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How To Calculate Standard Deviation

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First, you need to determine the mean. The mean of a list of numbers is the sum of those numbers divided by the quantity of items in the list (read: add all the numbers up and divide by how many there are).

Then, subtract the mean from every number to get the list of deviations. Create a list of these numbers. It's OK to get negative numbers here. Next, square the resulting list of numbers (read: multiply them with themselves).

Add up all of the resulting squares to get their total sum. Divide your result by one less than the number of items in the list.

To get the standard deviation, just take the square root of the resulting number

I know this sounds confusing, but just check out this example:

your list of numbers: 1, 3, 4, 6, 9, 19

mean: (1+3+4+6+9+19) / 6 = 42 / 6 = 7

list of deviations: -6, -4, -3, -1, 2, 12

squares of deviations: 36, 16, 9, 1, 4, 144

sum of deviations: 36+16+9+1+4+144 = 210

divided by one less than the number of items in the list: 210 / 5 = 42

square root of this number: square root (42) = about 6.48

Comments

PRINCE on February 26, 2014:

Thanks to my virtual tutor, simplicity is the ultimate sophistication

rebecca on February 14, 2014:

perfect explanation. thanks

hmm, on July 30, 2013:

divide by n or n-1?

rj on July 28, 2013:

thanks fr the help

Krishley on July 22, 2013:

Calculating standard deviation is not really difficult but can get quite complex when you have lots of numbers. I'm currently preparing for my A-Level statistics paper and i use this calculator to check my answers : http://standard-deviation-calculator.en.softonic.c...

Quintana on March 11, 2013:

If you calculate the standard deviation by hand, you will likely end up having to use a calculator anyway for the last step in computing the square root. So why not just use a calculator for the whole thing?

http://www.wolframalpha.com/input/?i=population+st...

http://www.had2know.com/academics/variance-standar...

or

http://www.calculatewhat.com/math/statistics/varia...

At least you can avoid computational errors. The only errors you might make are in entering the data. While it's good to know the general method and formula for standard deviation, I don't think calculating it by hand is practical or wise for a set with more than 10 numbers. You can also use something like Excel or Calc (open source spreadsheet program that comes with Open Office.)

sarDeemsgen on March 08, 2013:

We helpful to obtain at the top of living however as of late We have developed a new opposition.

Shannon on March 05, 2013:

THANK YOU! This has been the first time in over a year I have been able to understand this!!

sigmund on February 25, 2013:

very helpful. thanks.

bastillandude on February 08, 2013:

how could i compute the standard deviation of this data

no. of respondents

survey 3 - Very 2 - Satis 1 - Poor

question good factory

1 40 2 0

2 38 2 2

3 37 5 0

.

.

.

.

i need asap

thanks

1.

Jack Knowell on February 07, 2013:

Its interesting wish it was explained with some graphics and more examples as it was explained on http://www.jpfis.com

Goldenserenity on January 15, 2013:

Thank you! I read the chapter on this in my math book and I didn't have a clue. I went through your post once and I understand it. You are awesome!!

nkiran on January 14, 2013:

I don't understand why to divide the sum of squares by "by one less than the number of items in the list" and not by "number of items in the list". I find the latter logical.

Ramin on December 16, 2012:

Thanks a lot saved me the test :D

Trevor on November 27, 2012:

agabrans, it is 14.2640

hanaa ibrahim on November 15, 2012:

i have a question says : if the standard deviation of the numbers X1, X2, .....,Xn is 4 , find the standard deviation of the numbers : X1 -2, X2-2,.....,Xn-2

InterestCaptured on November 04, 2012:

This is perfectly explained, thanks a lot

roboporno on October 28, 2012:

every calculator i go on says when i square a negative it comes with a bigger negative, help?

thegoofyblue on October 09, 2012:

omg this is the best thing ever! THANK YOUUUUUU

Vincent on September 29, 2012:

YEs i finally knows now

Andrew Laughton on September 25, 2012:

Thanks so much for the explanation!

agabrans on September 20, 2012:

I'm trying to figure out this problem from school. Your explanation of Standard Deviation was awesome. I seriously understand it now! Is it possible to figure out this problem? I don't see the mean or list of numbers that I THINK I need. Thanks!

You have collected 11 water samples. The variance of the sample is 47.3084. What is the standard deviation of the sample (4 decimal places)?

Sal Stephens on September 17, 2012:

You rock Sonia! Great explanation!!!!!

kabirdes on September 12, 2012:

The explanation was Awsm sonia :)

Can u plz explain where do we use it..?

Lucy on September 05, 2012:

Thank you! I completely understand now. Based on the comments, it seems like lots of people are doing this in High School maths. I wonder why we need it in 6th grade maths? Also, is there a real-world use for standard deviation?

ben on September 04, 2012:

no thats wrong you don't divided by one less than the number of items in the list you just divide by the same number of numbers in the list

that the right way.

lauren on August 20, 2012:

Thanks so much! that was easy to understand

MiddleSchooler on July 25, 2012:

I find it interesting everyone seems to be in high school or going into college. I'm in 6th grade and learning this. Lol, too many big words from other websites. I CAN FINALLY GET MY HOMEWORK DONE! Thank you.

Lee on June 19, 2012:

I would like you help me about my final report free test. I don't know how can I do students deviation or standard deviation below:

360 all of students for free test in final report

108 group of sample in final report

score of group sample for free test there are 15 questions.

list of students score:

22 students got 1 score

23 students got 2 score

30 students got 5 score

10 students got 7 score

10 students got 9 score

10 students got 13 score

3 students got 15 score

Please use determine the standard deviation by using the function bellow:

SD= ?(( N?fx^2 - (?f?x)?^2) )/?( 2@N))

FOSTOM on June 12, 2012:

MIND WIDE OPEN!

GREAT JOB......

sean on June 11, 2012:

cheers, very easy to understand

michaelheemson on June 10, 2012:

hi there tommy if your still knoking around i think this is the contact

and some info , they have a deal on at the mo , mention michael hiltams put you on

RESPECT BOY on June 05, 2012:

PLS CAN YOU ARRANGE EVERYTHING IN ROLLS AND COLUME,AND SHOW THE FORMULAR DETAIL?

Dawnysha on May 31, 2012:

Why do you divide by 5 and not 6? (at the end)

Tommy on May 24, 2012:

Thank you, OMG

Nunu Norbert on May 22, 2012:

u just saved me from failing my stats examx thanx a lot

Afshan on May 19, 2012:

awesome!!! you're the man!

Math Gal on May 16, 2012:

this is incorrect!

you do not divide by one number less than the number of items in the list, you divide by the number of items in the list.

the correct standard deviation for this problem is approximately 5.916.

Daina on May 15, 2012:

thanks 4 such a great effort itx really help 4 d readerx

Jonathan on April 29, 2012:

Is 42 the variance in this case?

Apples on April 28, 2012:

You should verify that this is the formula that your class is using though, another common version divides the second part by just the "number of things" instead of "number of things - 1".

This gives us 1890/10=189

root(189) = 13.something, probably around 13.7.

Apples on April 28, 2012:

55 53 80 53 56 90 70 78 80 85

Mean = 700/10=70

difs = 15 17 10 17 14 20 0 8 10 15

squares of difs = 225 289 100 289 198 400 0 64 100 225

sum of squares of difs = 1890

sum of squares of difs/(number of things-1) = 1890/9 = 210

standard deviation = root(210) = slightly less than 14.5

Do not hold me to mathematical accuracy, I am not currently with a calculator (hence the approximation of the final answer)

Olivia on April 28, 2012:

Pls i have this assignment help me. 55+53+80+53+56+90+70+78+80+85 in a standard deviation

des donnelly from Co Tyrone.... on April 23, 2012:

mine is better :-) https://owlcation.com/stem/deviation

thanks this kinda helped on April 19, 2012:

this was pretty good

Math is terrible on April 16, 2012:

Thank's Sonia! God bless you! It helped a lot when i did my math homework!

Anil Singh on March 21, 2012:

Thanks Sonia. Now i can say its simple to calculate SDEV.

geezel on March 20, 2012:

thank you so much.it really helps me.

ellie on March 12, 2012:

Thanks!!It was easy to understand because you broke it down step by step THANKS for the help!!

yuva on February 27, 2012:

very interesting and nice example thank u

Garry on February 23, 2012:

Superb. Nice and easy to follow, thank you

ganesh on February 20, 2012:

X f dx dx2 fdx fdx2

40 12 10 100 120 1200

Here every thing i understand 3rd raw dx i don't understand any body help for this...

EE on February 16, 2012:

Thanks

Lene on February 14, 2012:

that was awesome.

short,sweet,to the point. AND easy to understand.

Sarah N on February 13, 2012:

The only reason I will be passing my stats test tonight is because of you! Thank you so much. I wish my teacher could break it down like this :/

Jye on February 06, 2012:

Thanks so much. Helping out my year 12 math.

I do your mum on January 31, 2012:

No one will me read my comment;)

Devon on January 25, 2012:

Stupid

SUCK MY EAR on January 23, 2012:

THIS IE A PIE AND IT IS LIKE A LLAMA WHO EATS YOUR UNDERWEAR WHEN YOU ARE NOT LOOKING

In otherwords, this is EPIC POTATO :)

Hetty on January 23, 2012:

Thank you for this!

Helped me with applied science coursework!

Buddy on January 19, 2012:

I'm sure it's been mentioned but I didn't read every post. This is the method for finding a sample standard deviation. If you're finding a population standard deviation you divide by the number of observations, not n-1.

SATHISH on January 08, 2012:

SIMPLY EXCELLENT

hehehehe on January 04, 2012:

THANK YOU SO MUCH!!!!!!!!

The example school gave was kind of confusing, and when I was reviewing I couldn't understand.

j on January 02, 2012:

can the standard deviation and the range of a set of data ever equal??

easin on December 21, 2011:

thank u so so so so so so much

Ikram Ahmad on December 19, 2011:

Thanks

Max on December 09, 2011:

Thank you very much.

Maria on December 08, 2011:

THANK YOU SO much! I have an exam tomorrow and ALL semester I NEVER got it until now :)

gangsta on December 02, 2011:

omg its thugtastic yo

Dabee on November 12, 2011:

I need some one to help me how to caculate SD when i have 2 value in a sample

For exemple:

A=3.3 and 3.5

B=3.2 and 3.4

C=3.4 and 3.4

D=3.3 and 3.1

If some one know please help me and send me the formula on my e-mail address as dabee_chiakua@hotmail.com

Thank you very much

Musa simon on November 07, 2011:

Interesting!!!

Prasad Ajinkya on November 07, 2011:

We have written a post on how Standard Deviation is applicable in Finance, especially when doing an analysis in Excel. Do check it out - http://www.edupristine.com/blog/what-is-standard-d...

Comments and feedback invited!

Anon on November 05, 2011:

well there is a catch. we only divide by one less than the numbers in the list only if our list is a sample. we divide by the exact number of items in our list if our list is the population, not a sample

Lee on October 31, 2011:

Good Job, Thumbs up from me, needed clarification i was doing this right, and you did a great job of it

AngaNaa on October 25, 2011:

OMG!!!! Thank you SO much. This was very helpful and now I can work on mastering this technique for my Operations Management Test that is coming up. I had forgotten how to do standard deviations and lost my notes from my statistics class to refer back to it! What a life saver!!

kaye on October 25, 2011:

Thanks for that explanation, you make it sound so easy, i am currently doing an Ass. Deg. and this statistics is killing me, but u certainly cleared that up for me, by the way after doing all of that what is the next step.

emma on October 24, 2011:

wow! awesome! you're AMAZING

Jeff on October 23, 2011:

The description was simple and the example you provided helped tie it all together. Great job.

Esther Ripplinger on October 20, 2011:

I agree with the other one who said,"This is exactly what I needed!" I misinterpreted the formula. It left me wondering, but walking me through it is exactly what I needed. Thank you!

zainab on October 16, 2011:

That is exactly what I was looking for!!

Thanks a lot :)

Amanda on October 09, 2011:

I totally prayed to understand how to calculate standard deviations and found this site! AMAZING example!!! I could not understand how to calculate standard deviations until reading this right now! Love it, thanks a bunch! Jesus is AWESOME!

irene on October 09, 2011:

I can my exam comfortably thank you Sonia

Samuel on October 03, 2011:

Thanks a lot. i passed my test thanks to you.

Deedee on September 30, 2011:

That formula had me soooo confused but you made it so simple! Thanks so much!

Melissa on September 28, 2011:

Just wanted to add to the long list of THANK YOUS! Doing a bloody stats unit :)

cin on September 25, 2011:

it's very clear thank you sooo much!

Andrea Noisternig on September 23, 2011:

Oh my god, soo good explained! If only all math teachers could be that detailed.

THANKS

mars on September 20, 2011:

wow, lot of comments on here.

Statlover on September 08, 2011:

that method is epic so cool

it is sorta different in school. This is nice and easy.

Magarac on September 04, 2011:

Wow Sonja, you rock. YOU should be teaching my Statistics course, not the doofus who's getting paid by my university to do it! :-)

Thank you!

Sunday John Bosco on September 04, 2011:

I have got the point but why do you "divided by one less than the number of items in the list" I am confused there.

madds on September 03, 2011:

omg thank you so much!! this is great!

:) on September 01, 2011:

Thanks. Helped out tons

Dennis on August 27, 2011:

Hey thnx I was very impress with d way u explained SD

ELVIS SHIVHALO on August 16, 2011:

JA WELL,THAX 4 HELPIN ME,BUT HEY THERE IS STILL SOME CONFUSION IN THIS,MAYBE SOMEBODY CAN TRY ANOTHER METHOD

Adham on July 22, 2011:

Thanks for the easy to understand example; I just have one wonder which I am clear about:

Why we need to divided by one less than the number of items in the list: 210 / 5 = 42??? Is it just the role; to always divide the total of all deviations by one less than the total/sum of all items in the list? Please clarify.

Thanks a lot,

Adham

jesereth on July 12, 2011:

It was simple yet helpful information. Thanks!

lenieke on June 15, 2011:

Hey great, many difficult explanation on the web, but this example makes sense :)

Qasim on June 08, 2011:

Sonia! Thank you...

C2W on June 03, 2011:

For me,it is the easiest and the best tanx...Good work

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