Graphing Frequency Distributions

Introduction to Frequency Distributions

Frequency distributions summarize quantitative data by grouping values into classes and counting how many data points fall into each class. Visualizing these distributions helps in understanding the data's structure and patterns.

• Frequency Distribution: A table that displays the number of occurrences (frequency) of each class or group of data values.

• Graphical Representation: Graphs such as histograms provide a visual summary of the frequency distribution, making it easier to interpret the data.

Frequency Histograms

A frequency histogram is a type of bar graph used to display quantitative data organized into classes. It is distinct from common bar graphs in several ways:

• Bars represent classes: Each bar corresponds to a class interval, with the bar's height indicating the frequency of that class.

• Continuous bars: There are no gaps between bars, as the classes are adjacent and cover the entire range of data without overlap or omission.

• Axes labeling: The vertical axis is labeled "frequency," and the horizontal axis is labeled with the variable name and units.

• Cutpoints: The left and right endpoints of each bar represent the lower and upper boundaries (cutpoints) of the class interval.

• Class midpoints: Sometimes, class midpoints are used instead of cutpoints, with each midpoint placed at the center of the bar's base.

• Scale selection: The vertical axis should start at zero, and the scale should be chosen to accommodate the highest frequency (e.g., counting by 5s or 10s).

• Empty classes: If a class has zero frequency, no bar is drawn for that class, which may create apparent gaps.

Example: A histogram showing the percentage of people aged 65 and over in US states might have class intervals such as 3.0–5.0%, 5.0–7.0%, etc., with the height of each bar representing the number of states in each interval.

Relative Frequency Histograms

A relative frequency histogram is similar to a frequency histogram, but the vertical axis represents the relative frequency (the proportion or percentage of data in each class) instead of the absolute frequency.

• Relative frequency: Calculated as

• Vertical axis: Labeled as "relative frequency" and scaled appropriately (e.g., 0.10, 0.20, etc.).

• Interpretation: Shows the proportion of the dataset in each class, making it easier to compare distributions with different sample sizes.

Example: If the largest relative frequency is 0.36, the vertical axis might be marked in increments of 0.10 up to 0.40.

Single-Valued Classes

For data where each class represents a single value (such as the number of people in a household), histograms are drawn with each value at the center of the corresponding bar.

• No cutpoints: The single value is placed at the midpoint of the bar's base.

• Tick marks: A tick mark below each bar indicates the value it represents.

Example: A histogram for the number of people in 66 households would have bars centered at 1, 2, 3, etc., with heights corresponding to the number of households for each value.

Distribution Shapes

Describing Distribution Shapes

The shape of a distribution provides insight into the data's characteristics. By connecting the tops of the bars in a histogram, one can visualize the overall pattern or shape of the data.

• Bell-shaped (Normal) Distribution: Symmetric with a single peak (mode) in the center. The sides "level off" gradually, forming a smooth curve.

• Triangular Distribution: Also symmetric with a single peak, but the sides drop off sharply, "hitting bottom suddenly" rather than gradually.

• Right-skewed Distribution: The right tail (higher values) is longer than the left. Most data are concentrated on the left, with a few larger values stretching the distribution to the right.

• Left-skewed Distribution: The left tail (lower values) is longer than the right. Most data are concentrated on the right, with a few smaller values stretching the distribution to the left.

• Bimodal/Multimodal Distribution: The distribution has two (bimodal) or more (multimodal) peaks or modes.

Example: Income data are often right-skewed, with most people earning moderate amounts and a few earning much more. Heights of adult humans are typically bell-shaped (normal distribution).

Summary Table: Common Distribution Shapes

Key Points for Drawing Histograms

• Label axes clearly, including units.

• Choose appropriate scales for both axes.

• Bars must touch (no gaps) for continuous data.

• Start the vertical axis at zero.

• For single-valued data, center bars at each value.

Additional info:

• Histograms are foundational for understanding more advanced statistical concepts, such as probability distributions and inferential statistics.

• Recognizing distribution shapes is important for selecting appropriate statistical methods in later analyses.