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How To Calculate Standard Deviation

Updated on December 4, 2006

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

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      PRINCE 3 years ago

      Thanks to my virtual tutor, simplicity is the ultimate sophistication

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      rebecca 3 years ago

      perfect explanation. thanks

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      hmm, 4 years ago

      divide by n or n-1?

    • profile image

      rj 4 years ago

      thanks fr the help

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      Krishley 4 years ago

      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...

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      paxwill 4 years ago

      It just goes to show you how much emphasis search engines place on old content. This hub shows up higher in searches for standard deviation because it's over 6 years old, but there are much better s.d. tutorials on HubPages, even if they happen to be younger.

      For instance, this hub doesn't explain the difference between population and sample s.d., that is, when you divide by n and when you divide by n-1 before taking the square root. Here are hubs that offer better explanations:

      https://hubpages.com/education/Statistics-Made-Eas...

      http://calculus-geometry.hubpages.com/hub/Statisti...

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      Quintana 4 years ago

      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.)

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      sarDeemsgen 4 years ago

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

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      Shannon 4 years ago

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

    • profile image

      sigmund 4 years ago

      very helpful. thanks.

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      calculus-geometry 4 years ago

      Just noticed my statistics tutorials are getting a lot of incoming visits from this article. The comment thread here is 100 times longer than the actual hub, lol.

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      bastillandude 4 years ago

      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.

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      Jack Knowell 4 years ago

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

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      Goldenserenity 4 years ago

      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!!

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      nkiran 4 years ago

      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.

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      Ramin 4 years ago

      Thanks a lot saved me the test :D

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      Trevor 4 years ago

      agabrans, it is 14.2640

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      hanaa ibrahim 4 years ago

      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 profile image

      InterestCaptured 4 years ago

      This is perfectly explained, thanks a lot

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      roboporno 4 years ago

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

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      thegoofyblue 4 years ago

      omg this is the best thing ever! THANK YOUUUUUU

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      Vincent 4 years ago

      YEs i finally knows now

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      Andrew Laughton 4 years ago

      Thanks so much for the explanation!

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      agabrans 5 years ago

      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)?

    • profile image

      Sal Stephens 5 years ago

      You rock Sonia! Great explanation!!!!!

    • profile image

      kabirdes 5 years ago

      The explanation was Awsm sonia :)

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

    • profile image

      Lucy 5 years ago

      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?

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      ben 5 years ago

      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.

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      lauren 5 years ago

      Thanks so much! that was easy to understand

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      MiddleSchooler 5 years ago

      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.

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      Lee 5 years ago

      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))

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      FOSTOM 5 years ago

      MIND WIDE OPEN!

      GREAT JOB......

    • profile image

      sean 5 years ago

      cheers, very easy to understand

    • profile image

      michaelheemson 5 years ago

      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

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      RESPECT BOY 5 years ago

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

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      Dawnysha 5 years ago

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

    • profile image

      Tommy 5 years ago

      Thank you, OMG

    • profile image

      Nunu Norbert 5 years ago

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

    • profile image

      Afshan 5 years ago

      awesome!!! you're the man!

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      Math Gal 5 years ago

      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.

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      Daina 5 years ago

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

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      Jonathan 5 years ago

      Is 42 the variance in this case?

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      Apples 5 years ago

      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.

    • profile image

      Apples 5 years ago

      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)

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      Olivia 5 years ago

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

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      des donnelly 5 years ago from NYC....

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      thanks this kinda helped 5 years ago

      this was pretty good

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      Math is terrible 5 years ago

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

    • profile image

      Anil Singh 5 years ago

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

    • profile image

      geezel 5 years ago

      thank you so much.it really helps me.

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      ellie 5 years ago

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

    • profile image

      yuva 5 years ago

      very interesting and nice example thank u

    • profile image

      Garry 5 years ago

      Superb. Nice and easy to follow, thank you

    • profile image

      ganesh 5 years ago

      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...

    • profile image

      EE 5 years ago

      Thanks

    • profile image

      Lene 5 years ago

      that was awesome.

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

    • profile image

      Sarah N 5 years ago

      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 :/

    • profile image

      Jye 5 years ago

      Thanks so much. Helping out my year 12 math.

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      I do your mum 5 years ago

      No one will me read my comment;)

    • profile image

      Devon 5 years ago

      Stupid

    • profile image

      SUCK MY EAR 5 years ago

      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 :)

    • profile image

      Hetty 5 years ago

      Thank you for this!

      Helped me with applied science coursework!

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      Buddy 5 years ago

      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.

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      SATHISH 5 years ago

      SIMPLY EXCELLENT

    • profile image

      hehehehe 5 years ago

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

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

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      5 years ago

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

    • profile image

      easin 5 years ago

      thank u so so so so so so much

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      Ikram Ahmad 5 years ago

      Thanks

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      Max 5 years ago

      Thank you very much.

    • profile image

      Maria 5 years ago

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

    • profile image

      gangsta 5 years ago

      omg its thugtastic yo

    • profile image

      Dabee 5 years ago

      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

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      Musa simon 5 years ago

      Interesting!!!

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      Prasad Ajinkya 5 years ago

      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!

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      Anon 5 years ago

      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

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      Lee 5 years ago

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

    • profile image

      AngaNaa 5 years ago

      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!!

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      kaye 5 years ago

      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.

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      emma 5 years ago

      wow! awesome! you're AMAZING

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      Jeff 5 years ago

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

    • profile image

      Esther Ripplinger 5 years ago

      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!

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      zainab 5 years ago

      That is exactly what I was looking for!!

      Thanks a lot :)

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      Amanda 5 years ago

      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!

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      irene 5 years ago

      I can my exam comfortably thank you Sonia

    • profile image

      Samuel 5 years ago

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

    • profile image

      Deedee 5 years ago

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

    • profile image

      Melissa 5 years ago

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

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      cin 5 years ago

      it's very clear thank you sooo much!

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      Andrea Noisternig 6 years ago

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

      THANKS

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      mars 6 years ago

      wow, lot of comments on here.

    • profile image

      Statlover 6 years ago

      that method is epic so cool

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

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      Magarac 6 years ago

      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!

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      Sunday John Bosco 6 years ago

      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.

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      madds 6 years ago

      omg thank you so much!! this is great!

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      :) 6 years ago

      Thanks. Helped out tons

    • profile image

      Dennis 6 years ago

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

    • profile image

      ELVIS SHIVHALO 6 years ago

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

    • profile image

      Adham 6 years ago

      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

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      jesereth 6 years ago

      It was simple yet helpful information. Thanks!

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      lenieke 6 years ago

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