1. Descriptive Statistics

Descriptive statistics summarize and describe features of a dataset using measures of central tendency, dispersion, and data distribution.

1.1 Measures of Central Tendency

These statistics describe the center of the data:

• Mean (xˉ\bar{x}): The arithmetic average of a dataset.

  • Formula:

    xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}

  • The mean is sensitive to outliers.

• Median: The middle value when data is sorted in ascending order.

  • If is odd, the median is the middle value.

    nn

  • If is even, the median is the average of the two middle values.

    nn

  • Less sensitive to outliers than the mean.

• Mode: The most frequently occurring value(s) in a dataset.
  • A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (multiple modes).

1.2 Measures of Dispersion

Dispersion measures the spread of data points around the center.

• Variance (σ2\sigma^2): Measures the average squared deviation from the mean.

  • Population Variance:

    σ2=∑(xi−xˉ)2n\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}

  • Sample Variance:

    s2=∑(xi−xˉ)2n−1s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}

• Standard Deviation (σ\sigma): The square root of variance, representing data spread in the same unit as the data.

  • Formula:

    σ=σ2\sigma = \sqrt{\sigma^2}

• Range: The difference between the maximum and minimum values in the dataset.

  • Formula:

    Range=max⁡(x)−min⁡(x)\text{Range} = \max(x) - \min(x)

• Interquartile Range (IQR): The range of the middle 50% of data, calculated as:

  • IQR=Q3−Q1IQR = Q3 - Q1
  • Where:
    • Q1Q1 (first quartile) is the 25th percentile.
    • Q3Q3 (third quartile) is the 75th percentile.

2. Data Distributions

Distributions describe how data is spread.

2.1 Normal Distribution

• A bell-shaped and symmetric distribution defined by:

  • Mean ()

    μ\mu

  • Standard deviation ()

    σ\sigma

• Empirical Rule (68-95-99.7 Rule):

  • 68% of values fall within

    μ±σ\mu \pm \sigma

  • 95% of values fall within

    μ±2σ\mu \pm 2\sigma

  • 99.7% of values fall within

    μ±3σ\mu \pm 3\sigma

2.2 Skewness and Kurtosis

• Skewness: Measures asymmetry in a distribution.
  • Positive Skew: Right tail is longer.
  • Negative Skew: Left tail is longer.