Histograms are the statistics topic students most often confuse with bar charts. They look similar — but they are not the same chart, and at GCSE Higher tier the difference is the whole point of the question. This guide explains exactly what makes a histogram a histogram, how frequency density works, and how to answer every question type that appears on the Higher paper.
Histograms vs Bar Charts — The Critical Distinction
A bar chart shows the frequency of discrete categories. The bars all have the same width, and the height of each bar is the frequency.
A histogram shows the frequency distribution of continuous numerical data grouped into class intervals — and here is where it diverges from a bar chart:
- The class intervals can be different widths
- The height of each bar is frequency density, not frequency
- The area of each bar represents the frequency
This is the entire point of a histogram: when class widths differ, using frequency density makes the visual comparison fair.
What Is Frequency Density?
Frequency density is the formula you must memorise for every histogram question:
frequency density = frequency ÷ class width
That is it. Memorise that formula. Every histogram question — without exception — uses it.
Example
A class interval of length 10 cm has a frequency of 30 students. Its frequency density is 30 ÷ 10 = 3.
Another class interval of length 5 cm has a frequency of 25 students. Its frequency density is 25 ÷ 5 = 5.
Even though the second class has a smaller total frequency, its frequency density is higher — because the data is more "concentrated" in that interval.
Drawing a Histogram from a Table
This is the most common histogram question on Higher papers. You are given a frequency table with class intervals (often of varying widths) and asked to draw the histogram.
Worked Example
The heights of plants in a garden, in cm:
| Height h (cm) | Frequency |
|---|---|
| 0 < h ≤ 10 | 8 |
| 10 < h ≤ 20 | 14 |
| 20 < h ≤ 40 | 30 |
| 40 < h ≤ 60 | 24 |
| 60 < h ≤ 100 | 16 |
Step 1: Find each class width.
- 0 < h ≤ 10: width 10
- 10 < h ≤ 20: width 10
- 20 < h ≤ 40: width 20
- 40 < h ≤ 60: width 20
- 60 < h ≤ 100: width 40
Step 2: Calculate frequency density for each class.
| Class | Frequency | Width | Frequency density |
|---|---|---|---|
| 0 < h ≤ 10 | 8 | 10 | 0.8 |
| 10 < h ≤ 20 | 14 | 10 | 1.4 |
| 20 < h ≤ 40 | 30 | 20 | 1.5 |
| 40 < h ≤ 60 | 24 | 20 | 1.2 |
| 60 < h ≤ 100 | 16 | 40 | 0.4 |
Step 3: Draw the histogram.
On graph paper:
- x-axis: the variable (height in cm) — choose a scale that fits all classes
- y-axis: frequency density — choose a scale that fits the maximum value (1.5)
- Draw each bar with width equal to the class width and height equal to the frequency density
The first bar is 10 wide and 0.8 tall. The third bar is 20 wide and 1.5 tall. And so on.
Important: label both axes clearly, including the units, and call the y-axis "Frequency density" (not "Frequency"). Examiners look for this label.
Reading a Histogram — Finding Frequencies
The other big question type goes the other way: you are given the histogram and asked to find the frequency for a class interval, or for a custom range.
The key rearrangement of the formula:
frequency = frequency density × class width
Worked Example
A histogram shows the frequency density for ages at a music festival. The bar for ages 25–30 has a frequency density of 8.
Frequency = 8 × 5 = 40 people aged 25–30.
Finding the Frequency for Part of a Bar
This is where students lose marks. The question asks something like: "Estimate the number of people aged between 27 and 35."
This range covers part of one bar and part of another. You need to:
- Identify which classes the range covers
- Take only the relevant portion of each bar
- Multiply each portion's width by the frequency density
Continued example: The bar for ages 25–30 has frequency density 8. The bar for ages 30–40 has frequency density 5.
- Ages 27–30: that is 3 years of the first bar. Frequency = 8 × 3 = 24.
- Ages 30–35: that is 5 years of the second bar. Frequency = 5 × 5 = 25.
Total estimate: 24 + 25 = 49 people aged between 27 and 35.
Note the word "estimate". When you read frequencies from a histogram for a partial bar, you are assuming the data is evenly distributed across the class. That assumption is rarely exactly true — hence "estimate".
Finding the Median from a Histogram
Some Higher papers ask you to estimate the median from a histogram. The technique:
- Find the total frequency by adding all the frequencies together (or multiplying each frequency density by its class width)
- Halve it to find the position of the median
- Work out which class the median falls in
- Estimate where in that class by interpolation
Worked Example
Total frequency from the plant heights table earlier: 8 + 14 + 30 + 24 + 16 = 92.
The median is at position 92 ÷ 2 = 46.
Running totals:
- After 0–10: 8
- After 10–20: 22
- After 20–40: 52
The median sits inside the class 20 < h ≤ 40 (because 22 < 46 ≤ 52).
How far into this class? We need 46 − 22 = 24 more values. The class has 30 values spread across a width of 20.
Estimated median = 20 + (24 ÷ 30) × 20 = 20 + 16 = 36 cm.
Common Question Types Summarised
Type 1: Complete the table or histogram. You are given some data and asked to fill in the missing frequencies or frequency densities. Use the formula in both directions.
Type 2: Estimate the number in a range. Use frequency = frequency density × width. Watch for partial classes.
Type 3: Estimate the median. Find total frequency, halve it, locate the class, interpolate.
Type 4: Compare two distributions. You will have two histograms or two tables. Compare them using mean, median, range and the visual shape (skewed, symmetric, where the peak is).
Type 5: Find the proportion / percentage above or below a value. Find the frequency above/below the value, then divide by total frequency. Multiply by 100 for a percentage.
Common Mistakes That Lose Marks
Using frequency on the y-axis instead of frequency density. This is the most common error. If your class widths differ, the chart you draw is not a histogram — it is a misleading bar chart. Always use frequency density.
Wrong class width. Read the boundaries carefully. The class 20 < h ≤ 40 has width 20, not 40 and not 19.
Forgetting the "estimate" caveat. When you read data from inside a class, you are estimating. Mark schemes look for the word "estimate" in your final answer.
Mixing up the formula direction. When drawing: divide frequency by width. When reading: multiply frequency density by width. Write the formula down at the start of each question to fix the direction in your mind.
Sloppy graph drawing. If your bars do not touch the right values, you lose marks. Use a ruler. Label the axes. Title the histogram.
What Examiners Are Marking
On a typical 5-mark histogram question, the marks usually break down as:
- 1 mark for the correct formula or method (frequency = density × width)
- 2 marks for correctly calculating two frequencies (one for each relevant class)
- 1 mark for handling the partial class correctly
- 1 mark for the final answer, correctly summed and clearly stated
You can earn 4 of the 5 marks even if you make a single arithmetic error, provided you show your method clearly. Always write the formula, then substitute, then calculate.
Where Histograms Fit in the Higher Statistics Topic Area
Histograms are part of the Statistics strand of the GCSE specification, which also includes:
- Cumulative frequency — a different way to handle grouped data, useful for medians and quartiles
- Box plots — visualise the quartiles and spread you find from cumulative frequency
- Scatter graphs — pair-wise data, not grouped data — different chart, different method
Of these four, histograms are the most consistently tested at Higher tier. The other three appear roughly every year too, so revise them as a set.
Quick Self-Check
Can you do these in under 60 seconds each?
- A class 0 < x ≤ 30 has frequency 60. What is its frequency density?
- A class 30 < x ≤ 35 has frequency density 12. What is its frequency?
- A class 40 < x ≤ 60 has frequency density 5. How many values lie in the range 50 < x ≤ 60?
Answers: (1) 2; (2) 60; (3) 50.
If those felt fast, you have got the core formula. The rest is just careful reading.
Practise GCSE Higher histogram questions with AI marking → — instant feedback on every step of your working. Free for all students.
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