frac{4}{25}) = 0.16 (16%)
    • RF(70) = (\frac{7}{25}) = 0.28 (28%)
    • RF(71) = (\frac{5}{25}) = 0.20 (20%)
    • RF(72) = (\frac{5}{25}) = 0.20 (20%)
    • RF(73) = (\frac{3}{25}) = 0.12 (12%)
  • Relative Frequency Percentages (RF%):
    • RF%(68) = 4\%
    • RF%(69) = 16\%
    • RF%(70) = 28\%
    • RF%(71) = 20\%
    • RF%(72) = 20\%
    • RF%(73) = 12\%
  • Sanity check: sum of RF% = 100\%.
  • Cumulative relative frequency percentages (as you go up):
    • CF%(68) = 4\%
    • CF%(69) = 4 + 16 = 20\%
    • CF%(70) = 20 + 28 = 48\%
    • CF%(71) = 48 + 20 = 68\%
    • CF%(72) = 68 + 20 = 88\%
    • CF%(73) = 88 + 12 = 100\%
  • These calculations illustrate how cumulative frequency and relative frequency build upon f and N to describe the distribution.

Connections to Principles and Real-World Relevance

  • Frequency distributions connect raw data to interpretable summaries that underpin many statistical methods (e.g., hypothesis tests, estimates, and graphical summaries).
  • The shape of the distribution informs the choice of statistical procedures (e.g., normal-theory methods assume approximate normality).
  • Relative frequency and percentiles provide standardized measures of position within a distribution, enabling comparisons across samples with different sizes.
  • Understanding cumulative frequency and percentiles is essential for interpreting exam scores, test results, and many real-world datasets (e.g., thermostats, performance metrics).

Ethical, Philosophical, and Practical Implications

  • Accurate representation of data is critical; mislabeling a frequency (e.g., confusing f with RF) can lead to incorrect conclusions.
  • Data visualization choices (bar vs histogram vs polygon) influence interpretation; select the graph type that faithfully reflects the data type and distribution.
  • Recognize the limitations of normal-curve approximations for highly skewed or bimodal data; rely on appropriate nonparametric methods or data transformations when needed.

Key Formulas and Notation (Summary)

  • Frequency distribution table: list of scores X with frequencies f.
  • Total sample size: N=∑fN = \sum f
  • Relative frequency: RF=fN\text{RF} = \frac{f}{N}
  • Relative frequency (percent): RF%=fN×100%\text{RF}_{\%} = \frac{f}{N} \times 100\%
  • Cumulative frequency: CF(x)=∑s≤xf(s) ,\text{CF}(x) = \sum_{s \le x} f(s)\,, i.e., number of scores at or below x.
  • Percentile (concept): the percent of all scores below a given score x; equivalently, Percentile(x)=100⋅CF(x)N\text{Percentile}(x) = 100 \cdot \frac{\text{CF}(x)}{N}, or via area under the normal curve: P(X≤x)=∫<em>−∞xf</em>X(t) dt=Φ(x−μσ) .P(X \le x) = \int<em>{-\infty}^{x} f</em>X(t)\,dt = \Phi\left(\frac{x-\mu}{\sigma}\right)\,.