▁▃▇ · Frequency & Histograms
Bin them, and the distribution's shape rises out of the pile.
A roster of 25 classmates' heights — does reading 25 numbers reveal the class?
Consider bin width together with frequency.
Two data sets, same average, utterly different faces.
Open the immersive version →A reading path containing this lesson: What a collection of numbers tells us · 2/3
Why does this question matter?
Thirty test scores, 25 heights, a thousand customer ages — data arrives as piles of individual numbers, and reading them yields no judgment.
The fix: choose intervals (classes) and count how many fall in each (frequency). As a table it's a frequency table; as bars, a histogram — the pile becomes a shape.
Once the shape shows, questions become possible: where does it cluster (center), how wide (spread), is it lopsided (skew)? Statistical thinking starts here.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — read the roster of 25 classmates' heights (25 raw numbers). Does the class's character show at a glance?
Bin the roster
Below is the actual roster. Press to bin into 5cm classes and count.
Read the idea from another angle
A histogram is the data's face.
A mean is just the data's address; the histogram shows the whole visage. Two groups with equal means can wear utterly different faces — the habit of looking at distributions is half of statistics.
Bin width is resolution.
Too narrow and you see only jagged noise; too wide and features smear away. The same data tells different stories at different widths — choosing the width IS analysis.
Now read the definitions with their conditions
What the height-roster experiment confirmed, in mathematical language.
Class and frequency
A class is a bin holding data; frequency counts what falls inside — their table is the frequency table.
Relative frequency
Frequency divided by the total — the common language for comparing groups of different sizes. Always sums to 1.
The histogram
The frequency table drawn — the bars' silhouette is the distribution's shape (bell, skew, peaks).
Make one more distinction
Binning loses information, so it's a loss.
You lose individual values but gain the more important thing: the distribution's shape. Where the data clusters and how it spreads never shows in a roster — summarizing is translation, not loss.
The group with the taller bar has the higher share in that class too.
With different group sizes, comparing counts is meaningless. 7 of 25 (28%) beats 20 of 100 (20%) in share — groups compare by relative frequency, not frequency.
Where else can this idea help?
Score distributions
'Average 72' says little. Draw the histogram and you see whether it's twin-peaked (studied vs gave up) or bell-shaped — why teachers read distributions before averages.
Population pyramids
Two age histograms (male/female) in 5-year bins, placed back to back — aging societies read off a single shape.
Camera exposure histograms
A photo app's histogram is the frequency distribution of pixel brightness. Piled to the right means overexposed — photographers read one per shot.
Game rank distributions
An online game's tier chart is a histogram of player skill. Where you sit on the bell is your percentile.
Explain it in your own words
Try another question
Q1Which heights belong to the class [155, 160)? Does 160 count?Show answer ▾
155 up to but not including 160 — 160 goes to the next class [160, 165). The 'from-inclusive, to-exclusive' agreement keeps boundary values from being counted twice.
Q2If 6 of 25 students fall in [150,155), what's the relative frequency?Show answer ▾
6 ÷ 25 = 0.24 (24%). All relative frequencies sum to exactly 1 — a built-in check.
Q3What happens to the histogram with very narrow (1cm) bins?Show answer ▾
Nearly 25 scattered, jagged bars — the shape dissolves. Too wide (30cm) and everything smears into one bar. A moderate width keeps the shape alive — width choice is part of the analysis.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: What a collection of numbers tells us · 2/3
Previous concept
Averages
Feel one number's limits (the mean) and the need for shape (distribution) appears.
← Averages labLeads to next
Box Plots
Another portrait — summarizing a distribution in five numbers instead of the full shape. The weapon of group comparison.
Go to the Box Plots lab →Related lab
Averages
Summarizing the center in one number — complete only alongside the shape.
Go to the Averages lab →Related lab