Frequency Distributions in Statistics

Chapter 2: Frequency Distributions

Learning Outcomes

  • Understand how frequency distributions are used.

  • Organize data into a frequency distribution table and into a grouped frequency distribution table.

  • Know how to interpret frequency distributions.

  • Organize data into frequency distribution graphs.

  • Know how to interpret and understand graphs.

Tools You Will Need

  • Proportions (Math Review, Appendix A)

  • Fractions

  • Decimals

  • Percentages

  • Scales of Measurement (Chapter 1)

    • Nominal, ordinal, interval, and ratio

  • Continuous and Discrete Variables (Chapter 1)

  • Real Limits (Chapter 1)

2.1 Frequency Distributions

  • Definition of Frequency Distribution:

    • An organized tabulation that shows the number of individuals located in each category on the scale of measurement.

  • Forms:

    • Can be represented as either a table or a graph.

  • Components:

    • Always shows:

    • The categories that make up the scale.

    • The frequency (the number of individuals) in each category.

Frequency Distribution Tables

  • Structure of Frequency Distribution Table:

    • Categories listed in a column (often ordered from highest to lowest).

    • Frequency count adjacent to each category.

    • Total Frequency: ext{Σ}f = N, where (N) is the total number of subjects.

  • To compute (ΣX) from a table:

    • Convert table back to original scores, or

    • Compute (ΣfX) where (X) is the score.

Proportions and Percentages

  • Proportions:

    • Measures the fraction of the total group associated with each score, referred to as relative frequencies.

    • Relative frequency formula: (p = \frac{f}{N}), where (f) is frequency and (N) is total.

  • Percentages:

    • Expresses the relative frequency out of 100.

    • Percentage formula: (\text{percent} = p(100)).

Example of Frequency, Proportion, and Percent

  • For specific values:

    • ( X \, f \, p \, \text{percent} )

    • 5: ( p = \frac{1}{10} = 0.10, \text{percent} = 10\% )

    • 4: ( p = \frac{2}{10} = 0.20, \text{percent} = 20\% )

    • 3: ( p = \frac{3}{10} = 0.30, \text{percent} = 30\% )

    • 2: ( p = \frac{3}{10} = 0.30, \text{percent} = 30\% )

    • 1: ( p = \frac{1}{10} = 0.10, \text{percent} = 10\% )

Learning Checks

  • Learning Check 1:

    • Use the Frequency Distribution Table to determine how many subjects were in the study. The distribution is represented by:

    • ( X \, f : 5 \, 2 \, 4 \, 4 \, 3 \, 1 \, 2 \, 0 \, 1 \, 3 )

2.2 Grouped Frequency Distribution Tables

  • Purpose:

    • If the number of categories is extensive, categories are combined (grouped) to simplify understanding.

    • Note: Grouping can result in loss of information, as individual scores cannot be retrieved. The wider the grouping interval, the more information lost.

Rules for Constructing Grouped Frequency Distributions

  • Requirements (Mandatory Guidelines):

    • All intervals must be of the same width.

    • The lower score of each interval should be a multiple of the interval width.

  • Rules of Thumb (Suggested Guidelines):

    • Use ten or fewer class intervals.

    • Choose a “simple” number for interval width (e.g., 2, 5, 10).

Discrete Variables in Frequency or Grouped Distributions

  • Constructing frequency distributions for discrete variables is straightforward as each individual score represents an exact measurement.

Continuous Variables in Frequency Distributions

  • Understanding continuous variables:

    • A recorded score represents an interval, which indicates the score may encompass various actual values within its real limits.

    • Rounding occurs to the nearest middle value between the score's real limits.

  • When grouping continuous variables, group several intervals, affecting the apparent limits of the grouped class, which are always one unit smaller than the real limits.

Learning Checks and True/False Statements

  • Learning Check 2: Find the width of the interval 20-29 in a grouped frequency distribution table:

    • Answer: Width = 10

  • True/False Statements: Once data is grouped, original scores cannot be reconstructed from the frequency distribution table.

2.3 Frequency Distribution Graphs

  • General Characteristics:

    • Frequency distribution graphs provide a visual representation of data organized in tables.

    • All graphs consist of two axes:

    • X-axis (abscissa): Typically depicts measurement categories increasing from left to right.

    • Y-axis (ordinate): Typically shows frequencies increasing from bottom to top.

    • General Principles: Each axis should value 0 at the point they meet, and the height of the graph should be roughly 2/3 to 3/4 of the length of the axis.

Data Graphing Questions

  • Before graphing, consider:

    • Level of measurement (nominal, ordinal, interval, ratio).

    • Whether data is discrete or continuous.

    • Whether you're describing samples or populations.

Frequency Distribution Histogram

  • Requires numeric data (interval or ratio) and includes:

    • Scores on the X-axis covering minimum to maximum observed values.

    • Bars drawn above each score (interval), with height corresponding to frequency and width corresponding to score real limits.

Example of Frequency Histogram

  • Figure 2.1 shows quiz scores (number correct) represented in a histogram.

Grouped Frequency Distribution Histogram

  • Similar to the frequency distribution histogram, but bars are created above each grouped class interval.

    • The bar width is based on class interval real limits, extending apparent limits out one-half score unit at each interval's end.

Modified (Block) Histogram

  • A block histogram represents frequencies by stacking blocks to show counts.

Frequency Distribution Polygons

  • Constructed by listing all numeric scores on the X-axis, including scores with a frequency of zero.

  • A dot is placed above each center of the interval corresponding to frequency, connected with a line to form a polygon.

Graphs for Nominal or Ordinal Data

  • Utilizes a bar graph similar to a histogram, with spacing indicating discrete categories for nominal data (no order) and ordinal data (non-measurable width).

Population Distribution Graphs

  • When the population size is small, scores for each member create a histogram. For large populations, relative frequency graphs using smooth curves indicate approximate scores.

  • Normal Distribution: Symmetrical with the highest frequency at the center, common in many variable data.

Frequency Distribution Shape

  • Descriptions of Distribution Shape:

    • Symmetrical Distribution: Both sides mirror each other.

    • Skewed Distribution: Scores are clustered on one side, tapering off on the other;

    • Positive Skew: Tail on the right (high scores).

    • Negative Skew: Tail on the left (low scores).

Learning Checks for Distribution Shapes

  • Learning Check 3: Determine the shape of a distribution provided in a question.

  • Evaluate the truth of given statements regarding variables and their graphical representations.

Conclusion

  • All discussions relate to the entire methodology of organizing, interpreting, and visually presenting data sets. It's vital to understand each component's definitions, structures, and usage in the context of frequencies, distributions, and their implications.