Statistics 101: Analyzing and Visualizing Numerical Data

Understanding numerical data is crucial in statistics. This blog post covers key topics discussed in the Week 3 statistics-1 lectures of the IITM BS Degree in Data Science and Applications.

Types of Numerical Data

Frequency Tables for Numerical Data

Frequency tables help organize data by counting occurrences. Let's look at some examples:

Example: Household Sizes

Consider survey data from 15 individuals:

Guidelines for Organizing Data into Classes

Key Terms in Grouped Data

Example: Student Marks

Graphical Summaries

Histogram

A histogram represents frequency distributions visually. Here's an example of student test scores:

Stem-and-Leaf Plot

A stem-and-leaf plot organizes data by separating stems (leading digits) from leaves (trailing digits).

1 | 0, 4, 5
2 | 2, 3, 5, 8, 9
3 | 1, 6
4 | 5

Measures of Central Tendency

Mean

The arithmetic mean is calculated as:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Example:

For grouped data, we use:

$\bar{x} = \frac{\sum f_i m_i}{\sum f_i}$

where $f_i$ is the frequency and $m_i$ is the midpoint of each class.

Median

The median is the middle value in ordered data:

Mode

The mode is the most frequent value:

Measures of Dispersion

Range

$\text{Range} = \text{Maximum} - \text{Minimum}$

Variance & Standard Deviation

The sample variance is calculated as:

$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

The standard deviation is:

$s = \sqrt{s^2}$

Interquartile Range (IQR)

$\text{IQR} = Q_3 - Q_1$

Advanced Statistical Concepts

Properties of Mean

Example of Mean Properties

Properties of Median and Mode

Statistical Formulas

Population Parameters

$\text{Population Mean (µ)} = \frac{\sum_{i=1}^{N} x_i}{N}$

$\text{Population Variance (σ²)} = \frac{\sum_{i=1}^{N} (x_i - µ)^2}{N}$

Sample Statistics

$\text{Sample Mean (}\bar{x}\text{)} = \frac{\sum_{i=1}^{n} x_i}{n}$

$\text{Sample Variance (s²)} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

Five-Number Summary

Calculating Percentiles