Describing and Organizing Data: Frequency Distributions, Graphs, and Classes

Describing Data

Introduction

Once data has been collected through surveys or experiments, it must be organized into a manageable form for analysis. This process transforms raw data into meaningful summaries using various statistical techniques. The following notes outline key methods for organizing both qualitative and quantitative data, including frequency distributions, graphical representations, and the construction of classes for continuous variables.

Organizing Qualitative Data

Frequency Distributions

A frequency distribution lists each category of qualitative data and the number of occurrences for each category.

• Definition: A table that displays the count (frequency) of observations in each category.

• Example: A physical therapist records the body part requiring rehabilitation for 30 patients. The frequency distribution shows which body part is most commonly injured.

Body Part	Tally	Frequency
Back	|||| |||| ||	12
Wrist	||	2
Elbow	|	1
Hip	|	1
Shoulder	||||	4
Knee	||	2
Hand	||	2
Groin	|	1
Neck	|	1

Caution: Frequency distributions for qualitative data simply represent counts, not measurements.

Relative Frequency Distributions

The relative frequency is the proportion or percentage of observations within a category, calculated as:

$\text{relative frequency} = \frac{\text{frequency}}{\text{sum of all frequencies}}$

• Relative frequency distribution: Lists each category with its relative frequency.

Body Part	Frequency	Relative Frequency
Back	12	0.4
Wrist	2	0.067
Elbow	1	0.033
Hip	1	0.033
Shoulder	4	0.133
Knee	5	0.167
Hand	2	0.067
Groin	1	0.033
Neck	1	0.033
Total	30	1

Bar Graphs

A bar graph visually represents categorical data by drawing rectangles of equal width for each category. The height of each rectangle corresponds to the frequency or relative frequency.

• Categories are labeled on one axis; frequencies on the other.

• Bars do not touch each other.

Pareto Charts

A Pareto chart is a bar graph where categories are ordered by decreasing frequency or relative frequency. This helps highlight the most significant categories.

Pie Charts

A pie chart is a circle divided into sectors, each representing a category. The area of each sector is proportional to the frequency or relative frequency of the category.

• Degree measure for each sector: $\text{Degree} = \text{Relative Frequency} \times 360^\circ$

• Example: For 'Not a high school graduate' with relative frequency 0.089, the sector is $0.089 \times 360^\circ \approx 32^\circ$.

Educational Attainment	Frequency	Relative Frequency	Degree Measure
Not a high school graduate	20,054	0.089	32
High school diploma	62,547	0.279	100
Some college, no degree	33,455	0.149	54
Associate's degree	23,477	0.105	38
Bachelor's degree	52,805	0.235	85
Graduate/professional degree	32,232	0.144	52

Pie Chart vs Bar Chart

• Bar charts can convey more detailed data, especially when comparing multiple groups.

• Pie charts are best for showing proportions of a whole.

Organizing Quantitative Data

Discrete vs Continuous Data

• Discrete data: Consists of countable values (e.g., number of arrivals).

• Continuous data: Can take any value within a range (e.g., IQ scores).

Organizing Discrete Data

• Class: A specific discrete value.

• Class frequency: Number of observations in a class.

• Frequency distribution: List of classes with their frequencies.

• Relative frequency (RF): $\text{RF} = \frac{\text{class frequency}}{\text{total number of observations}}$

• Class percentage: $\text{Percentage} = \text{RF} \times 100$

Number of Customers	Tally	Frequency	Relative Frequency
1	|	1	0.025
2	||||| |	6	0.15
3	|	1	0.025
4	||||	4	0.1
5	||||| ||	7	0.175
6	||||| |	5	0.125
7	||	2	0.05
8	|	1	0.025
9	|	1	0.025
10		0	0.0
11	|	1	0.025

Histograms

A histogram is constructed by drawing rectangles for each class of data. The height of each rectangle is the frequency or relative frequency, and the width is the same for all rectangles. Unlike bar graphs, the rectangles in histograms touch each other, indicating the continuity of the data.

Organizing Continuous Data

• Classes consist of intervals of numbers.

• Lower class limit: Smallest value in the class.

• Upper class limit: Largest value in the class.

• Class width: Difference between consecutive lower class limits.

Histogram for a Numerical Variable

Example: IQ scores for 60 students are grouped into intervals (classes), and a histogram is constructed to visualize the distribution.

Class	Count
80-89	3
90-99	10
100-109	14
110-119	17
120-129	11
130-139	3
140-149	2

Same Data, Different Classes

The choice of class intervals can affect the appearance and interpretation of histograms. Different class widths or boundaries may highlight different patterns in the data.

Classes for Continuous Data: Rules

• Rule 1: Determine the number of classes ($k$). Generally, use between 5 and 20 classes.

• Rule 2: Use classes with equal size (width). Compute class width ($w$) as: $\text{Class width} = \frac{\text{largest data value} - \text{smallest data value}}{\text{number of classes}}$

• Rule 3: Classes should be inclusive and non-overlapping. Each observation must belong to one and only one class, and class boundaries should be clearly defined.

Remarks on Frequency Distributions

• There is not one correct frequency distribution for a data set.

• Some distributions better illustrate patterns; constructing them is partly an art.

• Choose the distribution that best summarizes the data for your purpose.

Summary Table: Educational Attainment (1990 vs 2021)

Educational Attainment	1990	2021
Not a high school graduate	39,344	20,054
High school diploma	47,643	62,547
Some college, no degree	29,780	33,455
Associate's degree	20,792	23,477
Bachelor's degree	29,832	52,805
Graduate/professional degree	11,478	32,232
Totals	158,870	224,580

Key Questions for Review

1. Why do we use relative frequency as the measure?

2. What are some advantages of the relative frequency graph over the frequency table? What deductions can you make about education level in 1990 vs 2021?

Additional info:

• Relative frequency allows for comparison between groups of different sizes and is essential for interpreting proportions.

• Bar graphs and histograms are fundamental tools for visualizing categorical and numerical data, respectively.

• Pie charts are best for showing parts of a whole, but bar charts provide more detailed comparisons.