Statistics Chapter: Measures of Center

Measures of Center

• Mean and median are the two most common measures of center.
• Mean is the average, calculated by dividing the sum of observations by the number of observations. It acts as a balancing point in the dataset.
• Median represents the middle value of an ordered data set.

Key Variables

• X denotes the variable being measured, representing the sample data.
• n indicates sample size, which signifies the number of observations within the sample.
• X₁ is the first observation, while Xn is the last observation in the sample.

Summation and Mean Notation

• ∑x refers to the total sum of all x values in the dataset.
• x̅ (x-bar) represents the sample mean, whereas µ (mu) denotes the population mean, which is the average of all values in the population.

Types of Mean

• Weighted mean is found by multiplying each value by its corresponding weight, summing these products, and dividing by the total sum of weights.

Impact of Outliers

• Outliers can significantly affect the mean, making it less representative of the dataset.

Mode and Range

• Mode is defined as the most frequently occurring observation in the dataset.
• Range measures the spread of the dataset, calculated as the largest value minus the smallest value.

Measures of Variation

• Common measures of variation are based on deviations from the mean.
• Sample deviation is the difference between an observation and the sample mean.
• Variance quantifies the degree of spread by calculating the sum of squared deviations from the sample mean divided by (n-1).
• Sample standard deviation is the square root of the sample variance, while δ represents population standard deviation, and δ² denotes population variance.

Quartiles and Distribution

• Q₁ is the lower quartile, marking the median of the lower half of sorted data, while Q₃ is the upper quartile for the upper half.
• Interquartile range (IQR) is the difference between Q₃ and Q₁, indicating the dispersion of the middle 50% of data.

Summary and Visualization

• The five-number summary includes minimum, Q₁, median, Q₃, and maximum values.
• Boxplot visually represents the five-number summary to easily convey data distribution and detect outliers.