If you want to know how to find the standard deviation, you’re in for a treat! In this article, we’ll walk you through everything you need to know about standard deviation and how to find it. As you know, these calculations can get a bit complex when applied in real-life applications. For that reason, in the real world, statisticians don’t calculate them by hand in the majority of cases. Also, it goes without saying that finding the standard deviations by hand takes a much longer time. That’s why they’re mostly calculated using excel spreadsheets and other computer programs to do the maths. However, by understanding how standard deviation is measured, you’ll have a much better understanding of how it works and when to apply it. So without further ado, let’s dive in!
How to Find the Standard Deviation
Now that you know more about the standard deviation formula, it’s time to break these steps down with a simple example. For example, let’s find the standard deviation of the following numbers: 3, 7, 9, 2, 4.
Step #1: Understand Standard Deviation
Before heading into the juicy stuff, you first need to understand what is the standard deviation and what it measures.
Let’s start by defining the standard deviation and then explain this definition a piece by piece. It might sound a bit confusing in the beginning, but it’ll get easier and clearer as we simplify it more and more.
The standard deviation is a statistical term that measures the dispersion and variation of a set of values when compared to its mean. Now that we cleared the rough definition out of the way, let’s explain it in a simpler way.
If you have a bunch of different values within a group, you can simply get the average value of these numbers. This can be done simply by adding all these numbers and then dividing it by the number of values included.
The standard deviation is the number that tells us how far these values are spread out from their average (mean).
A low standard deviation means that the majority of these values are actually close to the average value. On the other hand, the higher the standard deviation, the more these numbers are far spread out.
Standard deviations are represented in many forms. The most common ones are simply abbreviating it to “SD” or using the lowercase greek letter σ, which is pronounced “sigma”.
It’s also the square root of another statistical term known as the “variance”. Although variance is much simpler, it’s much less robust in practical representation. But more on that later.
Step #2: Understand Standard Deviation Mathematical Formula
To find the standard deviation, you need to understand its formula. In fact, there are different versions of the standard deviation formula. So, let’s clear that out of the way.
Here’s the general formula of standard deviation:
At first glance, this formula might look extremely complex. However, each part of it explains a simple step to find the standard deviation. Let’s explain the different part of the formula:
- σ = the value of standard deviation
- n = the total number of values in a sample
- μ = the mean value of the group (pronounced Miu)
- Σ = the sum (pronounced Sigma)
- xi = the value of each point in the data set
Don’t worry! As we break the formula down, it’ll start to slowly make sense, the previous formula explains the step by step process we’re going to do to find the standard deviation. So, here’s a quick roundup of these steps:
- Find the mean value of the given sample
- Find the distance of each value from the mean (by subtracting it) and square it
- Work the means of the values from step number 2
- Take the square root of the entire process and you have the standard deviation!
Step #3: Find the Mean (μ)
In the previous formula, the mean was represented as μ. It means the average number of all the values in our sample.
To get it, you sum all the values and divide them by their number (n). In that case, they’re 5 values.
This means that n = 5.
This means that the mean equals: (3+7+9+2+4)/5 = 25/5.
So, μ = 5.
Step #4: Find the Distance of Each Value from the Mean
This part sounded a bit tricky in the roundup. However, once you apply it, you’ll notice how simple it is. This part is represented in the formula above as (xi – μ).
So, xi stands for each individual value in the sample we started with. Again, these numbers were 3, 7, 9, 2, 4.
We apply this step to each one of them as follows:
- 3 – 5 = -2
- 7 – 5 = 2
- 9 – 5 = 4
- 2 – 5 = -3
- 4 – 5 = -1
Don’t worry about the negative sign, as it’s going to be canceled out in the following step.
Step #5: Square the Previous Values
Now that you found the distance of each number from the mean, it’s time to square these values just like the formula (xi – μ)².
Simply, work the square of each value of the previous step as follows:
- (-2)² = 4
- (2)² = 4
- (4)² = 16
- (-3)² = 9
- (-1) ² = 1
So, here are the final values of this step, 4, 4, 16, 9, 1.
Step #6: Find the Sum of the New Values
By now, we reached the part where comes into play. This figure means “the sum of the following” where i represented each one of the previous steps.
So, let’s sum the previous values: 4 + 4 + 16 + 9 + 1 = 34
Step #7: Divide by the number of samples
This part changes depending on whether you’re measuring the value of a population or a sample.
It’s represented in the equation as for population or for a sample. The choice between sample or population depends on the purpose of your study. You should also know that each one gives a different standard deviation.
In our case, the formula used is the one for samples. If you’re measuring it for a sample the formula will be , as we have 5 values.
Step #8: Take a Square Root
After working all the different parts of the equation, you’ve got yourself what is known as the “variance”. To get the standard deviation, you should take the square root of the entire value.
Here’s how the formula looks by now:
If we’re measuring standard deviation for a population, the formula will be
So, the final standard deviation here is 2.91 for the sample or 2.6 for the population. That’s all!
The Difference Between Standard Deviation and Variance
As you already know, standard deviation is the square root of variance. This means that when we reached step 4 of the previous step by step guide, we had the variance.
So why not stop there? And what does standard deviation have that makes it better at representing the variation in a group of values within a sample?
Here are some of the differences between variance and standard deviation
Standard deviation is based on the same units as the mean is, whereas the variance is expressed in different units (squared units)
Combining Standard Deviation
You can calculate a combined standard deviation of more than two groups of the same kind of values. This isn’t possible with other forms of measures, such as the variance.
Finding the Coefficient of Variation
If you want to compare the variability of two or more groups or distributions, you’ll need to use the coefficient of variation.
The coefficient of variation is based on standard deviation, as it’s the mean value of all the standard deviations involved.
Another benefit of standard deviation over variance is that standard deviation is less affected by the fluctuation of the samples. On the other hand, the squared value of variance is greatly affected by small changes in values.
Used For Other Statistical Measures
In addition to the coefficient of variation, there are other statistical works that are based on standard deviation, such as measuring skewness and correlations.
The Drawbacks of Standard Deviation
Despite all its merits, the standard deviation has some drawbacks. Here are some of them:
- Standard deviation always assumes a pattern of normal distribution.
- It doesn’t display a full range of data.
- As you could notice, it takes a bit of time to calculate by hand.
Alternatives for Calculating by Hand
With too many processes, the risk of making a mistake is too high when calculating by hand. Luckily, there are multiple methods that can do all the work for you, including:
- Using scientific calculators
- Using excel spreadsheets
- Using online calculators
From weather, sports, all the way to economics, the standard deviation is a valuable statistical measure that’s crucial for measuring different statistical aspects. With that said, you now know how to find the standard deviation in both samples and populations.
As you can see, the formula might seem a bit difficult to understand. But, once you break it down into simple steps, you’ll realize how easy it actually is!