Histograms

Histograms

What is it?

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.

In simple words

• 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.

Important terms

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 fh: When widths are unequal, height must be frequency density fh, otherwise a wide class would look more important only because it is wide.

What you should learn

• 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.

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

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.

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.

Full school-level explanation

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.

Equal width is the school default. Unequal width needs density:

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.

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.

Remember

• same
• height
• f

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.

Remember

• unequal
• fh
• area

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

Remember

• adjacent
• area
• frequency
• boundaries
• frequency density fh
• number line
• proportional to
• touch

How the ideas connect

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'.

Deeper understanding

Go beyond memorising Histograms. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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'.

Common mistakes

Exam focus

WHAT

Key Points

HOW

Examples

Exam focus

Quick revision

• adjacent
• area
• frequency
• boundaries
• 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.

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.

For long answers on Histograms: define clearly → explain the main parts → give one example → state one exam caution.