March 16, 2025
Descriptive statistics summarize and organize characteristics of a data set.
Two main types:
Measures of central tendency (mean, median, mode)
Measures of dispersion (range, variance, standard deviation)
Benefits:
Simplifies large amounts of data
Identifies patterns and trends
Foundation for inferential statistics
Three main measures:
Each measure provides different insights about the data’s central location.
Dispersion tells us how spread out the data is:
Higher values indicate greater data spread.
For a dataset \(X = \{x_1, x_2, \ldots, x_n\}\) with \(n\) observations:
Mean: \[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i\]
Sample Variance: \[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2\]
Sample Standard Deviation: \[s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}\]
Properties of the Mean:
If \(X = \{x_1, x_2, \ldots, x_n\}\) and \(c\) is a constant:
Linear Transformation: \[\overline{aX+b} = a\bar{X} + b\]
Sum of Means: \[\overline{X+Y} = \bar{X} + \bar{Y}\]
Minimization Property: \[\sum_{i=1}^{n}(x_i - \bar{x})^2 \leq \sum_{i=1}^{n}(x_i - c)^2\]
for any value \(c \neq \bar{x}\)
# Create 3D data
set.seed(789)
n <- 100
x <- rnorm(n, mean = 10, sd = 2)
y <- rnorm(n, mean = 15, sd = 3)
z <- 2*x + 3*y + rnorm(n, mean = 0, sd = 5)
plot_data <- data.frame(x = x, y = y, z = z)
# Create 3D scatter plot
plot_ly(plot_data, x = ~x, y = ~y, z = ~z,
marker = list(size = 5, color = ~z, colorscale = 'Viridis',
opacity = 0.8)) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'X Variable'),
yaxis = list(title = 'Y Variable'),
zaxis = list(title = 'Z Variable')),
title = "3D Scatter Plot of Variables")
Key Takeaways:
Descriptive statistics provide numeric and visual summaries
Different measures reveal different aspects of data
Critical foundation for data analysis
Applications:
Business: Financial performance metrics
Healthcare: Patient outcome measurements
Education: Student performance analysis
Research: Experimental results summary