Imagine tipping out a whole year of a coffee shop's receipts onto the table and asking one question: how busy is a normal day? You could shout back a single number — "about 240 cups" — but that throws almost everything away. Some days sold 90 cups; some sold 400. Weekends differ from Tuesdays. The average hides all of that.

A distribution keeps it. A distribution is the full picture of how often each value — or each range of values — shows up in your data: not just the middle and not just the spread, but the entire shape of what's common and what's rare. And the good news is you can see a distribution. A histogram is a distribution you can look at.

Frequency: counting how often

Start with something simple to count. Ask 40 friends how many pets they own, and tally the answers:

• 0 pets — 14 people
• 1 pet — 13 people
• 2 pets — 8 people
• 3 pets — 4 people
• 4 pets — 1 person

Each count is a frequency — literally, how frequently that value turned up. Draw a bar for each value with height equal to its frequency and you have drawn the distribution of "pets per person." At a glance you can see the story: most people have zero or one pet, and the big households thin out fast.

Divide each frequency by the total (40) and you get the relative frequency — the proportion of the data in each category. Here 14/40 = 0.35 of people own no pets. Relative frequencies always add up to 1 (that's just "everybody is somewhere"), which makes distributions from differently-sized samples comparable.

Binning: taming continuous data

"Number of pets" was easy: the values are whole numbers, so each gets its own bar. But most measurements aren't like that. Heights, weights, waiting times, temperatures — these are continuous, and no two measurements are ever exactly equal. If you gave every distinct height its own bar, every bar would have height 1 and the picture would be useless.

The fix is binning: chop the number line into equal-width intervals — say heights of 150–155 cm, 155–160 cm, 160–165 cm, and so on — and count how many measurements fall in each bin. Now each bar spans a range, and its height is the frequency for that range. That is exactly how a histogram turns a messy list of continuous numbers into a visible shape.

Collect more and more data and shrink the bins ever narrower, and the jagged staircase of bars settles toward a single smooth curve — the density curve of the distribution. It is the idealised shape the histogram is forever reaching for. Below, a fine-binned relative-frequency histogram of a fixed dataset is drawn behind its smooth density curve; watch how the tops of the bars trace out the bell.

Area means proportion

Once the bars become a density curve, we read the data through area, not height. The area under the curve between two values is the proportion of the data that falls there — equivalently, the probability that a randomly chosen value lands in that range.

Because every value lands somewhere, the total area under the curve is 1:

This is why the curve's height is called a density and not a count: a tall region means values are densely packed there, but it is the area — height times width — that turns into a proportion.

Worked example 1 — reading proportions off a histogram

A biologist measures the wingspans of 200 sparrows and bins them in 1 cm intervals. The bar heights (counts) come out as:

• 20–21 cm: 10 birds  •  21–22 cm: 30  •  22–23 cm: 60
• 23–24 cm: 55  •  24–25 cm: 30  •  25–26 cm: 15

Question: what fraction of sparrows have a wingspan of at least 24 cm?

Add the bars from 24 cm upward: 30 + 15 = 45 birds. As a proportion of the whole sample, 45 / 200 = 0.225. So about 22.5% — a bit more than one sparrow in five. Notice we never needed the average or the spread; we just counted the bars in the range and divided by the total. That is the whole trick of reading a distribution.

Worked example 2 — guessing the shape before you plot

You can often predict a distribution's shape from what the variable means. Three cases:

• The sum of two dice. The total runs from 2 to 12, but not evenly: there is only one way to roll a 2 (1+1) and one way to roll a 12 (6+6), yet six ways to roll a 7. So the distribution rises to a peak at 7 and falls away symmetrically — a triangular shape.
• Adult human heights. Most people cluster near a central height with fewer very short and very tall people, symmetric on both sides — a bell shape.
• Household income. Most households earn modest amounts, but a long thin tail of very high earners stretches far to the right — a right-skewed shape. The average gets dragged up by that tail, which is exactly why "average income" can be misleading and the median is often reported instead.

Same idea — a picture of how often each value occurs — but three genuinely different shapes. The shape is the information.

Worked example 3 — the same data, different bins

Here is something that surprises people. Take one fixed list of 100 exam marks and plot it three ways, changing only the bin width:

• Very wide bins (say, two bins: "under 50" and "50 or over"). You get two fat bars. You learn the pass rate and nothing else — all the internal structure is crushed flat.
• Sensible bins (steps of 10). A clean shape emerges: a peak around the 60s, tails on both sides. This is the useful picture.
• Very narrow bins (steps of 1). Now most bins hold 0, 1, or 2 marks and the histogram becomes a spiky comb — random ups and downs that mean nothing. The shape drowns in noise.

The underlying data never changed — only our choice of bin width did. Reading a histogram well means choosing bins wide enough to smooth out the randomness but narrow enough to keep the real structure.

Data's distribution vs. an idealised model

Keep two things apart. There is the distribution of your actual data — the real histogram of the sparrows you actually measured or the marks your class actually scored. It is bumpy, finite, and specific to that sample. Measure a fresh 200 sparrows and it will come out slightly different.

Then there is a theoretical (probability) distribution — a clean mathematical idealisation, like the perfectly smooth bell curve, defined by a formula rather than by counting. It is the shape an infinite amount of data would settle into. We use it as a model: we say "these sparrow wingspans are approximately normal" and then borrow all the tidy mathematics of the ideal curve to reason about the messy real sample. The first up-and-coming example of such an idealised curve is the normal distribution.