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A beginner’s guide to standard deviation and standard error

Posted on 26th September 2018 by

Tutorials and Fundamentals
Stick person, confused, with 2 equations either side of head

What is standard deviation?

Standard deviation tells you how spread out the data is. It is a measure of how far each observed value is from the mean. In any distribution, about 95% of values will be within 2 standard deviations of the mean.


How to calculate standard deviation

Standard deviation is rarely calculated by hand. It can, however, be done using the formula below, where x represents a value in a data set, μ represents the mean of the data set and N represents the number of values in the data set.

The steps in calculating the standard deviation are as follows:

  1. For each value, find its distance to the mean
  2. For each value, find the square of this distance
  3. Find the sum of these squared values
  4. Divide the sum by the number of values in the data set
  5. Find the square root of this


What is standard error?

When you are conducting research, you often only collect data of a small sample of the whole population. Because of this, you are likely to end up with slightly different sets of values with slightly different means each time.

If you take enough samples from a population, the means will be arranged into a distribution around the true population mean. The standard deviation of this distribution, i.e. the standard deviation of sample means, is called the standard error.

The standard error tells you how accurate the mean of any given sample from that population is likely to be compared to the true population mean. When the standard error increases, i.e. the means are more spread out, it becomes more likely that any given mean is an inaccurate representation of the true population mean.


How to calculate standard error

Standard error can be calculated using the formula below, where σ represents standard deviation and n represents sample size.


Standard error increases when standard deviation, i.e. the variance of the population, increases. Standard error decreases when sample size increases – as the sample size gets closer to the true size of the population, the sample means cluster more and more around the true population mean.



Image 1: Dan Kernler via Wikipedia Commons: https://commons.wikimedia.org/wiki/File:Empirical_Rule.PNG 

Image 2: https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-spread-distributions/a/calculating-standard-deviation-step-by-step

Image 3: https://toptipbio.com/standard-error-formula/










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No Comments on A beginner’s guide to standard deviation and standard error

  • CPAk Emelda Nyongesa

    Good explanation

    13th October 2022 at 8:23 am
    Reply to CPAk
  • Dale Brethauer

    Does the standard deviation length make a difference. What length should I use on a 3 month daily chart?

    4th October 2022 at 7:46 pm
    Reply to Dale
  • Elia Mascolo

    You say:
    “In any distribution, about 95% of values will be within 2 standard deviations of the mean”.
    That’s not true. It is expected for gaussian (or “normal”) distributions. It’s doesn’t apply to “any distribution”.

    You can find this even in your references
    or elsewhere

    24th March 2022 at 6:24 pm
    Reply to Elia
  • deng

    simple, yet so profound

    24th November 2021 at 1:09 am
    Reply to deng
  • Bright Amenorhu


    21st November 2021 at 8:05 pm
    Reply to Bright
  • DG Shankar

    Very Good.

    13th October 2021 at 9:39 am
    Reply to DG
  • ritik bisht

    the exam is just after an hour and I swear I understood this whole topic within 2 minutes, thanks a lot.

    20th September 2021 at 8:42 am
    Reply to ritik
  • Musa sale

    How do i differentiate variance,standard deviation and standard error

    14th March 2021 at 4:42 pm
    Reply to Musa
  • Kennedy Nashiana

    This is well simplified.

    19th November 2020 at 11:07 am
    Reply to Kennedy
  • Wesley

    Hi, Thank you! I’m pretty sure the formula for standard deviation has a mistake in it. The denominator should be n-1.

    23rd October 2020 at 6:25 am
    Reply to Wesley
  • Rohit

    Standard Deviation is the square root of variance, so its kind of trivial to state the conclusion about the increasing standard error with respect to standard error.
    Also please look into the symbol of sigma mentioned in the explanation of standard error.

    It is a good write up

    6th October 2020 at 4:15 pm
    Reply to Rohit
    • Emma Carter

      Thank you for flagging the symbol errors on the page Rohit. These have been updated now. Many thanks, Emma.

      6th October 2020 at 4:56 pm
      Reply to Emma
  • Guru

    I was not able to understand standard error. The procedures for calculating is given but i cant understand the process of calculation.

    4th August 2020 at 8:47 am
    Reply to Guru
  • Mylse Corpe

    It’s a good attempt and i will like to no more.

    24th June 2020 at 9:35 pm
    Reply to Mylse
  • Osama Elbahr

    Thanks for your illustrations.

    But, can you clarify when to incorporate SE in our research results and how to interpret ?

    Thanks again

    19th February 2020 at 1:46 pm
    Reply to Osama
  • Mustapha

    How do you then determine the sample size with the most minimal acceptable standard error. Because you need to have obtained the sample before you can determine standard deviation?

    9th December 2018 at 10:05 am
    Reply to Mustapha
  • Sayyid

    excellent explanation of the concepts

    5th November 2018 at 9:35 pm
    Reply to Sayyid
  • Terje Soerdal

    Very simply and nicely explained. Thanks!

    2nd October 2018 at 5:28 am
    Reply to Terje

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