# 2.1 What is an empirical distribution?

#### Overview
What is an empirical distribution? The empirical distribution of a variable is able to be characterised by considering each unique numerical value observed for that variable in a given sample of size $n$. If certain values are repeated, then we simply tally the number of each unique value obtained. These tallies are essentially ***raw frequencies*** of the values. We can order the values obtained for the variable from smallest to largest and then look at these frequencies cumulatively, as a percentage of the entire sample. A plot of these ***cumulative percentages*** as a function of the ordered values in the sample is known as the ***empirical cumulative distribution function***.

#### Description
More formally, suppose we have $n$ independent and identically distributed random variables, $Y_1, Y_2, \ldots, Y_n$, with a  common (but unknown) probability density function (pdf) of $f(y)$ and cumulative distribution function (cdf) of $F(y) = \text{Pr} \lbrace Y \leq y \rbrace $.

For the discrete case, we have $F(y) = \sum_{t \leq y} f(t)$.

For the continuous case, we have $F(y) = \int_{-\infty}^y f(t) \cdot dt$.


We obtain corresponding observed values $y_1, y_2, \ldots, y_n$ in a sample of size $n$.  Now, let $I(y_i \leq t)$ be an indicator that takes the value of $1$ if $y_i \leq t$ is true, and zero otherwise. The empirical cumulative distribution function $\hat{F}_ n(t)$ is defined as the proportion of data points in the sample that are less than or equal to $t$, i.e.,

$$
\hat{F}_ n(t) = \frac{1}{n} \cdot \sum_{i=1}^n I(y_i \leq t)
$$ 
 
This is therefore a step function, continuous from the right, that jumps up by a quantity of $1/n$ at each of the $n$ data points. Its shape gives us a basic visual understanding of the distributional shape of the data values.

#### A small example
Suppose we had the following data with a sample size of $n = 10$ for a variable, $Y$:
| Sample | *y* |
| :----- | :-: |
|1|2|
|2|7|
|3|10|
|4|12|
|5|2|
|6|5|
|7|5|
|8|8|
|9|10|
|10|3|

The raw frequencies are as follows:
| Value of *y* | Frequency |
| :----- | :-: |
|2|2|
|3|1|
|5|2|
|7|1|
|8|1|
|10|2|
|12|1|

Looking at these values cumulatively, we have:
| Value of *y* | Cumulative frequency |
| :----- | :-: |
|2|2|
|3|3|
|5|5|
|7|6|
|8|7|
|10|9|
|12|10|

Expressing these frequencies as ***cumulative proportions*** of the total sample, we have:
| Value of *y* | Cumulative proportion |
| :----- | :-: |
|2|0.2|
|3|0.3|
|5|0.5|
|7|0.6|
|8|0.7|
|10|0.9|
|12|1.0|

These cumulative proportions comprise the **empirical cdf**. A plot of this empirical distribution (a step function) is shown below, with open circles being used to show the discontinuities (i.e., at the point of each step).

[![01._A_small_example_[i].png](https://learninghub.primer-e.com/uploads/images/gallery/2025-12/scaled-1680-/01-a-small-example-i.png)](https://learninghub.primer-e.com/uploads/images/gallery/2025-12/01-a-small-example-i.png)

The above plot was obtained by running **Plots** > **Empirical Distribution Plot...** in PRIMER 8, and ticking the option to: '$\checkmark$Express as proportions: [0, 1]'. Note that you can alternatively look at an empirical cdf with the values expressed as percentages instead of proportions (i.e., a plot of $100 \times \hat{F}_ n(t)$ *versus* $t$), which is the default.

As an aside, a related graphical tool for examining distributional shapes of variables is a **histogram**. A histogram is a plot of the raw frequencies (as bars on the y-axis) *vs* the empirical values (on the x-axis). For a histogram, we would typically pool together (sum) the raw frequencies into larger 'bin' sizes (instead of having one bin for every unique value in the dataset), which can be very useful if $n$ is large. That's what the **Plots** > **Histogram Plot...** function in PRIMER 8 does. Once you get a histogram, you can also change the bin size by clicking **Graph** > **Special**.