Statistics and Probability - Measures of Central Tendency (Mean, Median, Mode)

Measures of Central Tendency are statistical constants that represent the center point or 'typical' value of a dataset. Imagine a dot plot where data points cluster together; these measures help identify the specific value around which the data is centered.

The Mean is the arithmetic average, calculated by summing all data points and dividing by the total count. Visually, it acts as the 'balance point' of a histogram; if the distribution was a physical object on a pivot, the Mean is where it would perfectly balance.

The Median is the middle-most value when a dataset is arranged in numerical order. In a list of numbers, if you cross out values from both ends until you reach the center, that center is the Median. On a box plot, the Median is represented by the vertical line inside the box, dividing the data into two equal halves.

The Mode is the value that appears most frequently in a dataset. In a bar chart or frequency graph, the Mode is the category or value associated with the tallest bar. A dataset can be bimodal (having two peaks) or multimodal if multiple values share the highest frequency.

Outliers are extreme values that lie far away from the rest of the data points. On a number line, an outlier is an 'isolated' point. Outliers significantly pull the Mean toward them (either much higher or much lower), whereas the Median remains relatively unchanged and is often a better measure for skewed data.

The Range, while a measure of spread rather than central tendency, describes the distance between the highest and lowest values ($x_{max} - x_{min}$). A large range suggests the data is widely dispersed, while a small range suggests the data is tightly packed around the measures of central tendency.

Data Distribution affects which measure is most useful. In a perfectly symmetrical 'Bell Curve' (Normal Distribution), the Mean, Median, and Mode are all equal and located at the exact center of the curve.