Histograms and empirical distributions · RiskAtoms Academy

By the end you will be able to

1Build a histogram: choose bins and count the values in each.
2Tell frequency from density, and convert between them.
3Explain how the bin width changes the picture.
4Read the share of values at or below any level from the empirical distribution.

Why you need it

A list of 250 numbers tells you very little at a glance. Sort them into bins and draw the counts, and the shape jumps out: where the middle is, how wide it spreads, and how far the extremes reach.

These Core Path lessons use it:

L2.5.02: Return distributions
Histograms show the shape of past returns: the centre, the spread and the tails.
L2.5.07: Historical VaR
It reads the share of past outcomes below a level straight from the empirical distribution.

The idea

Sort into bins, then count

Cut the number line into equal pieces, called bins. Take each value in turn and drop it into the bin it belongs to. The piles you end up with are a histogram.

Sort values into bins

0 of 20 dropped · bins 1% wide

0.4−1.30.92.1−0.20.6−0.81.40.1−2.40.7−0.51.10.3−1.70.2−0.11.8−0.60.5

Press the button to drop the first value.

Most values pile up near 0, and a few sit out at the edges. That outline, tall in the middle and thin at the ends, is the shape of the data.

The rules

Bins and counts. A histogram needs one choice from you, the bin width, and one convention for values on an edge.

Frequency or density? The bar height can be the count (the frequency), the share, or the density. Density is the one to use when you compare histograms with different bin widths or sample sizes, or lay a curve such as the normal from M-C06: The normal distribution on top.

The empirical distribution. A histogram answers “how many in this bin?”. The question “what share is below this level?” is answered directly, without bins, by the empirical distribution function.

Read the empirical distribution

the same 20 values

Value x−1%

At or below x3 of 20

F̂(x)15%

The rules, for reference

n_k = \#\{\,i : b_{k-1} \le x_i < b_k\,\} \qquad f_k = \frac{n_k}{n} \qquad d_k = \frac{n_k}{n\,h} \qquad \sum_k h\,d_k = 1

\hat F(x) = \frac{\#\{\,i : x_i \le x\,\}}{n}

The # sign means “the number of”. Reading $\hat F$ backwards, from a share to a value, gives a quantile: see M-C09: Quantiles.

Try it

Slide the bin width from very narrow to very wide: when does the shape look clearest? Switch to density and change the number of days: the bars stay on the same scale. Then move x into the left tail and read the share of days at or below it.

Try itHistogram explorer

Descriptive statistics calculator →

Bin width h0.5%

Mark a value x−2%

Bar height

Number of days

250 illustrative daily returns, in %. Shaded: days at or below x.

Number of bins16

Tallest bar61 days

Total area of the bars (density)1

Days at or below x8 of 250

Empirical share F̂(x)3.2%

Worked examples

1. A histogram by hand

Step 1 of 3

1
The set-up
Twelve daily returns, in % (illustrative): −1.6, −0.4, 0.3, 1.2, −0.7, 0.1, 2.3, −0.2, 0.8, −1.1, 0.5, 0.6. Use bins 1% wide, starting at a whole number.

2. Same data, different bins

Step 1 of 3

1
Half-width bins (h = 0.5)
Bin Count Density = count ÷ (12 × 0.5)
−2% to −1.5% 1 0.167
−1.5% to −1% 1 0.167
−1% to −0.5% 1 0.167
−0.5% to 0% 2 0.333
0% to 0.5% 2 0.333
0.5% to 1% 3 0.5
1% to 1.5% 1 0.167
1.5% to 2% 0 0
2% to 2.5% 1 0.167

Bin	Count	Density = count ÷ (12 × 0.5)
−2% to −1.5%	1	0.167
−1.5% to −1%	1	0.167
−1% to −0.5%	1	0.167
−0.5% to 0%	2	0.333
0% to 0.5%	2	0.333
0.5% to 1%	3	0.5
1% to 1.5%	1	0.167
1.5% to 2%	0	0
2% to 2.5%	1	0.167

3. What share of days was below −2%?

Step 1 of 4

1
The set-up
Take the first 250 illustrative daily returns from the Try it. What share of days had a return of −2% or worse?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 2

40 of 200 values fall in a bin 0.5 wide. What is the density height of that bar?

Question 2 · objective 3

In count mode, you double the bin width. What usually happens?

AFewer, taller bars: each bar now gathers about twice as many values.BMore, shorter bars.CThe bars keep their heights; only the labels change.DThe total number of values doubles.

Question 3 · objective 4

Of 250 daily returns, 12 are at or below −2%. What is the empirical share F̂(−2%), in %?

%

Question 4 · objective 2

What do the bars of a density histogram add up to?

ATheir areas add up to 1.BTheir heights add up to 1.CTheir heights add up to n.DTheir areas add up to n.

Question 5 · objective 4

Data: 3, 7, 1, 9, 4. What share of the values is at or below 4? (Answer as a decimal.)

Glossary

Histogram: A bar chart of data grouped into bins: each bar shows how many values (or what share) fall in its bin.
Bin: One interval of the axis, such as from 0% up to 1%. Here a value on an edge goes in the bin to its right.
Bin width: The length of each bin, h. Narrow bins show more detail and more noise; wide bins smooth both away.
Frequency and relative frequency: The count of values in a bin, n_k. Its share of all the data, n_k / n, is the relative frequency.
Density: Bar height n_k / (n h), chosen so that each bar's area is its share of the data and all the areas add up to 1.
Empirical distribution function: F̂(x): the share of the data at or below x. A staircase that climbs by 1/n at each value.