There are many ways to quantify variability, however, here we will focus on the most common ones: **variance**, **standard deviation**, and **coefficient of variation**.

In the field of statistics, we typically use different formulas when working with population data and sample data.

**Sample Formulas vs Population Formulas**

When we have the whole population, each data point is known so you are 100% sure of the measures we are calculating.

When we take a sample of this population and compute a sample statistic, it is interpreted as an approximation of the population parameter.

Moreover, if we extract 10 different samples from the same population, we will get 10 different measures.

Statisticians have solved the problem by adjusting the algebraic formulas for many statistics to reflect this issue. Therefore, we will explore both population and sample formulas, as they are both used.

**The Mean, Median and Mode**

You must be asking yourself why there are unique formulas for the **mean**, **median** and **mode**. Well, actually, the **sample** **mean** is the average of the sample data points, while the **population** **mean** is the average of the population data points. As you can see in the picture below, there are two different formulas, but technically, they are computed in the same way.

After this short clarification, it’s time to get onto **variance**.

## Variance Formula: Sample Variance and Population Variance

**Variance** measures the dispersion of a set of data points around their **mean** value.

**Population variance**, denoted by *sigma* squared, is equal to the sum of squared differences between the observed values and the **population** **mean**, divided by the total number of observations.

**Sample variance**, on the other hand, is denoted by s squared and is equal to the sum of squared differences between observed **sample** values and the **sample** **mean**, divided by the number of sample observations minus 1.

**A Closer Look at the Formula for Population Variance**

When you are getting acquainted with statistics, it is hard to grasp everything right away. Therefore, let’s stop for a second to examine the formula for the population and try to clarify its meaning. The main part of the formula is its *numerator*, so that’s what we want to comprehend.

The sum of differences between the observations and the **mean**, squared. So, this means that the closer a number is to the **mean**, the lower the result we obtain will be. And the further away from the **mean** it lies, the larger this difference.

**Why do we Elevate to the Second Degree**

Squaring the differences has two main purposes.

- First, by squaring the numbers, we always get non-negative computations. Without going too deep into the mathematics of it, it is intuitive that dispersion cannot be negative. Dispersion is about distance and
*distance cannot be negative*.

If, on the other hand, we calculate the difference and do not elevate to the second degree, we would obtain both positive and negative values that, when summed, would cancel out, leaving us with no information about the dispersion.

- Second, squaring amplifies the effect of large differences. For example, if the
**mean**is 0 and you have an observation of 100, the squared spread is 10,000!

**Putting the Population Formula to Use**

Alright, enough dry theory. It is time for a practical example. We have a population of five observations – 1, 2, 3, 4 and 5. Let’s find its **variance**.

We start by calculating the **mean**: (1 + 2 + 3 + 4 + 5) / 5 = 3.

Then we apply the formula which we just discussed:

((1 – 3)^{2} + (2 – 3)^{2}+ (3 – 3)^{2} + (4 – 3)^{2 }+ (5 – 3)^{2}) / 5.

When we do the math, we get 2. So, the **population variance** of the data set is 2.

**Calculating the Sample Variance**

But what about the **sample variance**? This would only be suitable if we were told that these five observations were a sample drawn from a population. So, let’s imagine that’s the case. The **sample** **mean** is once again 3. The numerator is the same, but the denominator is going to be 4, instead of 5.

This gives us a **sample variance** of 2.5.

### Why the Results are not the Same

To conclude the **variance** topic, we should interpret the result. Why is the **sample variance** bigger than the **population variance**? In the first case, we knew the population. That is, we had all the data and we calculated the **variance**. In the second case, we were told that 1, 2, 3, 4 and 5 was a sample, drawn from a bigger population.

**The Population of the Sample**

Imagine that the population of the sample were the following 9 numbers: 1, 1, 1, 2, 3, 4, 5, 5 and 5.

Clearly, the numbers are the same, but there is a concentration around the two extremes of the data set – 1 and 5. The **variance** of this population is 2.96.

So, our **sample variance** has rightfully corrected upwards in order to reflect the higher **potential** variability. This is the reason why there are different formulas for sample and population data.

## Standard Deviation Formula: Sample Standard Deviation and Population Standard Deviation

While **variance** is a common measure of data dispersion, in most cases the figure you will obtain is pretty large. Moreover, it is hard to compare because the unit of measurement is squared. The easy fix is to calculate its square root and obtain a statistic known as **standard deviation**.

In most analyses, **standard deviation** is much more meaningful than **variance**.

**The Formulas**

Similar to the **variance** there is also **population** and **sample standard deviation**. The formulas are: the square root of the **population** **variance** and square root of the **sample** **variance** respectively. I believe there is no need for an example of the calculation. Anyone with a calculator in their hands will be able to do the job.

**The Coefficient of Variation (CV)**

The last measure which we will introduce is the **coefficient of variation**. It is equal to the **standard deviation**, divided by the **mean**.

Another name for the term is **relative standard deviation**. This is an easy way to remember its formula – it is simply the **standard deviation** relative to the **mean**.

As you probably guessed, there is a population and sample formula once again.

**Why We Need the Coefficient of Variation**

So, **standard deviation** is the most common measure of variability for a single data set. But why do we need yet another measure such as the **coefficient of variation**? Well, comparing the **standard deviations** of two different data sets is meaningless, but comparing **coefficients of variation** is not.

Aristotle once said:

“Tell me, I’ll forget. Show me, I’ll remember. Involve me, I’ll understand.”

## Examples of Comparing Standard Deviations

To make sure you remember, here’s an example of a comparison between **standard deviations**. Let’s take the prices of pizza at 10 different places in New York. As you can see in the picture below, they range from 1 to 11 dollars.

Now, imagine that you only have Mexican pesos. To you, the prices will look more like 18.81 pesos to 206.91 pesos, given the exchange rate of 18.81 pesos for one dollar.

Let’s combine our knowledge so far and find the **standard deviations** and **coefficients of variation** of these two data sets.

**Sample or Population Data**

- First, we have to see if this is a sample or a population. Are there only 11 restaurants in New York? Of course not. This is obviously a sample drawn from all the restaurants in the city. Then we have to use the formulas for sample
*measures of variability*.

**Finding the Mean**

- Second, we have to find the
**mean**. The**mean**in dollars is equal to 5.5 and the**mean**in pesos to 103.46.

**Calculating the Sample Variance and the Standard Deviation**

- The third step of the process is finding the
**sample variance**. Following the formula that we went over earlier, we can obtain 10.72 dollars squared and 3793.69 pesos squared. - The respective sample
**standard deviations**are 3.27 dollars and 61.59 pesos, as shown in the picture below.

**A Few Observations**

Let’s make a couple of observations.

First, **variance** gives results in squared units, while **standard deviation** in original units, as shown below.

This is the main reason why professionals prefer to use **standard deviation** as the main measure of variability. It is directly interpretable. Squared dollars mean nothing, even in the field of statistics.

Second, we got **standard deviations** of 3.27 and 61.59 for the same pizza at the same 11 restaurants in New York City. However, this seems wrong. Let’s make it right by using our last tool – **the coefficient of variation**.

**The Advantage of the Coefficient of Variation**

We can divide the **standard deviations** by the respective **means**. As you can see in the picture below, we get the two **coefficients of variation**.

The result is the same – 0.60.

**Important:** Notice that it is not dollars, pesos, dollars squared or pesos squared. It is just 0.60.

This shows us the great advantage that the **coefficient of variation** gives us. Now, we can confidently say that the two data sets have the same variability, which was what we expected beforehand.

In the picture above, you can see the main advantages of the **coefficient of variation**.

The article first appeared on: https://365datascience.com/coefficient-variation-variance-standard-deviation/