Descriptive Statistics, visually explained.

Before summarizing numbers, we must understand what those numbers represent. Data is any collection of information, observations, or measurements. Broadly, data falls into two categories:

Descriptive Statistics focuses on summarizing, organizing, and visualizing these datasets (e.g., finding the average score on a test). Conversely, Inferential Statistics uses data from a small sample to make predictions or draw conclusions about a larger population.

Measures of central tendency point us toward the "center" or middle region of a dataset. The three most common summaries are:

1. Mean (Arithmetic Average): Sum of all values divided by the number of values. Represented by x̄ (sample mean) or μ (population mean):

2. Median: The middle value when values are sorted in ascending order. If the dataset size is even, the median is the average of the two middle values. Median is robust; it resists the pulling power of outliers.

3. Mode: The value that appears most frequently in the dataset. A dataset can have one mode, multiple modes (bimodal/multimodal), or no mode at all if all values occur with equal frequency.

Two datasets can have the exact same mean but look completely different. Dispersion measures how spread out or varied the data points are. The key metrics are:

1. Range: The difference between the highest and lowest value:

2. Variance (s²): The average of the squared differences from the mean. Divisor is n - 1 for sample variance to correct statistical bias (Bessel's correction):

3. Standard Deviation (s): The square root of variance. Standard deviation returns the dispersion metric back to the original unit of the data, making it directly comparable:

An outlier is an observation point that is distant from other observations in a dataset. Outliers can occur due to measurement error or actual extreme variability (like super-wealthy individuals).

Outliers create skewness — a lack of symmetry in distributions. When a distribution has a long tail trailing off to the right, it is right-skewed (positively skewed). Here, the mean is pulled right, resulting in Mean > Median.

The Normal Distribution (or Bell Curve) is a perfectly symmetrical curve centered around the mean. Many real-world variables, like human heights or exam scores, closely approximate a normal shape.

To compare different raw scores on a standard scale, we calculate their Z-score. A Z-score tells us exactly how many standard deviations a score lies from the mean:

Grade Curving: Educators use normal distributions to curve scores. By assuming grades follow a normal curve, they assign letters based on standard deviation limits: A ($\ge +1.2\sigma$), B ($+0.4\sigma$ to $+1.2\sigma$), C ($-0.4\sigma$ to $+0.4\sigma$), D ($-1.2\sigma$ to $-0.4\sigma$), and F ($< -1.2\sigma$).