Welcome to Histograms and Frequency Polygons!
Welcome to one of the most powerful and visual topics in GCSE Statistics! When working with large sets of continuous data—like the heights of students, the time spent revising, or the weights of parcels—we group the data into classes. But how do we represent this data visually so it gives a fair, accurate picture?
In this chapter, you will learn how to construct and interpret Histograms and Frequency Polygons. Don't worry if this seems tricky at first; once you master one simple formula, you will be able to solve any histogram question with confidence!
---1. Histograms vs. Bar Charts: What's the Difference?
Many students confuse histograms with regular bar charts. Let's clear that up straight away so you never mix them up in an exam!
Bar Charts:
• Used for discrete or categorical data (e.g., shoe size, favourite colour).
• The bars have equal widths.
• There are gaps between the bars.
• The height of the bar shows the frequency.
Histograms:
• Used for continuous grouped data (e.g., time, length, mass).
• The bars can have unequal (different) widths.
• There are no gaps between bars (unless a class has a frequency of \(0\)).
• The area of the bar represents the frequency, NOT the height!
Analogy to remember: Imagine spreading jam on toast. If you have a wide slice of bread (a wide class interval), the jam is spread out thinner (lower height). If you have a narrow slice (a narrow class interval), the same amount of jam will pile up higher! The total amount of jam (frequency) depends on both the width and the height.
Key Takeaway: In a histogram, Area = Frequency.
---2. The Golden Concept: Frequency Density
When the class widths of grouped data are not all the same, plotting frequency directly on the vertical axis is misleading because wider groups would look artificially massive. To fix this, we use a special measure on the vertical axis called Frequency Density (FD).
The Core Formulas
1. Finding Class Width:
\(\text{Class Width} = \text{Upper Boundary} - \text{Lower Boundary}\)
2. Finding Frequency Density:
\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)
3. Finding Frequency (from a drawn histogram):
\(\text{Frequency} = \text{Frequency Density} \times \text{Class Width} = \text{Area of the Bar}\)
Memory Trick: You can use a formula triangle just like in science!
Put Frequency (F) at the top, and put Frequency Density (FD) and Class Width (CW) at the bottom:
• Cover \(F\): \(F = FD \times CW\)
• Cover \(FD\): \(FD = \frac{F}{CW}\)
• Cover \(CW\): \(CW = \frac{F}{FD}\)
Did you know? "Density" tells us how tightly packed the data is. A tall, narrow bar means a lot of data is squeezed into a very small interval!
Key Takeaway: Always check the vertical axis. If it says Frequency Density, the area of the bar gives the frequency!
---3. Step-by-Step: How to Draw a Histogram
Let's walk through drawing a histogram from a grouped frequency table step-by-step.
Worked Example:
The table below shows the time, \(t\) (in minutes), taken by \(60\) runners to complete a \(10\text{ km}\) race.
• Class \(30 \le t < 40\): Frequency = \(15\)
• Class \(40 \le t < 50\): Frequency = \(25\)
• Class \(50 \le t < 70\): Frequency = \(20\)
Step 1: Calculate the Class Width for each interval
• For \(30 \le t < 40\): \(\text{Class Width} = 40 - 30 = 10\)
• For \(40 \le t < 50\): \(\text{Class Width} = 50 - 40 = 10\)
• For \(50 \le t < 70\): \(\text{Class Width} = 70 - 50 = 20\)
Step 2: Calculate the Frequency Density for each interval
• For \(30 \le t < 40\): \(\text{Frequency Density} = \frac{15}{10} = 1.5\)
• For \(40 \le t < 50\): \(\text{Frequency Density} = \frac{25}{10} = 2.5\)
• For \(50 \le t < 70\): \(\text{Frequency Density} = \frac{20}{20} = 1.0\)
Step 3: Set up your axes
• Horizontal axis (\(x\)-axis): Label this with the continuous variable, including units (e.g., Time (minutes)). Choose a sensible linear scale starting from at least \(30\) up to \(70\).
• Vertical axis (\(y\)-axis): Always label this Frequency Density. Choose a scale that accommodates your highest value (here, \(2.5\)).
Step 4: Draw the bars
• Draw the first bar from \(30\) to \(40\) with a height of \(1.5\).
• Draw the second bar from \(40\) to \(50\) with a height of \(2.5\).
• Draw the third bar from \(50\) to \(70\) with a height of \(1.0\).
• Ensure there are no spaces between adjacent bars.
Key Takeaway: Create two extra columns in your exam table: one for Class Width and one for Frequency Density before you start drawing!
---4. Reading and Interpreting Histograms
Exam questions will frequently ask you to work backwards: you are given a histogram and must calculate frequencies or estimate the number of values in a specific range.
Example 1: Finding Total Frequency
To find the total number of items represented in a histogram, calculate the area of every single bar and add them together:
\(\text{Total Frequency} = \sum (\text{Class Width} \times \text{Frequency Density})\)
Example 2: Estimating Frequencies for a Subset of Data
Suppose you are asked: "Estimate the number of runners who took longer than \(55\) minutes."
Looking at our example:
• The group \(50 \le t < 70\) has a frequency density of \(1.0\).
• The part of this bar that is greater than \(55\) minutes runs from \(t = 55\) to \(t = 70\).
• The width of this section is: \(70 - 55 = 15\text{ minutes}\).
• Estimated frequency = \(\text{Width} \times \text{Frequency Density} = 15 \times 1.0 = 15\text{ runners}\).
Why is this an estimate? Because we assume the data values are evenly distributed (spread out uniformly) across the entire class interval.
Key Takeaway: To find the frequency of a partial interval, find the width of the relevant section and multiply it by that bar's frequency density.
---5. Frequency Polygons
A frequency polygon is another great visual tool for displaying grouped data. It is a line graph created by joining points with straight line segments.
How to Construct a Frequency Polygon:
Step 1: Find the Midpoint of each class
\(\text{Midpoint} = \frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}\)
Example: For the interval \(20 \le x < 30\), \(\text{Midpoint} = \frac{20 + 30}{2} = 25\).
Step 2: Plot the points
• If working with equal class widths: Plot (Midpoint, Frequency).
• If drawing a polygon from an unequal class width histogram: Plot (Midpoint, Frequency Density) at the top centre of each bar.
Step 3: Join the plotted points
• Use a ruler to join adjacent points with straight line segments.
• Do not join the first and last points back down to zero on the horizontal axis unless specifically instructed to do so!
Comparing Distributions Using Frequency Polygons
Frequency polygons are especially useful when you want to compare two different data sets on the same graph (for example, test scores of Class A vs. Class B).
When comparing two polygons, comment on two main features:
1. Average / Peak: Which distribution is shifted further to the right (higher values)?
2. Spread: Which polygon is wider (more spread out) or narrower (more consistent)?
Key Takeaway: Frequency polygons are plotted at the midpoint of each group and joined with straight lines using a ruler.
---6. Common Mistakes to Avoid in Exams
Mistake 1: Plotting frequency instead of frequency density.
Always check if the class widths are unequal. If they are, you must calculate and plot frequency density on the vertical axis.
Mistake 2: Plotting at the start or end of the interval for a polygon.
Frequency polygons must always be plotted at the midpoint of the interval, never at the lower or upper boundary.
Mistake 3: Drawing curves instead of straight lines on a frequency polygon.
A polygon has straight sides. Always use a ruler to connect point to point!
Mistake 4: Forgetting axis labels and units.
Always write the label and measurement unit (e.g., Mass (\(\text{kg}\)), Time (\(\text{s}\))) on the horizontal axis and Frequency Density on the vertical axis.
7. Quick Review Summary
• Histogram: Used for continuous data; no gaps between bars.
• Key Rule: \(\text{Area} = \text{Frequency}\).
• Key Formula: \(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\).
• Partial Bar Frequency: \(\text{Width of Section} \times \text{Frequency Density}\).
• Frequency Polygon: Plot at \((\text{Midpoint}, \text{Frequency})\) or \((\text{Midpoint}, \text{Frequency Density})\) and join with straight lines using a ruler.
• Comparison: Compare two distributions on the same axes by looking at their peaks (averages) and spread (consistency).