Sample Standard Deviation Formula

Sample Standard Deviation Formula
Go back to  'Data'

Joel was browsing through the internet while working on his assignments.

He was given an assignment to calculate the increase in the population of the city he lived in. 

He chose to use the concept of standard deviation to find the increase in population. 

Standard deviation measures the spread of the values in a given set of data from the mean or average.

Standard deviation finds its applications in finance, opinion polling, and so on.

There are two types of standard deviation. They are:

  • Population Standard Deviation
  • Sample Standard Deviation

Sometimes the data for which we calculate standard deviation is a whole lot or the full population.

In such cases, we use population standard deviation and in some cases, we would be interested in knowing the standard deviation for a defined set of data or a sample, then we opt for the sample standard deviation.

Population and Sample standard deviation

In this mini-lesson, we will discuss about standard deviation formula, standard deviation meaning, standard deviation, and variance.

Lesson Plan

Try out the sample standard deviation calculator given below!

Enter the sample values separated by commas and click the "Calculate" button to know the sample standard deviation value. 


What Is Standard Deviation?

Standard deviation meaning: Standard deviation is the measure to find out how much a value in a data set has deviated from the mean value.

We can show this using a bell curve, where the center portion holds the mean or average value. 

The values to the left and right of the bell curve are the dispersions from the mean.

The values of mean 0 and standard deviation 1 means that the mean values are not too high or low.

Mean - Bell curve


What Is Sample Standard Deviation Formula?

\(\begin{align}s\:=\:\sqrt{\dfrac{\sum(X-\bar{X})^2}{(n-1)}}\end{align}\)

where, 

\(\begin{align}s\:=\: sample\:standard\: deviation\end{align}\)

\(\begin{align}X\:=\: data\: item\:in\: the\: list\end{align}\)

\(\begin{align}\bar{X}\:=\:Mean\end{align}\)

\(\begin{align}n\:=\:Number\:of\:items\:in\:the\:list\end{align}\)


How to Use Standard Deviation Formula?

Follow the steps below to use the standard deviation formula.

Step 1: Calculate the mean of the data. (\(\begin{align}\bar{X}\:=\:Mean\end{align}\))

Step 2:  Subtract the mean value from each value in the given set of data. These values are the deviations. It can be either positive or negative.

Step 3: Find the squares of all the deviation values.

Step 4: Find the sum of all the squared deviation values.

Step 5: Divide the result with (total number of values in the data set - 1). The obtained value is the variance.

Step 6: Find the square root of the variance. This is called the standard deviation.

 
important notes to remember
Important Notes
  1. The sample standard deviation cannot be negative.
  2. Sample standard deviation is a measure of the spread of data that is taken from a few samples.
  3. Standard deviation is obtained by taking the square root of variance.
  4. A bell curve is also called a normal distribution curve and it shows how far the standard deviation is from the mean value.

Solved Examples

Example 1

 

 

Find the mean of the following data sample: 3,5,7,9,11.

Solution

\(\begin{align}Mean\:=\:\dfrac{3+5+7+9+11}{5}\end{align}\)

\(\begin{align}Mean\:=\:7\end{align}\)

Mean\(\begin{align}\:=\:7\end{align}\)
Example 2

 

 

Calculate the variance of the following data set.

5,12,13,14,16

Solution

Let us find out the value of the mean.

\(\begin{align}Mean\:=\:\dfrac{5+12+13+14+16}{5}\end{align}\)

\(\begin{align}Mean\:=\:\dfrac{60}{5}\end{align}\)

\(\begin{align}Mean\:=\:12\end{align}\)

Now let us subtract each value from the mean. 

\(\begin{align}5-12\:=\:-7\end{align}\)

\(\begin{align}12-12\:=\:0\end{align}\)

\(\begin{align}13-12\:=\:1\end{align}\)

\(\begin{align}14-12\:=\:2\end{align}\)

\(\begin{align}16-12\:=\:4\end{align}\)

Now square all the values.

\(\begin{align}-7^2\:=\:49\end{align}\)

\(\begin{align}0^2\:=\:0\end{align}\)

\(\begin{align}1^2\:=\:1\end{align}\)

\(\begin{align}2^2\:=\:4\end{align}\)

\(\begin{align}4^2\:=\:16\end{align}\)

Now sum up all the values.

\(\begin{align}49+0+1+4+16\end{align}\)

=\(\begin{align}70\end{align}\)

Divide the result by (total number of data set values - 1). 

Here, the number of values in the data set = 5.

On dividing we get,

\(\begin{align}\dfrac{70}{5-1}\end{align}\)

\(\begin{align}\dfrac{70}{4}\end{align}\)

\(\begin{align}=17.5\end{align}\)

\(\begin{align}Variance\:=\:17.5\end{align}\)
Example 3

 

 

Bob owns a book shop.

He wants to find out the genre of books that are preferred by most people.

Comics 20
Fiction 35
Adventure 25
Self-Help 15

Solution

The formula to find the sample standard deviation is as follows:

\(\begin{align}s\:=\:\sqrt{\dfrac{\sum(X-\bar{X})^2}{(n-1)}}\end{align}\)

where, 

\(\begin{align}s\:=\: sample\:standard\: deviation\end{align}\)

\(\begin{align}X\:=\: data\: item\:in\: the\: list\end{align}\)

\(\begin{align}\bar{X}\:=\:Mean\end{align}\)

\(\begin{align}n\:=\:Number\:of\:items\:in\:the\:list\end{align}\)

The sample data set is \(\begin{align}20,35,25,15\end{align}\)

Step 1:\(\begin{align}Mean\:=\:\dfrac{20+35+25+15}{4}\end{align}\)

                                          \(\begin{align}Mean\:=\:23.75\end{align}\)

   

Step 2: Subtract each value from the mean.

Value Deviation
\(\begin{align}20\end{align}\) \(\begin{align}20 - 23.75\: =\:-3.75\end{align}\)
\(\begin{align}35\end{align}\) \(\begin{align}35 - 23.75\:=\:11.25\end{align}\)
\(\begin{align}25\end{align}\) \(\begin{align}25 - 23.75\:=\:1.25\end{align}\)
\(\begin{align}15\end{align}\) \(\begin{align}15 - 23.75\:=\:8.75\end{align}\)

Step 3: Find the square of each value of deviation.

\(\begin{align}\:(-3.75)^2\: =\: 14.0625 \end{align}\)

\(\begin{align}\:(11.25)^2\: =\: 126.5625 \end{align}\)

\(\begin{align}\:(1.25)^2\: =\: 1.5625 \end{align}\)

\(\begin{align}\:(8.75)^2\: =\: 76.5625 \end{align}\)

Step 4: Sum up all the squared deviation values.

\(\begin{align}14.0625+126.5625+1.5625+76.5625\end{align}\)

\(\begin{align}218.75\end{align}\)

Step 5: Divide the result by total number of data set values - 1

\(\begin{align}\dfrac{218.75}{4-1}\end{align}\)

\(\begin{align}\dfrac{218.75}{3}\end{align}\)

\(\begin{align}\dfrac{219}{3}\end{align}\)

\(\begin{align} = 73\end{align}\) 

Step 6: Find the square root of variance. 

\(\begin{align}\sqrt{73}\end{align}\)

\(\begin{align}8.54\end{align}\)

Sample standard deviation is \(\begin{align}8.54\end{align}\)           

Sample standard deviation is \(\begin{align}8.54\end{align}\)           

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
Challenge your math skills
Challenging Questions
  1. Which of the following data sets has the second largest sample standard deviation?

    A = {First five whole numbers} 
    B = {First five natural numbers}
    C = {First five even numbers}
    D = {First five odd numbers}
Let's Summarize

The mini-lesson targeted the concept of sample standard deviation. By now you would have understood the concept of sample standard deviation how to calculate it. Going through the solved examples and interactive questions will give you more clarity on calculating the sample standard deviation. 

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. 


Frequently Asked Questions(FAQs)

1. How do you find the sample standard deviation on a calculator?

Find the mean of the values of the data set. Next, subtract each data value from the mean and square the result values. Add them all and divide the result with 1 less than the number of data values. Finally, find the square root of variance.

2. How do you know the standard deviation is high or low?

If the standard deviation values are closer to the mean, then we can say that the standard deviation is low.

On the contrary, if the standard deviation values are far away from the mean or spread out from the mean, we can say that the standard deviation value is high.

3. What does a standard deviation of 1 mean?

A standard deviation of 1 means the data values are not too high or low. They are neither widely spread or very close to the mean.

  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus