Measures of Central Tendency in Statistics

Study Notes

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

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 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

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

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

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

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 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

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

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