Review of Statistical Concepts

Overview

• Review of key statistical concepts learned so far.
• Focus on fitting concepts together with calculations using real numbers.

Concepts and Terminology

1. Central Tendency

• Mean (Average): Sum of all data points divided by the number of data points.
  • Population Mean (μ): Sum of all data points in the population divided by N (number of data points).
  • Sample Mean (x̄): Sum of all data points in the sample divided by n (number of data points in the sample).
• Median and Mode: Other measures of central tendency.

2. Variance

• Population Variance (σ²): Average of the squared differences from the mean.
  • Formula: ( \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} )
• Sample Variance (s²): Unbiased estimate of population variance, divide by n-1.
  • Formula: ( s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} )
  • Provides a better estimate of population variance.

3. Standard Deviation

• Population Standard Deviation (σ): Square root of the population variance.
  • ( \sigma = \sqrt{\sigma^2} )
• Sample Standard Deviation (s): Square root of the sample variance.
  • Note: Not an unbiased estimator.
• Units: Standard deviation is more relatable because it is in the same units as the original data, unlike variance.

Example Calculation

Calculating Mean, Variance, and Standard Deviation

Data Set: 1, 2, 3, 8, 7 (Assume it’s a Population)

1. Mean
   • ( \text{Mean} = \frac{1 + 2 + 3 + 8 + 7}{5} = 4.20 )
2. Variance
   • Calculate squared differences from the mean:
     • ( (1 - 4.20)^2, (2 - 4.20)^2, (3 - 4.20)^2, (8 - 4.20)^2, (7 - 4.20)^2 )
   • Sum of squared differences: 38.80
   • ( \text{Variance} = \frac{38.80}{5} = 7.76 )
3. Standard Deviation
   • ( \text{Standard Deviation} = \sqrt{7.76} = 2.79 )

If the Data Was a Sample

• Sample Variance:
  • ( \text{Sample Variance} = \frac{38.80}{4} = 9.70 )
• Sample Standard Deviation:
  • ( \sqrt{9.70} = 3.11 )

Conclusion

• Understanding these calculations helps solidify comprehension of statistical measures.
• Further exploration into expected values and unbiased estimators will follow in future lectures.