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 the 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 larger population sample, 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 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 centimeters is 9, 12, 13, 11, 12, 8, 10, and 11. Calculate the sample standard deviation of the 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 the 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 its 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:
