Standard deviation is used to measure the amount of variation in a process. This is one of the most common measures of variability in a data set or population.

There are 2 types of equations: Sample and Population.

## What is the difference between Population and Sample.

Population refers to ALL of a set and sample is a subset. We most often have a sample and are trying to infer something about the whole group. However, if we want to know a truth of a subset of a whole population, use the Population equation.

### Use Population When:

- You have the entire population.
- You have a sample of a larger population, but you are only interested in this sample and do not wish to generalize your findings to the population.

### Use Sample When (Most often):

- If all you have is a sample, but you wish to make a statement about the population standard deviation from which the sample is drawn, you need to use the sample SD.

**Remember:** It is impossible to have a negative standard deviation.

## How to Measure the Standard Deviation for a Sample (s)

Standard Deviation for a Sample (s)

- Calculate the mean of the data set (x-bar)
- Subtract the mean from each value in the data set
- Square the differences found in step 2.
- Add up the squared differences found in step 3.
- Divide the total from step 4 by (n – 1) for sample data
- (Note: At this point you have the variance of the data).
- Take the square root of the result from step 5 to get the SD

### Example of Standard Deviation for a Sample (s)

The length of 8 bars in centimetres is 9,12,13,11,12,8,10 and 11. Calculate the sample standard deviation of length of the bar.

- Calculate the mean of the data set

(9+12+13+11+12+8+10+11)/8=86/8=10.75

2. Subtract the mean from each value in the data set

- 10.75-9=1.75
- 10.75-12=-1.25
- 10.75-13=-2.25
- 10.75-11=-0.25
- 10.75-12=-1.25
- 10.75-8=2.75
- 10.75-10=0.75
- 10.75-11=-0.25

3. Square the differences found in step 2.

- (1.75)
^{2}=3.06 - (-1.25)
^{2}=1.56 - (-2.25)
^{2}=5.06 - (-0.25)
^{2}=0.06 - (-1.25)
^{2}=1.56 - (2.75)
^{2}=7.56 - (0.75)
^{2}=0.56 - (-0.25)
^{2}=0.06

4. Add up the squared differences found in step 3.=19.50

5. Divide the total from step 4 by (n – 1) for sample data=19.5/8-1=2.79, so variance is 2.79

6. Take the square root of the result from step 5, SD of sample bar length = 1.67

## How to Measure the Standard Deviation for a Population (σ)

Standard Deviation for a Population (σ)

- Calculate the mean of the data set (μ)
- Subtract the mean from each value in the data set
- Square the differences found in step 2.
- Add up the squared differences found in step 3.
- Divide the total from step 4 by N (for population data).
- (Note: At this point you have the variance of the data).

- Take the square root of the result from step 5 to get the SD

### Example of Standard Deviation for a Population (σ)

Nana’s Bakery wants to optimize the consistency of their cakes. The recipe calls for a certain number of eggs. The problem is that there is variation in egg sizes. Six eggs were randomly selected and the following weights were recorded (measured in ounces).

2.25; 1.75; 2.0; 2.5; 2.3;1.8

What is the SD of the egg weights?

## Standard Deviation and Variance

Variance is Std Dev ^2.

Std Dev = Sqrt(variance)

## The Uniform Distribution

A uniform distribution is a continuous probability distribution. It describes the condition where all possible outcomes of a random experiment are equally likely to occur. For the uniform distribution, the probability density function f(x) is constant over the possible values of x.

##### The formula for Mean and standard deviation of uniform distribution

##### Probability density function

##### The Area between p and q

##### Area right of x

##### Area left of x

**Example:** The number of mobile phones sold by Beta stores is uniformly distributed between 6 and 20 per day. Then find

- Mean
- Standard deviation
- Probability that the daily sales fall between 10 and 12
- Probability that the Beta stores will sell at least 16

Let X be the number of mobiles sold daily by beta stores, X follows the uniform distribution over (6,20). Thus the probability density function is

## Comments (23)

Thanks a lot!

Very welcome, Roberto.

I really appreciate your explanations.

You’re welcome, Sonia. Happy to help!

How is this related to Six Sigma where we are expected to see a 99.9996 ?

Sashi, the sigma in Six Sigma refers to standard deviation. Six Sigma refers to what percentage is under the curve at six sigmas – or standard deviations from center. Does that help?

I am understanding now. thanks

Glad to hear it, Sharon!

Please advise vsf targets and minimum vs maximum Range and if no maximum Range do we have ranking for ex..

1 to 2 is controlled

3 to 4 is out of control

4 to 5 is above cout of control

I’m not sure what you mean here, Nessma. Can you elaborate?

When d2 changes the cp and cpk values are varrying. If so what will be the spec for each subgroup size.

Hi Omy,

See our answer below in the other comments.

Best, Ted.

i have a question

Let’s assume that based on a customer survey, the weight of a product is approved at the best acceptable level of 120 grams +- 12 grams . standard deviation of each product labled .If the standard deviation of a product is 6, what is the level of the sigma? If it is the 10 what the sigma level

So you want to determine the Sigma level of a process given a Standard Deviation? What have you tried?

Hello Alireza,

Actually we need process mean to compute sigma value.

Cpk is a measure to show how many standard deviations the specification limits are from the center of the process.

Process sigma = 3* Cpk.

LSL = 108

USL= 132

SD=6 or 10

Process mean= ??

• Cplower = (Process Mean – LSL)/(3*Standard Deviation)

• Cpupper = (USL – Process Mean)/(3*Standard Deviation)

• Cpk is smallest value of the Cpl or Cpu: Cpk= Min (Cpl, Cpu)

You can refer more about process sigma at https://sixsigmastudyguide.com/project-baseline-sigma/

Thanks

Estimation of standard deviation depends on what

Specification limits

Target/Nominal value

Observed data

None of the above

This is a neat question, Arun.

To answer I recommend looking at each of the options you listed and asking what each has to do with Standard Deviation, if anything.

What are your thoughts?

Thanks Ted for the well-written guide.

In all of my studies, I still struggle with visualizing/understanding the difference between variance. I get standard deviation, but variance throws me for a loop.

Can you help close the gap?

Thanks for the complements, Tanner. They are very much appreciated.

Variance is very similar to Standard Deviation. It’s just the squaring of standard deviation.

Another way to understand variance is to forget Std Dev all together. Imagine you had a police line up. I’m thinking of the one from the movie Usual Suspects. There’s going to be variation on heights of the people on the line up. You could easily calculate the mean height, right? So visualize that mean height as a bar going across the suspects in the line up. That bar will be over the heads of the shorter people and through the face or body of the taller people, right? Think of variance as the total amount of that distance from the mean line squared divided by how ever many people are in the line up.

Perhaps a one-sentence explanation could be the a measure of how far off of mean samples are in aggregate where greater distance from the mean is amplified (because mathematically we are squaring it).

Does that help?

Missing the 2.3 in the original data in the example

Yes Richard, we have updated it!

Hi! shouldn’t the equation for Six Sigma be Y=F(x)+σ ?

here, ‘Y’ being the end product/result,

‘x’ being the variables/inputs used in order to achieve ‘Y’

and ‘σ’ being the defects/wastes that show up in the process?

How can six sigma be represented as only Y=F(x) ?

Hi Rakesh,

y = f(x) is a bit of an abstraction. What it means is that outputs are generally governed by some kind of process and are usually not completely random. If processes are deterministic, then we can identify how to optimize them. More on this in our article on Causal Theory here.

Best, Ted.