Constructing Graphical and Tabular Displays of Data: Histograms and Distribution Shapes

Constructing Graphical and Tabular Displays of Data (3.4)

Introduction

This section introduces the construction and interpretation of graphical and tabular displays for numerical data, focusing on histograms and the characteristics of distributions. These tools are essential for summarizing, visualizing, and understanding data in statistics.

Histograms

Frequency and Relative Frequency

• Frequency of a class: The number of observations in a specific class interval.

• Relative frequency of a class: The proportion of the total observations that fall within a specific class interval.

• Sum of Relative Frequencies: For a numerical variable, the sum of the relative frequencies across all classes equals 1.

Constructing Histograms

Histograms are bar graphs that represent the frequency or relative frequency of data within specified intervals (classes).

1. Determine class intervals and their limits (e.g., 30, 40, 50, ..., 110).

2. Mark these intervals on the horizontal axis.

3. Mark frequencies (or relative frequencies) on the vertical axis, spaced appropriately (e.g., 0, 2, 4, ..., 10 for frequency).

4. Draw rectangles for each class interval, with width equal to the class width and height equal to the class frequency or relative frequency.

Example: Frequency and Relative Frequency Table

Class	Tally	Frequency	Relative Frequency
30-39	|	1	0.029
40-49		0	0.000
50-59		0	0.000
60-69	|||||	6	0.171
70-79	|||||	5	0.143
80-89	||||| ||||	10	0.286
90-99	||||| |||	8	0.229
100-109	|||	3	0.086
Total		35	1.000

Relative Frequency Histogram

• Similar to a frequency histogram, but the vertical axis represents relative frequency (e.g., 0.05, 0.10, ..., 0.30).

• Useful for comparing distributions with different sample sizes.

Density Histogram

• The area of each bar is equal to the relative frequency of the class.

• The total area of all bars is equal to 1.

Using Density Histograms to Find Proportions

• To find the proportion of data within a range, sum the areas of the bars corresponding to that range.

• Example: If the area of bars for $30 \leq x \leq 39.99$ is 0.20 and 0.13, the total proportion is $0.20 + 0.13 = 0.33$.

• For values less than a certain threshold, sum the areas of all bars below that value.

• For values at least as large as a threshold, subtract the area below the threshold from 1.

Distribution Shapes

Unimodal, Bimodal, and Multimodal Distributions

• Unimodal: One peak (mound) in the distribution.

• Bimodal: Two peaks (mounds).

• Multimodal: More than two peaks.

Skewness and Symmetry

• Skewed left: The left tail is longer than the right tail.

• Skewed right: The right tail is longer than the left tail.

• Symmetric: The left and right tails are mirror images.

Describing Distributions: Four Key Characteristics

1. Identify outliers: Correct or remove outliers due to errors; consider separate analysis for other outliers.

2. Determine the shape: Assess modality and skewness; consider subgroup analysis for bimodal/multimodal data.

3. Measure and interpret the center: Use mean, median, or mode as appropriate.

4. Describe the spread: Use range, interquartile range, variance, or standard deviation.

Models in Statistics

• A model is a mathematical description of a real-world situation, used to represent and analyze data.

Types of Variables and Diagram Benefits

Diagram	Types of Variables	Benefits
Frequency Bar Graph	One categorical variable	Compare frequencies of categories
Relative Frequency Bar Graph	One categorical variable	Compare a part to the whole
Multiple Bar Graph	Two categorical variables	Compare groups
Pie Chart	One categorical variable	Compare a part to the whole
Two-Way Table	Two categorical variables	Compare a part to the whole
Dotplot	One numerical variable	Describe individual values for a small or medium number of observations
Frequency Histogram	One numerical variable	Compare frequencies of classes
Relative Frequency Histogram	One numerical variable	Compare the frequencies of classes
Density Histogram	One numerical variable	Compare a part to the whole
Time-Series Plot	Two numerical variables	Find the association between two variables