Graphical Representation of Data

Graphical representations are visual tools used to display and analyze data. They help us understand patterns, trends, and relationships within a dataset more easily than simply looking at a table of numbers. This section focuses on three specific types of graphical representations commonly used in high school math: bar graphs, histograms, and frequency polygons.

Bar Graphs: Bar graphs are used to compare discrete categories or groups. Each bar represents a category, and the height of the bar corresponds to the value or frequency for that category. The bars are typically separated by spaces, emphasizing the distinct nature of each category.

Histograms: Histograms are similar to bar graphs, but they are used to represent continuous data grouped into intervals or bins. The bars in a histogram touch each other, indicating the continuous nature of the data. The area of each bar is proportional to the frequency of the data within that interval. Histograms can have uniform or varying widths for the bins. Uniform width means all bins have the same width, while varying width allows for different bin sizes, which can be useful for representing data with varying densities.

Frequency Polygons: A frequency polygon is a line graph that represents the same data as a histogram. It’s created by connecting the midpoints of the tops of the bars in a histogram with straight lines. Frequency polygons are particularly useful for comparing the distributions of different datasets.

Formulae

While there aren’t specific formulas in the same way as in algebra, understanding how to calculate the following is essential:

• Frequency: The number of times a particular value or value range (bin) appears in the dataset.
• Class Width: The range of values contained within a bin/interval in a histogram. Calculated as: Class Width = Upper Class Limit - Lower Class Limit
• Frequency Density (for histograms with varying bin widths): Calculated as Frequency Density = Frequency / Class Width. This is important when comparing intervals of unequal widths.

Examples

Example-1: Bar Graph

A survey was conducted to determine the favorite color of students in a class. The results are:

• Red: 10 students
• Blue: 15 students
• Green: 8 students
• Yellow: 12 students

A bar graph would represent this data with four bars, one for each color. The height of each bar would correspond to the number of students who chose that color. For example, the “Blue” bar would be taller than the “Green” bar.

Example-2: Histogram with Uniform Width

The following data shows the heights (in cm) of students in a class, grouped into intervals:

• 150-155 cm: 5 students
• 155-160 cm: 10 students
• 160-165 cm: 12 students
• 165-170 cm: 8 students

A histogram would display this data. Each bar would represent a height interval. The height of each bar would represent the number of students in that interval. All intervals (bins) have the same width (5cm).

Real Life Application

Graphical representations are used extensively in various real-life applications:

• Business and Economics: Companies use bar graphs and histograms to analyze sales figures, track market trends, and represent financial data.
• Science and Engineering: Scientists use graphs to visualize experimental results, analyze data, and identify patterns. For example, histograms are used in data analysis to understand the distribution of values.
• Healthcare: Doctors and researchers use graphs to monitor patient health, track disease outbreaks, and analyze medical data.
• Politics and Social Science: Politicians and social scientists use graphs to display survey results, track demographics, and illustrate trends in society.

Fun Fact

The earliest known form of graphical representation dates back to ancient Egypt, where they used simple drawings and symbols to represent data. Modern data visualization techniques have evolved significantly since then, becoming an indispensable tool for understanding and interpreting complex information.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the main difference between a bar graph and a histogram?

Check Solution

Ans: A

A bar graph is used for categorical data, and a histogram is used for numerical data, specifically grouped into intervals.

Q.2 A histogram has bars with varying widths. What does the *height* of each bar represent?

Check Solution

Ans: C

In a histogram with varying widths, the area of the bar represents the frequency. The height represents the frequency density (frequency divided by class width).

Q.3 The following data represents the scores of 20 students in a math test: 60, 65, 70, 70, 75, 75, 75, 80, 80, 80, 80, 85, 85, 90, 90, 90, 95, 95, 100, 100. If we construct a frequency polygon, which of the following statements is true about its shape?

Check Solution

Ans: B

The data is more or less symmetrically distributed. The frequencies are concentrated around the middle scores (75-90).

Q.4 The class intervals of a histogram are 10-20, 20-30, 30-40, and 40-50. Their corresponding frequencies are 5, 8, 12, and 6 respectively. If we were to represent this data with a frequency polygon, what would be the x-coordinate of the first point plotted?

Check Solution

Ans: B

The frequency polygon starts at the midpoint of the first class interval. The midpoint is calculated as $(10+20)/2=15$

Q.5 A histogram is constructed with uniform class widths. If the frequency of a class interval is doubled, what happens to the height of the bar representing that interval?

Check Solution

Ans: C

In a histogram with uniform class widths, height is directly proportional to frequency. If frequency doubles, then the height also doubles.

Next Topic: Measures of Central Tendency for Ungrouped Data