MathematicsChapter 11: Basic Statistics
Histograms
A histogram is a bar picture of grouped data.
Learning objectives
- Explain histograms in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
A histogram is a bar picture of grouped data. The rectangles are adjacent (no gaps between classes that touch at a boundary). Each base sits on the class boundaries. The area of a rectangle is taken as proportional to frequency. When every class has the same width h, height can stand for frequency, so taller means more. When widths are unequal, height must be frequency density fh, otherwise a wide class would look more important only because it is wide.
Where / symbols
- adjacent
- The rectangles are adjacent (no gaps between classes that touch at a boundary).
- area
- The area of a rectangle is taken as proportional to frequency.
- frequency
- The area of a rectangle is taken as proportional to frequency.
- boundaries
- Each base sits on the class boundaries.
- frequency density f/h
- When widths are unequal, height must be frequency density f/h, otherwise a wide class would look more important only because it is wide.
Overview
Equal width is the school default. Unequal width needs density:
Equal Width
If h is the same in every class, you may plot frequency on the vertical axis. Bases still use boundaries so the bars touch. A bar from 10.5 to 20.5 of height 8 means eight observations in that class. Do not leave a gap the way a bar chart of favourite colours would; a histogram of a continuous variable should look like joined blocks.
Examples Marks 0-10, 10-20, 20-30 with frequencies 4, 9, 7: three touching bars · Height of the middle bar is 9 if widths are equal · Using limits 1-10 and 11-20 without 10.5 would leave a false gap
Unequal Width
If widths are unequal, plotting raw f makes a wide class look more crowded than it is. Use frequency density = fh as the height. Then area = (fh) * h = f, so area still matches frequency. Label the vertical axis 'frequency density', not 'frequency', when you do this.
Examples Class width 5 with f = 10 => density 2 · Class width 10 with f = 10 => density 1 (same count, lower bar) · A double-width class with the same density has double the area, hence double the frequency
WHAT — Core idea
A histogram is a bar picture of grouped data. The rectangles are adjacent (no gaps between classes that touch at a boundary). Each base sits on the class boundaries. The area of a rectangle is taken as proportional to frequency. When every class has the same width h, height can stand for frequency, so taller means more. When widths are unequal, height must be frequency density fh, otherwise a wide class would look more important only because it is wide.
HOW — Rules, method, and examples
1. EQUAL WIDTH: If h is the same in every class, you may plot frequency on the vertical axis. Bases still use boundaries so the bars touch. A bar from 10.5 to 20.5 of height 8 means eight observations in that class. Do not leave a gap the way a bar chart of favourite colours would; a histogram of a continuous variable should look like joined blocks. 2. UNEQUAL WIDTH: If widths are unequal, plotting raw f makes a wide class look more crowded than it is. Use frequency density = fh as the height. Then area = (fh) * h = f, so area still matches frequency. Label the vertical axis 'frequency density', not 'frequency', when you do this.
WHY — Meaning and connections
A bar chart of favourite sports may have gaps and equal-width category bars. A histogram is for numerical classes on a number line. The ~ in 'area ~ frequency' means 'proportional to', the same similar-sign you met in geometry, used here for 'goes with'.
HOW — Answer in an exam
Histogram bars touch; bases are class boundaries. Equal width: height ~ f. Unequal width: height = frequency density fh so that area ~ frequency. Do not plot raw f on unequal classes. Do not confuse a histogram with a gap bar chart.
Explanation
A bar chart of favourite sports may have gaps and equal-width category bars. A histogram is for numerical classes on a number line. The ~ in 'area ~ frequency' means 'proportional to', the same similar-sign you met in geometry, used here for 'goes with'.
In simple words
Histogram bars touch; bases are class boundaries. Equal width: height ~ f. Unequal width: height = frequency density fh so that area ~ frequency. Do not plot raw f on unequal classes. Do not confuse a histogram with a gap bar chart.
Key points
- adjacent
- area
- frequency
- boundaries
- same
- height
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Histograms
Sticky Notes · Quick Revision
WHAT
- A histogram is a bar picture of grouped data.
Key Points
- A histogram is a bar picture of grouped data.
- If h is the same in every class, you may plot frequency on the vertical axis.
- If widths are unequal, plotting raw f makes a wide class look more crowded than it is.
- A bar chart of favourite sports may have gaps and equal-width category bars.
HOW
- 1.
- If h is the same in every class, you may plot frequency on the vertical axis.
- If widths are unequal, plotting raw f makes a wide class look more crowded than it is.
Examples
- Marks 0-10, 10-20, 20-30 with frequencies 4, 9, 7: three touching bars
- Height of the middle bar is 9 if widths are equal
- Using limits 1-10 and 11-20 without 10.5 would leave a false gap
- Class width 5 with f = 10 => density 2
- Class width 10 with f = 10 => density 1 (same count, lower bar)
Common Mistakes
- Correction: Histogram bars touch; bases are class boundaries.
- Correction: Equal width: height ~ f.
- Correction: Unequal width: height = frequency density fh so that area ~ frequency.
Exam Tip
- Histogram bars touch; bases are class boundaries. Equal width: height ~ f. Unequal width: height = frequency density fh so that area ~ frequency. Do not plot raw f on unequal classes. Do not confuse a histogram with a gap bar chart.
60-second revision
- A histogram is a bar picture of grouped data.
- If h is the same in every class, you may plot frequency on the vertical axis.
- If widths are unequal, plotting raw f makes a wide class look more crowded than it is.
- A bar chart of favourite sports may have gaps and equal-width category bars.
- Histogram bars touch; bases are class boundaries.
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