Ninth Grade Mathematics: Graphical Representation: Bar Graphs & Histograms
A table of numbers can convey precise data, but a graphical representation communicates trends, distributions, and comparisons with immediate visual clarity. In this chapter, you will master the two most important columnar visualizations in introductory statistics—bar graphs and histograms—learning when each is appropriate, how to construct them with accurate scale, and how to handle varying class widths.
The Structural Anatomy of a Bar Graph
A bar graph (or bar chart) is a visual display used primarily for categorical (qualitative) data or discrete numerical variables. It consists of a series of rectangular bars of uniform width erected along an axis.
Anatomy of a Bar Graph:
Number of Students (Frequency)
^
20 | +----+
| | |
15 | +----+ | |
| | | | | +----+
10 | | | | | | |
| | | | | | | +----+
5 | | | | | | | | |
+---+----+--+--+----+--+--+----+--+--+----+--->
Bus | Walk | Bicycle | Car (Transportation Mode)
GAPS between bars (Discrete categories)
Rules for Bar Graphs:
1. The width of each bar is arbitrary, but MUST BE UNIFORM for all bars.
2. The spaces (gaps) between consecutive bars MUST BE EQUAL, indicating separate, non-continuous categories.
3. The height of each bar is directly proportional to the frequency of that category.
The Anatomy of a Histogram
A histogram is a graphical representation used exclusively for continuous grouped frequency distributions. While a bar graph has gaps between bars, a histogram has NO GAPS between adjacent rectangles because the class intervals form an unbroken numerical continuum.
Anatomy of a Histogram (Continuous Data):
Frequency (f)
^
15 | +--------+
| | |
10 | +--------+ +--------+
| | | | |
5 | +------+ | | +------+
| | | | | | |
0 +-+------+--------+--------+--------+------+--->
0 10 20 30 40 50 (Marks)
Continuous boundaries: NO GAPS between rectangles!
Key Differences:
Feature Bar Graph Histogram
---------------------------------------------------------------------------------
Data Type Categorical or discrete data Continuous quantitative data
Gaps Between Bars Mandatory equal gaps between bars NO GAPS (bars touch each other)
Base Representation Arbitrary categories on base Continuous class intervals on x-axis
Proportionality Height represents frequency AREA of rectangle represents frequency
The Crucial Principle: Area Represents Frequency
In a standard histogram where all class intervals have equal width, the height of each rectangle is proportional to frequency. However, if a dataset contains unequal class widths, plotting frequencies directly as heights distorts the visual data!
The Fundamental Histogram Law:
The AREA of each rectangle in a histogram is proportional to its class frequency:
Area = Class Width * Height proportional to Frequency
Adjusting Heights for Unequal Class Widths
When class intervals have unequal widths, we must adjust the height of each rectangle by calculating its Frequency Density (modified length):
Frequency Density Formula:
Minimum Class Size
Adjusted Height = -------------------- * Class Frequency
Given Class Size
Example:
Minimum class size in table is 5.
For an interval of width 10 with frequency 14:
Adjusted Height = (5 / 10) * 14 = 7 units.
Adjusting the height ensures that wider rectangles do not visually exaggerate the data, preserving the fundamental truth that area represents frequency.
The Kink (Zigzag Break) on the Axis
When the first class interval does not begin at zero, drawing the graph to scale would leave an empty, wasted gap from 0 to the starting value. To indicate that the origin is broken or compressed, we draw a zigzag mark called a kink along the horizontal axis:
Horizontal Axis with Kink:
0 - - /\ - - 50 60 70 80 90
Kink indicates skipped distance between 0 and 50
Chapter Exercises
Exercise 1: The following data shows the favorite sports of 200 high school students: Soccer (65), Basketball (50), Volleyball (35), Track (30), Tennis (20). Describe the construction of an appropriate bar graph, stating whether gaps should be present.
Exercise 2: The weekly wages of 50 factory workers are grouped as follows: $300-$350 (8 workers), $350-$400 (15 workers), $400-$450 (18 workers), $450-$500 (6 workers), $500-$550 (3 workers). Describe how to construct a histogram for this data, explaining whether a kink is needed on the x-axis.
Exercise 3: Explain why a histogram cannot have spaces between adjacent bars, and describe how to calculate the adjusted height of a rectangle when an interval has double the width of the baseline class interval.
Solutions and Explanations
Solution 1: Because favorite sport is a qualitative categorical variable, a bar graph is the appropriate visual display. Erect five vertical rectangular bars along the horizontal axis, labeled Soccer, Basketball, Volleyball, Track, and Tennis. Choose a vertical scale of 1 cm = 10 students, with heights reaching 6.5 cm, 5 cm, 3.5 cm, 3 cm, and 2 cm respectively. All five bars must have identical width, and equal gaps must be left between adjacent bars to visually signify that the categories are separate, discrete entities with no intermediate numerical continuum.
Solution 2: Because weekly wages represent continuous quantitative data, a histogram is the appropriate display. The class intervals are continuous: 300 to 350, 350 to 400, etc. On the horizontal axis, mark the continuous wage boundaries: 300, 350, 400, 450, 500, 550. Because the data begins at 300 rather than 0, a zigzag kink must be drawn between the origin (0) and 300 to indicate that the first 300 units are compressed. On the vertical axis, choose a scale of 1 cm = 2 workers. Erect contiguous rectangular bars (with no gaps between them) over each class interval with heights corresponding to 8, 15, 18, 6, and 3 workers.
Solution 3: A histogram cannot have spaces between adjacent bars because the underlying variable is continuous, meaning every real number within the distribution's range belongs to a class interval. The upper boundary of one interval is identical to the lower boundary of the next interval (such as 30 to 40, and 40 to 50); leaving a gap would falsely imply that no values exist between 39.99 and 40.01. When a class interval has double the width of the baseline class interval (width = 2*w), keeping its height equal to frequency would double its area, visually exaggerating the frequency twofold. To maintain area proportionality, its height must be divided by 2: Adjusted Height = (Baseline Width / Given Width) * Frequency = (w / 2w) * Frequency = (1/2) * Frequency.