Lecture Notes on Statistics

Key Concepts Covered

• Binomial Distribution

  • Definition: Discusses a specific number of successes out of a total number of trials.
  • Trials considered as experiments; Beri experiments have only successes or failures.

• Central Limit Theorem (CLT)

  • Importance: Fundamental theorem in statistics.
  • Statement: The sample mean will approximate a normal distribution, regardless of the population distribution.
  • Example: Retail store sales analysis over 30 days—despite daily fluctuations, average daily sales will follow a normal distribution.
  • Tool: Use RND() to generate random samples and calculate the mean.

Interactive Examples

• Measures of Central Tendency and Distributions

  • Example: Business tracks new customers (data points: 50, 55, 60, 65, 70, 75, 80).
    • Calculate Mean and Standard Deviation using Excel.
    • Excel Functions: AVERAGE() for mean, STDEV() for standard deviation.

• Skewness and Kurtosis

  • Skewness: Direction in which data is skewed (right/positive or left/negative).
  • Kurtosis: Describes tails of distribution; high kurtosis means heavy tails.
  • Example: Retail sales either low or high suggests positive skewness and high kurtosis.

• Application of CLT

  • Manager tracks sales across 10 stores; CLT helps predict average weekly sales.
  • Distribution of average daily sales will be approximately normal.

Distribution Types

• Normal Distribution

  • Bell-shaped, symmetric; mean, median, and mode are equal.

• Right and Left Skewed Distributions

  • Right Skew: Tail on right side is longer; mean > median.
  • Left Skew: Tail on left side is longer; mean < median.

• Kurtosis

  • High Kurtosis: Heavy tails, more frequent extreme values.
  • Low Kurtosis: Less frequent extreme values.

Sampling

• Concept of Sampling
  • Taking averages of samples gets closer to the theoretical mean.
  • Example: Coin flip model illustrating sampling and average convergence to expected value.

Variance and Covariance

• Variance

  • Describes spread of a variable.

• Covariance

  • Measures how two variables change together.
  • Related to correlation: standardizes the relationship.
  • Conceptual analogy: Cat movements to explain variable interaction.

Correlation

• Correlation
  • Standardized measure of relationship between two variables.
  • Adjusts for size of variables, similar to co-variance but proportionate.

Practical Business Example

• Advertising and Sales Revenue
  • Covariance can show relationship between advertising spend and sales.

Next Steps

• Use Excel for calculating covariance and correlation.
• Practice and understand using real-world examples and data.

Note: Focus on the analogies used to better grasp statistical concepts and think about practical applications between now and next class.