Lecture Notes: Standard Deviation in Statistics

Introduction

• In real life, obtaining all data for a population is impractical or impossible.
• Instead, samples are used to approximate the population's standard deviation.

Population vs. Sample Concepts

Population

• Mean Symbol: ( \mu ) (Mu)
• Calculation: Sum of data divided by the number of data points, ( N ).

Sample

• Mean Symbol: ( \bar{X} ) (X bar)
• Calculation: Sum of sample values divided by the number of sample points, ( n ).

Standard Deviation

Population Standard Deviation

• Symbol: ( \sigma ) (Sigma)
• Calculation:
  • Compute deviations: ( x - \mu )
  • Square deviations: ((x - \mu)^2)
  • Average squared deviations by dividing by ( N )
  • Square root to correct for squaring

Sample Standard Deviation

• Symbol: ( s )
• Calculation:
  • Compute deviations from sample mean: ( x - \bar{X} )
  • Square deviations
  • Instead of dividing by ( n ), divide by ( n - 1 )

Importance of Differentiating Between Population and Sample

• Using ( n - 1 ) instead of ( n ) corrects for bias when estimating population standard deviation from a sample.
• It ensures the sample standard deviation is a good approximation of the population standard deviation.

Why Divide by ( n - 1 )?

• Aim is to measure the average distance from the population mean, ( \mu ).
• ( \bar{X} ) approximates ( \mu ) but is not identical.
• Dividing by ( n - 1 ) corrects the bias and provides a better estimate.
• More advanced statistics courses cover the detailed explanation of why ( n - 1 ) works.

Conclusion

• Understanding when to use population vs. sample formulas is crucial.
• Next example will show the calculation of sample standard deviation.