Displaying and Describing Quantitative Data

Introduction

Quantitative data analysis is essential in business statistics, providing methods to summarize, visualize, and interpret numerical information. This section introduces the primary tools and concepts used to describe and standardize quantitative variables, focusing on visualization, distribution description, and standardization.

Visualizing Quantitative Variables

Visualization helps clarify patterns and trends in raw quantitative data, making interpretation easier and more effective.

• Histograms: Graphical representations that divide data into intervals (bins) and display the frequency of observations in each bin. There are no categories; data are grouped by value ranges.

• Relative Frequency Histograms: Similar to histograms, but the y-axis shows the percentage of observations in each bin, allowing for comparison between datasets of different sizes. The shape remains unchanged; only the labeling differs.

• Example: A histogram of average stock prices over several years can reveal trends, peaks, and outliers in the data.

Describing Distributions

To describe a distribution, focus on three main aspects: shape, center, and spread.

Shape

• Modes: Peaks in a histogram. Types include:

  • Unimodal: One main peak.

  • Bimodal: Two main peaks.

  • Multimodal: Three or more peaks.

  • Uniform: All bars are approximately the same height; no clear mode.

• Symmetry: A distribution is symmetric if the left and right sides are mirror images. If one tail is longer, the distribution is skewed (right or left).

• Outliers: Observations that stand apart from the rest of the data. Outliers should be reported and considered, as they can affect statistical measures.

Center

• Mean: The arithmetic average of all data values.

• Median: The middle value when data are ordered. If the number of observations is even, the median is the average of the two middle values.

• Resistant Measures: The median is resistant to outliers and skewed data, while the mean is not.

Spread

• Range: The difference between the maximum and minimum values.

• Interquartile Range (IQR): The range of the middle 50% of the data.

  • Lower quartile (Q1): 25th percentile

  • Upper quartile (Q3): 75th percentile

• Standard Deviation: Measures the average distance of data values from the mean. Appropriate for symmetric distributions without outliers.

Choosing Measures of Center and Spread

• If the distribution is skewed or contains outliers, use the median and IQR.

• If the distribution is roughly symmetric and has no outliers, use the mean and standard deviation.

Standardizing Variables

Standardizing allows comparison of values from different distributions by expressing them in terms of standard deviations from the mean. The standardized value is called a z-score.

• Z-Score: Indicates how many standard deviations a value is above or below the mean.

• Comparing z-scores for different variables (e.g., price and square footage) helps determine which value is more unusual relative to its distribution.

• Example: If a house price has a z-score of 2.5 and square footage has a z-score of 1.2, the price is more unusual compared to its distribution.

Special Cases and Summary Statistics

• Identify modes to determine if data can be split into groups.

• Report outliers and consider their effect on measures of center and spread.

• Pair the median with the IQR, and the mean with the standard deviation for summary statistics.

Summary Table: Measures of Center and Spread