# Demystifying Standard Deviation: A Step-by-Step Guide for Beginners

Standard deviation is a statistic that measures how dispersed or spread out a set of data is from the mean. While it may seem intimidating at first calculating standard deviation is actually quite straightforward if you follow a simple 6 step process. In this beginner’s guide, I’ll walk you through how to calculate standard deviation by hand and explain what each step means in simple terms.

## Why Calculate Standard Deviation?

Before jumping into the calculation it’s helpful to understand why standard deviation is an important statistic

• It provides a numerical value of the amount of variation or dispersion in a data set. A low standard deviation means the data points tend to be very close to the mean, while a high standard deviation indicates the data is spread out over a wider range of values

• It allows you to compare the degree of variation between different data sets. For example, you can tell whose exam scores vary more between Class A and Class B by comparing their standard deviations.

• It is used in statistics to measure confidence in predictions made from sample data. A lower standard deviation means you can be more certain about conclusions drawn from the data.

• Many statistical tests use standard deviation in their formulas, like the t-tests and z-scores. So calculating standard deviation is a prerequisite for these techniques.

Now that you know why it’s useful, let’s go through how to calculate it step-by-step.

## Step 1: Find the Mean

The first step is to find the mean (average) of all the data points in your sample. Add up all the data values and divide by the total number of values.

For example, if your data set is:

15, 20, 45, 50, 75

The mean would be:

(15 + 20 + 45 + 50 + 75) / 5 = 41

Make sure you’re using the population mean if your data includes all possible values, and sample mean if your data is a subset of the total possible values.

## Step 2: Subtract the Mean from Each Score

Once you’ve found the mean, take each data value and subtract the mean from it.

Continuing the example above, subtracting the mean of 41 from each score gives you:

15 – 41 = -26
20 – 41 = -21
45 – 41 = 4
50 – 41 = 9
75 – 41 = 34

This gives you the deviation of each data point from the mean.

## Step 3: Square Each Deviation

For each deviation value calculated in Step 2, square it.

Squaring removes negative signs and gives you the squared deviation value. For our example, squaring each deviation gives:

(-26)^2 = 676
(-21)^2 = 441
4^2 = 16
9^2 = 81
34^2 = 1156

## Step 4: Add the Squared Deviations

Sum all the squared deviation values calculated in Step 3.

For our example data set, the sum of squared deviations is:

676 + 441 + 16 + 81 + 1156 = 2370

This gives you the sum of squared deviations from the mean.

## Step 5: Divide the Sum by n-1

Take the sum of squared deviations and divide it by n-1, where n is the number of data points.

For our example with 5 data points, divide the sum by 5-1 = 4.

2370 / 4 = 592.5

This division normalizes the value based on the sample size, giving you the variance.

## Step 6: Take the Square Root

As the final step, take the square root of the result from Step 5 to calculate the standard deviation.

For our example data set, the standard deviation is:

sqrt(592.5) = 24.3

The standard deviation of the sample is 24.3. The higher the standard deviation, the more spread out the data.

And that’s it! Those 6 steps are all you need to manually calculate the standard deviation. Let’s do a few more examples to get the hang of it.

## Example 1: Calculating Standard Deviation

Given the data set:

12, 7, 13, 16, 14

Let’s find the standard deviation:

Step 1) Mean = (12 + 7 + 13 + 16 + 14) / 5 = 12

Step 2) Deviations from mean:

12 – 12 = 0
7 – 12 = -5
13 – 12 = 1
16 – 12 = 4
14 – 12 = 2

Step 3) Square the deviations:

0^2 = 0
(-5)^2 = 25
1^2 = 1
4^2 = 16
2^2 = 4

Step 4) Sum squared deviations:

0 + 25 + 1 + 16 + 4 = 46

Step 5) Divide sum by n-1 = 46 / (5-1) = 46 / 4 = 11.5

Step 6) Take square root:

sqrt(11.5) = 3.39

The standard deviation is 3.39.

## Example 2: Calculating Standard Deviation

For the data set:

28, 26, 25, 29, 32

Step 1) Mean = (28 + 26 + 25 + 29 + 32) / 5 = 28

Step 2) Deviations:

28 – 28 = 0
26 – 28 = -2
25 – 28 = -3
29 – 28 = 1
32 – 28 = 4

Step 3) Square deviations:

0^2 = 0
(-2)^2 = 4
(-3)^2 = 9
1^2 = 1
4^2 = 16

Step 4) Sum of squares = 0 + 4 + 9 + 1 + 16 = 30

Step 5) 30 / (5-1) = 30 / 4 = 7.5

Step 6) sqrt(7.5) = 2.74

Standard deviation = 2.74

I hope these step-by-step examples help explain how to manually calculate standard deviation. While the formulas look intimidating at first glance, breaking it down into simple steps makes it very manageable. With a bit of practice, you’ll be able to calculate standard deviation for any data set in a few minutes!

## When to Use Sample vs Population Standard Deviation

An important distinction to understand is the difference between sample and population standard deviation:

• Sample standard deviation is calculated when your data is a subset of the entire population. For sample data, you divide the sum of squared deviations by n-1 in Step 5.

• Population standard deviation is calculated when your data includes the entire population. For population data, you divide the sum of squares by n in Step 5 rather than n-1.

In most real-world scenarios, you’ll be working with sample data rather than entire populations. That’s why the examples above use n-1 in the formula. Just remember this key difference when deciding which version of standard deviation to use.

## Using Technology to Calculate Standard Deviation

While it’s good to know how to calculate standard deviation manually, technology makes finding it much easier. Here are some ways to find standard deviation using software tools:

• Excel – Use the STDEV.P() function for population standard deviation, or STDEV.S() for sample standard deviation.

• Google Sheets – Use STDEVP() or STDEV() functions.

• Python – Use numpy.std() for sample standard deviation or numpy.std(ddof=0) for population.

• R – Use sd() function. The default is sample but set ddof=0 for population.

• MATLAB – Use std() for sample deviation or std(0) for population.

• Calculators – Many scientific and graphing calculators have built-in standard deviation functions. Just enter your data set and it does the calculations automatically.

Leveraging technology allows you to find standard deviations quickly and accurately without doing the manual work. So take advantage of spreadsheet programs, coding languages, and calculator functions when possible.

## Real World Examples of Using Standard Deviation

Now that you understand the calculation, let’s look at some examples of how standard deviation is useful in real-world situations:

• Education – Teachers use standard deviation to compare test score distributions and performance across classes. A class with a higher standard deviation has greater variance in scores.

• Finance – Analysts use the standard deviation of stock returns to quantify investment risk. Stocks with higher standard deviations have more price volatility.

• Science – Researchers report standard deviations alongside means to show the amount of variation in experimental measurements and samples. A higher standard deviation indicates less precision.

• Manufacturing – Engineers use standard deviation to monitor production quality. A high standard deviation in product dimensions would indicate poor consistency in the production process.

• Psychology – Therapists use questionnaires that produce standard deviations to measure traits and symptoms in patients. Higher standard deviations indicate more variability in responses.

As you can see, standard deviation has many useful applications across different industries. Understanding how to calculate and interpret it

## Key Properties of Standard Deviation

One key property of standard deviation is additivity, which means that the standard deviation of a sum of random variables. This property allows researchers to accurately quantify the variability of aggregated data and make meaningful comparisons between different groups or populations as opposed to only analyzing single points of data.

Another property of standard deviation is scale invariance. This is particularly useful in comparing the variability of datasets with different units of measurement. For example, if one dataset is measured in inches and another in centimeters, their standard deviations can still be compared directly without needing to convert units.

Last, standard deviation has properties of symmetry and non-negativity. This means a standard deviation is always positive and symmetrically distributed around the mean. This symmetry property implies that deviations above the mean are balanced by deviations below the mean, resulting in a total balance of the entire data set. The property of always being positive means a standard deviation has a higher degree of comparability when looking at standard deviations across data sets.

## What Does a High Standard Deviation Mean?

A large standard deviation indicates that there is a lot of variance in the observed data around the mean. This indicates that the data observed is quite spread out. A small or low standard deviation would indicate instead that much of the data observed is clustered tightly around the mean.