Measures of Central Tendency in Statistics

Measures of Central Tendency

Mean

• Also known as the arithmetic mean
• Calculated by summing all values and dividing by the number of values
• Formula: μ = (Σx) / n
• Sensitive to outliers, as a single extreme value can greatly affect the result
• Not a robust measure of central tendency

Median

• Middle value in a dataset when arranged in order
• If the dataset has an odd number of values, the median is the middle value
• If the dataset has an even number of values, the median is the average of the two middle values
• More robust than the mean, as it is less affected by outliers

Mode

• Most frequently occurring value in a dataset
• A dataset can have multiple modes (bimodal, trimodal, etc.) or no mode at all
• Not a robust measure of central tendency, as it can be influenced by a single extreme value

Measures of Variability

Standard Deviation (σ)

• Measures the spread or dispersion of a dataset
• Calculated as the square root of the variance
• Formula: σ = √[(Σ(x - μ)^2) / (n - 1)]
• A small standard deviation indicates that the data points tend to be close to the mean
• A large standard deviation indicates that the data points are spread out over a larger range

Measures of Central Tendency

• Mean: calculated by summing all values and dividing by the number of values, sensitive to outliers, and not a robust measure
• Formula for Mean: μ = (Σx) / n
• Robustness of Mean: a single extreme value can greatly affect the result

Median

• Definition: middle value in a dataset when arranged in order
• Calculation for Odd Number of Values: middle value when dataset has an odd number of values
• Calculation for Even Number of Values: average of the two middle values when dataset has an even number of values
• Advantage: more robust than the mean, less affected by outliers

Mode

• Definition: most frequently occurring value in a dataset
• Types: can have multiple modes (bimodal, trimodal, etc.) or no mode at all
• Limitation: not a robust measure, can be influenced by a single extreme value

Measures of Variability

Standard Deviation (σ)

• Definition: measures the spread or dispersion of a dataset
• Calculation: square root of the variance
• Formula: σ = √[(Σ(x - μ)^2) / (n - 1)]
• Interpretation: small standard deviation indicates data points tend to be close to the mean, while large standard deviation indicates data points are spread out over a larger range

Measures of Dispersion

• Measures the amount of variation or dispersion of a set of values
• Standard Deviation (σ) is the square root of the variance
• Formula: σ = √[(Σ(xi - μ)²) / (n - 1)]
• Interpretation of standard deviation:
  • 68% of data points fall within 1 standard deviation of the mean
  • 95% of data points fall within 2 standard deviations of the mean
  • 99.7% of data points fall within 3 standard deviations of the mean

Measures of Position

• Quartiles divide data into four equal parts, each containing 25% of the data
• Quartiles (Q1, Q2, Q3) represent the 25th, 50th, and 75th percentiles
• Interquartile Range (IQR) = Q3 - Q1, measures the spread of the middle 50% of the data
• Deciles divide data into ten equal parts, each containing 10% of the data
• Deciles represent the 10th, 20th,..., 90th percentiles
• Percentiles measure the value below which a certain percentage of the data falls
• Example: 75th percentile is the value below which 75% of the data falls

Measures of Skewness

• Skewness measures the asymmetry of a distribution
• Formula: Skewness = (Σ(xi - μ)³) / (n * σ³)
• Interpretation of skewness:
  • Positive skewness: distribution is skewed to the right (long tail on the right)
  • Negative skewness: distribution is skewed to the left (long tail on the left)
  • Zero skewness: distribution is symmetrical