Visualizing and Summarizing Data: Frequency Distributions and Histograms

Visualizing Qualitative vs. Quantitative Data

Introduction

In statistics, data can be classified as either qualitative (categorical) or quantitative (numerical). The way we visualize these types of data differs, and understanding the appropriate graphical representation is essential for effective data analysis.

Qualitative Data

• Definition: Data that consists of names, labels, or categories (e.g., eye color, nationality).

• Common Graphs:

  • Bar Chart / Pareto Chart: Displays frequency of categories with bars. Pareto charts order bars by frequency.

  • Pie Chart: Shows proportions of categories as slices of a circle.

• Example:

  • Eye Colors: Bar chart showing frequency of each color.

  • Nationalities: Pie chart showing percentage of each nationality.

Quantitative Data

• Definition: Data that consists of numerical measurements or counts (e.g., test scores, heights).

• Common Graphs:

  • Histogram: Bar graph for quantitative data, showing frequency of data within intervals (classes).

  • Frequency Polygon: Line graph version of histogram, connecting midpoints of each class.

  • Stemplots (Stem & Leaf): Displays data values in a way that shows their distribution and retains actual values.

• Example:

  • Test Scores: Histogram showing distribution of scores.

  • Heights: Stemplot showing individual height values.

Frequency Distributions

Introduction

A frequency distribution is a table that displays the frequency (count) of observations within specified intervals, called classes. It is a foundational tool for summarizing quantitative data.

Constructing a Frequency Distribution

• Steps:

  1. Divide the range of data into equal-width intervals (classes).

  2. Count the number of data points (frequency) in each class.

  3. Calculate relative frequency as a percentage of the total number of observations: $\text{Relative Frequency} = \frac{f}{n} \times 100\%$ where $f$ is the frequency in a class and $n$ is the total number of observations.

• Key Terms:

  • Lower Class Limit: Smallest value in a class.

  • Upper Class Limit: Largest value in a class.

  • Class Midpoint: Average of lower and upper class limits: $\text{Class Midpoint} = \frac{\text{Lower} + \text{Upper}}{2}$

  • Class Width: Difference between lower limits of consecutive classes or between upper limits: $\text{Class Width} = \text{Upper Limit of Class} - \text{Lower Limit of Class}$

Example Table: Frequency Distribution

Time Spent Studying (mins)	Frequency (f)	Relative Frequency (%)
20–29
30–39
40–49
50–59
60–69
70–79

Additional info: Students should fill in frequencies and calculate relative frequencies using the provided data.

Practice: Frequency Distribution Construction

Practice Problems

• Given a data set, construct a frequency distribution using specified class limits and widths.

• Calculate class width and midpoints from a frequency table.

• Organize raw data into classes and tally frequencies.

Example Table: Travel Time to Work

Travel Time to Work (mins)	Frequency (f)
5–15	156
16–26	343
27–37	249
38–48	172
49–60	98
71–81	40

How to Create Frequency Distributions

Step-by-Step Method

• Calculate class width: $\text{Class Width} = \frac{\text{max} - \text{min}}{\text{number of classes}}$ Round up to a convenient number if necessary.

• Find lower class limits:

  • First lower limit = data minimum

  • Next lower limits = previous lower limit + class width

• Find upper class limits:

  • First upper limit = previous lower limit + class width - 1

• Tally each data value into its appropriate class.

Example Table: Sales Data

Sales ($)
1223, 1136, 819, 1098, 1011, 997, 973, 1025, 1017, 1118, 943, 1196, 1061, 942

Additional info: Students should organize these values into 5 classes using the method above.

Histograms

Introduction

A histogram is a graphical representation of a frequency distribution using adjacent bars to show the frequency of data within each class interval. Histograms are used for quantitative data and help visualize the shape of the data distribution.

Constructing Histograms

• Classes/Bins: Shown on the horizontal axis, usually as intervals or class midpoints.

• Frequencies: Shown on the vertical axis, with each bar's height representing the frequency.

Example Table: Frequency Distribution for Histogram

Time Spent Studying (mins)	Class Midpoint	Frequency (f)
20–29	24.5
30–39	34.5
40–49	44.5
50–59	54.5
60–69	64.5
70–79	74.5

Shapes of Histograms

• Normal: Symmetrical, bell-shaped distribution.

• Skewed Right: Tail extends to the right (positive skew).

• Skewed Left: Tail extends to the left (negative skew).

• Uniform: All bars are approximately the same height.

Example Table: Histogram Shapes

Shape	Description
Normal	Symmetrical, peak in center
Skewed Right	Tail on right side
Skewed Left	Tail on left side
Uniform	Bars of equal height

Practice: Interpreting Histograms

• Determine the number of classes and class width from a histogram.

• Identify the shape of the distribution (normal, skewed, uniform).

Creating Histograms on a TI-84 Calculator

Step-by-Step Instructions

1. Input data as list L1.

2. Edit and enter each value.

3. Graph the default histogram:

   • Open STAT PLOT, select histogram, set Xlist to L1.

   • ZoomStat or set appropriate window.

   • Show class boundaries and frequencies.

4. Adjust class boundaries and width in WINDOW settings.

5. Set Xmin and Xmax to cover data range.

6. Optionally adjust Ymin, Ymax, Yscale values.

Example

• Given a data set, use a class width of 15 to create a histogram and determine the distribution shape.

Summary Table: Key Terms and Formulas

Term	Definition	Formula
Frequency	Number of observations in a class	—
Relative Frequency	Proportion of total observations in a class	$\frac{f}{n} \times 100\%$
Class Midpoint	Average of lower and upper class limits	$\frac{\text{Lower} + \text{Upper}}{2}$
Class Width	Difference between consecutive lower class limits	$\text{Upper} - \text{Lower}$

Additional info: These notes cover foundational concepts in descriptive statistics, including graphical representation and summarization of data. Students should practice constructing frequency distributions and histograms using sample data sets.