الرياضيات

What Is Frequency Density?

Q: Why is the height of a histogram bar not the frequency?

Because the classes have different widths. If height were frequency, a class covering 30 minutes would look the same as one covering 5 minutes with the same count, hiding the fact that the second is far more crowded. Dividing by width puts every class on the same footing, and the frequency is then the area.

Q: How do I find the class width for 150–159 cm?

Ten. Heights given to the nearest centimetre mean the class actually covers 149.5 cm up to 159.5 cm, and 159.5 − 149.5 = 10. Subtracting the written labels gives 9, which is the most common error in the whole topic and shifts every frequency density in the table.

Q: Can frequency density be a decimal or less than 1?

Yes, and it usually is. It has no requirement to be a whole number because it is not a count — it is a count per unit of the horizontal axis. A class of 18 spread over a width of 30 gives 0.6, meaning 0.6 people per minute on average across that class.

Q: What is the difference between a histogram and a bar chart?

A bar chart shows categories or discrete values, has gaps between the bars, and uses height for frequency. A histogram shows continuous data grouped into classes, has no gaps because the classes join, and uses area for frequency. They look similar and answer different questions.

Q: Do I need frequency density if all the classes are the same width?

Strictly the axis should still be frequency density, but when every width is equal the bars have the same relative heights either way, so many textbooks plot frequency directly. As soon as one class differs, the shortcut breaks and the whole diagram misleads.

Frequency divided by class width, worked across a table of unequal classes — and why the tallest bar on a histogram is not the one with most in it.

Frequency density

Frequency density is the frequency of a class divided by its width, and it is the quantity plotted up the vertical axis of a histogram.

أين يقابله الطلاب: Grade 9 or 10 statistics, and on the Higher tier of GCSE Mathematics, where histograms with unequal class widths first appear.

الإجابة باختصار
Frequency density is frequency divided by class width, and it is what goes up the vertical axis of a histogram. A class of 24 spread over 20 minutes has density 1.2, so it is drawn shorter than a class of 15 packed into 10 minutes, whose density is 1.5.

مثال

0 < t ≤ 10: f = 15, density 1.5 | 10 < t ≤ 30: f = 24, density 1.2 | 30 < t ≤ 60: f = 18, density 0.6

The middle class has the largest frequency of the three but not the tallest bar. Twenty-four people spread across twenty minutes are thinner on the page than fifteen people packed into ten. Divide each frequency by its own class width — 15 ÷ 10, 24 ÷ 20, 18 ÷ 30 — and the three bars can finally be compared with each other.

The frequency lives in the area

A histogram is not a bar chart with the gaps taken out. On a bar chart the height carries the information. On a histogram the area does, and the height is only there to make the area come out right.

That single fact explains everything else. It is why the vertical axis has to be labelled "frequency density" and not "frequency". It is why classes of different widths can sit side by side and still be compared honestly. And it is why you can read a frequency back off the graph by multiplying, rather than by looking at a number on the axis.

Getting the class width right

The arithmetic is one division per row, so nearly all lost marks come from the width rather than the sum. For a class written 10 < t ≤ 30, the width is 20, not 21 and not 19 — subtract the boundaries.

Rounded data is the trap. Heights recorded to the nearest centimetre and grouped as 150–159 do not stop at 159. Anything from 149.5 up to 159.5 was recorded in that class, so the width is 10, not 9. Ages behave differently again: a class labelled 20–29 years usually runs from the 20th birthday to the 30th, which is also a width of 10, because nobody rounds their age up.

10 < t ≤ 30 — width 20
150–159 cm, measured to the nearest cm — width 10, running from 149.5 to 159.5
20–29 years old — width 10, running from exactly 20 to exactly 30

Reading a frequency back off the graph

Given a finished histogram, reverse the operation: frequency equals frequency density multiplied by class width, which is just the area of the bar. A bar of height 1.2 covering 20 minutes holds 24 people.

Questions often ask for the number inside part of a class — how many took between 10 and 15 minutes when the class runs from 10 to 30. Take the matching fraction of the area: a quarter of the width gives a quarter of the frequency, so 6 people. Be honest about what that assumes. It treats the 24 people as evenly spread across the twenty minutes, which they almost certainly are not. The answer is an estimate, and exam papers say so.

The area of a histogram bar, which is the frequency of that class: width × densityThe area of a histogram bar, which is the frequency of that class

أسئلة شائعة

Why is the height of a histogram bar not the frequency?