IB Maths AI HLStatistics ToolkitPaper 1 & 2~6 min read
Histograms
A frequency histogram shows grouped continuous data as bars, where the height of each bar is the frequency of that class. It looks like a bar chart, but with one crucial difference: there are no gaps between the bars, because the data is continuous — each class runs straight into the next. Histograms make the shape of a distribution obvious at a glance: you can spot the modal class, see whether the data is symmetric or skewed, and judge whether it might follow a normal distribution. For IB AI HL, the classes always have equal width, which keeps the heights directly comparable.
📘 What you need to know
A frequency histogram displays grouped continuous data as bars; the height = frequency of the class.
No gaps between bars — the data is continuous, so each bar runs from the lower boundary to the upper boundary of its class.
Frequency is on the y-axis; the continuous variable is on the x-axis.
Classes have equal width (for IB AI HL), so taller bar = more frequent class.
The modal class is the class (bar) with the highest frequency — read it straight off the tallest bar.
Histogram vs bar chart: a bar chart is for qualitative or discrete data and has gaps; a histogram is for continuous data and has no gaps.
Histograms show the shape of the distribution — symmetry, skew, and whether it could be modelled by a normal distribution (symmetric, bell-shaped).
Histogram vs bar chart
This distinction is examined directly, and the “no gaps” rule is the giveaway. It follows from the type of data each chart represents.
Bar chart
gaps
Qualitative or discrete data (e.g. car colours, number of pets). Separate categories → gaps between bars.
Histogram
no gaps
Continuous grouped data (e.g. mass, height, time). Classes touch → bars touch, no gaps.
🤔 Why no gaps?
Continuous data has no “breaks” in it — a mass could be 7.9 kg, 8.0 kg, 8.01 kg, anything. The class 4 ≤ m < 8 ends exactly where 8 ≤ m < 12 begins, with nothing missing in between. So the bars must touch to show the variable flows continuously. A gap would falsely suggest there are values that don’t belong to any class. By contrast, “red cars” and “blue cars” are separate categories with nothing between them, so a bar chart leaves gaps.
Drawing a frequency histogram
🧭 Recipe — drawing a frequency histogram
Draw and label the axes: continuous variable on x, frequency on y, both with even scales and units.
Mark the class boundaries along the x-axis.
For each class, draw a bar from its lower boundary to its upper boundary.
Set each bar’s height to the class frequency (read off the y-axis).
Leave no gaps between adjacent bars.
frequency histogram — dolphin masses
Bars touch with no gaps (continuous data). The tallest bar, 12 ≤ m < 16 with frequency 19, is the modal class.
What histograms tell you
Beyond just displaying the data, a histogram is read for two things: the modal class and the overall shape.
What you read
How
Modal class
the class with the tallest bar (equal widths)
Symmetry / skew
look at where the tall bars sit and how the heights tail off
Could it be normal?
roughly symmetric and bell-shaped → possibly a normal distribution
🧠 Memory aid — “modal class, not mode”
With grouped data you can never name a single mode — the exact values are hidden inside the classes. The best you can do is the modal class: the interval with the highest frequency. So on a histogram, always answer “the modal class is 12 ≤ m < 16″, never “the mode is 14”.
Worked examples
WE 1
Draw a frequency histogram
The table shows the masses (kg) of newborn dolphins. Draw a frequency histogram.
mass m (kg)
4≤m<8
8≤m<12
12≤m<16
16≤m<20
20≤m<24
frequency
4
15
19
10
6
set up the axesx-axis: mass 4 to 24 kg with class boundaries; y-axis: frequency 0 to 20.draw the bars (heights = frequencies)heights: 4, 15, 19, 10, 6bars must touch — NO gaps, because mass is continuous data.see the diagram above
WE 2
Write down the modal class
Using the dolphin data from WE 1, write down the modal class.
modal class = class with highest frequencyfrequency 19 is the highest → 12 ≤ m < 16modal class: 12 ≤ m < 16say “modal class”, never “mode = 14”, for grouped data.
WE 3
Histogram or bar chart?
For each dataset, state whether a histogram or a bar chart should be used. (a) The favourite colour of 50 students. (b) The heights, in cm, of 50 students.
(a) favourite colourqualitative data → separate categories(a) bar chart (with gaps)(b) heights in cmcontinuous data → grouped into classes(b) histogram (no gaps)continuous ⇒ histogram; qualitative/discrete ⇒ bar chart.
WE 4
Total frequency & a proportion from a histogram
A histogram has bars of height (frequency) 3, 8, 14, 9, 6 across five equal classes. (a) How many data values are there in total? (b) What fraction lie in the modal class?
(a) total = sum of bar heights3 + 8 + 14 + 9 + 6 = 40(a) n = 40(b) modal class frequency = 14fraction = 14/40 = 0.35(b) 0.35 (35%)with equal-width classes, the bar heights ARE the frequencies — just add them.
WE 5
Describe the shape
A histogram of waiting times has bar heights 2, 6, 12, 7, 3 across equal classes. Comment on the shape of the distribution.
look at the pattern of heightsrises to a peak in the middle (12), then falls away on both sidesthe bars are fairly balanced about the peak — roughly symmetric and bell-shaped.approximately symmetric → could be normala symmetric, bell-shaped histogram suggests the data may follow a normal distribution.
💡 Top tips
No gaps between bars — this is the single most-tested feature. Continuous data means touching bars.
Frequency on the y-axis, the continuous variable on the x-axis — both labelled with units and even scales.
Bars span the full class width, from lower boundary to upper boundary.
Modal class = tallest bar (valid because the IB uses equal class widths).
Always say “modal class”, not “mode” for grouped data.
Total frequency = sum of the bar heights when classes are equal width.
Symmetric & bell-shaped histogram → the data could be modelled by a normal distribution.
⚠ Common mistakes
Leaving gaps between the bars. A histogram is for continuous data — the bars must touch.
Drawing bars that don’t span the class width. Each bar runs from the lower to the upper boundary.
Writing “mode = 14”. For grouped data there’s no single mode — give the modal class.
Confusing histogram and bar chart. Qualitative/discrete = bar chart (gaps); continuous = histogram (no gaps).
Unequal or unlabelled axis scales. Use consistent scales and label both axes with units.
Putting the variable on the y-axis. Frequency goes on y; the data variable goes on x.
Next up — Interpreting Data. You’ve now met every tool in the kit: averages, spread, frequency tables, box plots, cumulative frequency curves and histograms. The final skill is choosing which measure or diagram fits a situation, and comparing datasets properly — using the median & IQR when there are outliers, and the mean & standard deviation when the data is symmetric.
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