IB Maths AA SL Topic 4 — Statistics Toolkit Paper 1 & 2 ~9 min read

Histograms

A histogram is a bar chart for continuous data — bars touch each other, no gaps. They show how data is distributed across class intervals at a glance, and tell you whether the data is symmetric, skewed, or has a clear peak.

📘 What you need to know

A histogram shows the frequency of each class interval as a bar.
Used for continuous grouped data (heights, weights, times, etc.).
Bars touch each other — no gaps. (Bar charts have gaps; histograms don’t.)
x-axis = the variable (with units), y-axis = frequency.
The tallest bar tells you the modal class — the class with the highest frequency.
For IB AA SL, all class intervals are equal width — bar height = frequency.

What is a histogram?

A histogram is just a chart showing how many values fell into each class interval. The bars sit on a number axis (no gaps between them), and the height of each bar tells you how frequent that range was.

For example, if 19 students had heights between 12 and 16 cm, the bar above “12–16” would be 19 units tall. Done.

A typical histogram — newborn dolphin masses

Notice how the bars touch each other — no gaps. That’s the dead giveaway that you’re looking at a histogram and not a bar chart. It shows the data is continuous: a value of, say, 11.99 kg flows seamlessly into 12 kg.

How to draw a histogram

5 steps to draw a histogram

Get a frequency table with class intervals (or build one from raw data).
Draw the axes — x-axis = the variable (with units), y-axis = frequency. Both should have even scales.
Mark the class boundaries on the x-axis (e.g. 4, 8, 12, 16, 20, 24).
Draw a bar over each class — height = frequency, no gaps between bars.
Label your axes and give the histogram a clear title.

📍

The bar’s width = the class width

Each bar starts at the lower boundary of its class and ends at the upper boundary. So a class of “12 ≤ m < 16” has a bar from 12 to 16 — width 4. The bar’s height is just the frequency.

Histogram vs bar chart — what’s the difference?

Histogram

For continuous data (e.g. height, time, mass).
Bars touch — no gaps between them.
x-axis is a continuous number line.
Each bar represents a range (class interval).

Bar chart

For discrete or qualitative data (e.g. shoe size, eye colour).
Bars are separate — gaps between them.
x-axis is just labels or categories.
Each bar represents a single value or category.

🧠

Memory trick: “Continuous = Connected”

If the data flows continuously (like height: 165, 165.1, 165.2…), the bars must connect. If the data is in discrete chunks (shoe size 5, 6, 7…), the bars stay apart. Continuous → connected. That’s the only rule you need to remember.

What does the shape tell you?

Once drawn, the shape of the histogram tells you a lot about the data at a glance:

Symmetric / Bell-shaped

Data is roughly symmetric around the centre. Often suggests it can be modelled by a normal distribution.

Right-skewed

Most data on the left, tail stretching to the right. The mean is bigger than the median.

Left-skewed

Most data on the right, tail stretching to the left. The mean is smaller than the median.

Three common histogram shapes

📍

Bell-shaped histograms suggest a normal distribution

If your histogram looks roughly symmetric with a clear peak in the middle and tails fading out on both sides — that’s a hint that the data could be modelled with a normal distribution. You’ll meet this in detail later in the syllabus.

What can I do with a histogram?

Find the modal class — just look for the tallest bar.
Spot the shape of the distribution — symmetric, skewed left, or skewed right.
Compare two data sets visually by drawing two histograms side by side.
Estimate frequencies in a range that spans multiple classes by adding heights.

A histogram is great for seeing data, but not the best for calculating things like the median or quartiles. For those, use a cumulative frequency graph instead.

Worked examples

WE 1

Draw a frequency histogram and find the modal class

The table below shows the masses, in kg, of some newborn bottlenose dolphins.

Mass, m (kg)	Frequency
4 ≤ m < 8	4
8 ≤ m < 12	15
12 ≤ m < 16	19
16 ≤ m < 20	10
20 ≤ m < 24	6

(a) Draw a frequency histogram. (b) State the modal class.

Continuous data → bars must touch. Bar widths = 4 kg each (equal class intervals).part (a) x-axis: 4 to 24 kg. y-axis: frequency 0 to 20. Draw bars at heights 4, 15, 19, 10, 6. part (b) Highest frequency = 19, in the row 12 ≤ m < 16. Modal class = 12 ≤ m < 16 no gaps between bars — that’s the histogram rule for continuous data

WE 2

Read information from a histogram

The histogram below shows the time, in minutes, that 40 students spent on homework one evening.

(a) State the modal class. (b) How many students spent at least 30 minutes on homework? (c) What percentage spent less than 15 minutes?

n = 40 studentspart (a) — modal class Tallest bar = 14 students: Modal class = 30 ≤ t < 45 minpart (b) — at least 30 minutes Add bars from 30 onwards: 14 + 6 + 3 = 23 23 studentspart (c) — less than 15 minutes First bar = 5 students. Percentage = 540 × 100 = 12.5% 12.5% add the heights of all relevant bars to find frequencies in a range

WE 3

Identify the shape of the distribution

For each frequency table, describe the shape of its histogram.

(a) Heights of trees (cm): 5, 8, 12, 8, 5

(b) Salaries ($000): 18, 12, 7, 4, 2

(c) Test scores: 1, 3, 6, 12, 18

Look at where the highest frequencies sit and where the tail goes.part (a) Frequencies rise then fall: 5, 8, 12, 8, 5 Roughly mirror image around the middle. Symmetric (bell-shaped)part (b) Bars get smaller from left to right: 18, 12, 7, 4, 2 Tail stretches to the right. Right-skewed (positively skewed)part (c) Bars get larger from left to right: 1, 3, 6, 12, 18 Tail stretches to the left. Left-skewed (negatively skewed) “skewed” = the side where the tail trails off

WE 4

Estimate the mean from a histogram

Using the dolphin masses from WE 1, estimate the mean mass of a newborn dolphin.

Histogram → grouped data → use mid-interval values for the mean. Find each mid-interval: 6, 10, 14, 18, 22 Multiply by frequencies: Σfx = 4×6 + 15×10 + 19×14 + 10×18 + 6×22 = 24 + 150 + 266 + 180 + 132 = 752 Divide by total frequency: n = 4+15+19+10+6 = 54 Mean ≈ 75254 ≈ 13.93 Estimated mean ≈ 13.9 kg histograms come from grouped data → only estimates are possible

WE 5

Reconstruct a frequency table from a histogram

The histogram below shows ages of visitors to a museum.

Build a frequency table and find the total number of visitors.

Read the height of each bar to get the frequencies. Read off bar heights: 0–20: 8, 20–40: 18, 40–60: 14, 60–80: 6, 80–100: 2 Total visitors: 8 + 18 + 14 + 6 + 2 = 48 Total = 48 visitors to reverse-engineer a frequency table, just measure each bar height

💡 Top tips

No gaps between bars. If you see gaps, it’s a bar chart, not a histogram. Continuous data → connected bars.
Use class boundaries on the x-axis, not midpoints. The bar runs from the lower boundary to the upper boundary.
Use even, evenly-spaced scales on both axes, with units on the x-axis.
Modal class = the class corresponding to the tallest bar. Don’t say “the mode is 14” — say “modal class is 12 ≤ m < 16”.
For frequencies in a range, just add up the heights of the bars that span that range.
For the mean from a histogram, treat each bar’s middle as the value, then use Σfx ÷ n with mid-intervals.
Bell shape → roughly normal. Long tail to the right → positively skewed. Long tail to the left → negatively skewed.
For IB AA SL, all class intervals are equal width, so bar height = frequency. (Frequency density is for HL only.)

⚠ Common mistakes

Drawing gaps between bars. A histogram for continuous data has bars touching — no gaps.
Using midpoints as the bar’s edges. Bars run from the lower boundary to the upper boundary, not centred on the midpoint.
Saying “mode = 14” instead of “modal class = 12 ≤ m < 16”. Grouped data only gives a modal class.
Confusing histograms with bar charts. Bar charts are for discrete or qualitative data; histograms are for continuous.
Missing axis labels or units. Examiners deduct marks for unlabelled axes, even if your bars are correct.
Reading bar heights as values, not frequencies. The y-axis is “how many” — not “what value”.
Calling a left-skewed shape “right-skewed” (or vice versa). Check where the tail trails off, not where the peak is.
Trying to find the median directly from a histogram. You can’t easily — use a cumulative frequency graph instead.

Histograms are mostly visual — they’re great for quick shape recognition and finding the modal class. The next note (Interpreting Data) brings everything in this toolkit together: how to choose the right measure, the right diagram, and how to compare two data sets in context.

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