Frequency Tables and Plots

Frequency tables count how often each value occurs. They are the quickest useful summary of discrete quantitative data.

• Frequency plot: Graph the count for each value or interval to expose patterns and trends.

Histograms

A histogram groups numerical observations into bins and shows the frequency in each bin. Unlike a categorical bar chart, adjacent bins represent continuous intervals.

• How to Interpret a Histogram:
  • Bin width: Each bin covers a value range; changing the width can change the apparent pattern.
  • Bin height: Height encodes the frequency within that range.
  • Shape: The silhouette can reveal symmetric, skewed, or bimodal structure.
  • Outliers: Isolated observations may signal outliers, though a histogram can hide exact values.

Stem and Leaf Plots

Stem-and-leaf plots preserve individual values while exposing the distribution. Split each observation into a stem, usually its leading digits, and a leaf, usually its final digit.

• How to Read a Stem and Leaf Plot:
  • Stem: The left side holds the higher-order digits.
  • Leaf: The right side holds the lower-order digits.
  • Reading: Combine a stem and leaf to recover an observation.
  • Arrangement: Sort leaves within each stem to expose the distribution, median, and mode.

Each view emphasizes a different part of the data: exact values, shape, center, or spread.

Understanding Distributions in Data

Read a distribution by its shape and its departures from that shape.

Common Shapes of Distributions

• Normal distribution: A symmetric, bell-shaped distribution with one central peak.
• Skewed distribution: One tail extends farther than the other. The longer tail names the direction of skew.
• Uniform distribution: Values occur at roughly equal frequencies, producing a flat shape.
• Bimodal distribution: Two peaks may indicate two underlying groups.

Clusters, Peaks, Gaps, and Outliers

• Clusters: Concentrations of nearby values.
• Peaks: High-frequency regions, or modes.
• Gaps: Regions with few or no observations.
• Outliers: Values far from the main body; they may be genuine variation or collection errors.

Dot Plots

• Dot plots show every observation and suit small datasets.
• Comparing dot plots: Compare center, spread, range, clusters, gaps, and outliers.

Histograms

• Histograms scale to larger datasets by grouping values into bins.
• Comparing histograms: Compare shape, center, spread, skew, and the number of peaks.

Box Plots

• Box plots summarize quartiles, median, and potential outliers.
• Comparing box plots: Compare median, range, interquartile range, and outliers across groups.

No single plot tells the whole story. Choose the one that exposes the feature under inspection.

Line Graphs: Uses and Potential Misleading Nature

Line graphs show change across an ordered axis, usually time. Their apparent story depends heavily on scale and selection.

Common Uses of Line Graphs

1. Trend Analysis: Line graphs are excellent for showing changes and trends over time.
2. Comparing Multiple Series: They allow for the comparison of multiple data series within the same graph, making it easy to compare trends between different groups or categories.
3. Highlighting Continuity: Line graphs emphasize the continuity of the data, particularly useful in cases where the data is collected over regular intervals.

How Line Graphs Can Be Misleading

1. Manipulating Axis Scale: If the scale of the y-axis is manipulated (either compressed or expanded), it can exaggerate or downplay trends.
2. Cherry-Picking Data Points: Selecting specific data ranges while omitting others can lead to misleading conclusions.
3. Not Starting the Y-Axis from Zero: Starting the y-axis from a value other than zero can dramatically alter the appearance of the graph, making changes seem more significant than they are.
4. Using Too Many Data Points: Overloading a line graph with too many data points or lines can make it cluttered and difficult to interpret.
5. Ignoring Confounding Variables: Not accounting for external factors that might affect the data can lead to incorrect interpretations of trends.

Use honest scales, show the relevant range, and state what the graph omits.