Average

Study Notes – Average

Quantitative Aptitude (MCAT)
MIT‑Level, From First Principles

1. Introduction

The average is a concept that appears in every branch of mathematics, statistics, physics, economics, and biology. In the context of the MCAT, “average” usually refers to the arithmetic mean, but a rigorous treatment reveals a zoo of related averages and a rich theoretical backbone.

Goal – Understand the why behind each average, how it behaves in edge cases, and why it is useful for MCAT problems.

2. First Principles

2.1 Arithmetic Mean of a Finite Set

Let (x_1, x_2, \dots, x_n \in \mathbb{R}).
The arithmetic mean (average) is

[ \bar{x} ;=; \frac{1}{n}\sum_{i=1}^{n} x_i . ]

• Derivation:
  The mean is the unique number (m) that minimizes the sum of squared deviations: [ m ;=;\arg\min_{m\in\mathbb{R}} \sum_{i=1}^{n}(x_i-m)^2 . ] Differentiating with respect to (m) and setting to zero: [ \frac{d}{dm}\sum_{i=1}^{n}(x_i-m)^2 = -2\sum_{i=1}^{n}(x_i-m)=0 ;\Rightarrow; m=\frac{1}{n}\sum_{i=1}^{n}x_i . ] This gives a rigorous justification for the formula.

• Properties:

  1. Linear: (\overline{ax+by}=a\bar{x}+b\bar{y}) for constants (a,b).
  2. Translation invariance: (\overline{x+c} = \bar{x}+c).
  3. Symmetry: The mean is unchanged under any permutation of the data.

2.2 Weighted Arithmetic Mean

When observations carry different importance, we assign non‑negative weights (w_i).

[ \bar{x}w ;=; \frac{\sum{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}\quad \bigl(\sum w_i \neq 0\bigr). ]

• Derivation:
  Minimize (\sum w_i (x_i-m)^2) → (m = \bar{x}_w).

• Edge Cases:

  • If all (w_i=0) → undefined (division by zero).
  • If some (w_i<0) → the “average” can lie outside the convex hull of the data, which is useful in some physics contexts but problematic for a statistical mean.

2.3 Harmonic and Geometric Means

These are natural when averaging rates or ratios.

Both are not linear, but satisfy useful inequalities (AM ≥ GM ≥ HM).

Proof of AM ≥ GM (for positive (x_i)):

[ \ln G = \frac{1}{n}\sum_{i=1}^{n}\ln x_i \le \ln!\left(\frac{1}{n}\sum_{i=1}^{n}x_i\right)=\ln \bar{x} ] by Jensen’s inequality for the concave function (\ln). Exponentiating gives (G \le \bar{x}).

2.4 Median and Mode

• Median: the middle value after sorting (or the average of the two middle values if (n) is even).
  Robust: insensitive to outliers; useful in MCAT when data can be skewed.

• Mode: the most frequent value.
  Discrete nature makes it well‑suited for categorical data.

3. Deeper Theory

3.1 Average as Expectation

In probability theory, the expected value (E[X]) of a random variable (X) is the average over an infinite sample space:

[ E[X] = \int_{\Omega} X(\omega), dP(\omega). ]

• For a discrete distribution with probabilities (p_i) and values (x_i):

[ E[X] = \sum_{i} p_i x_i . ]

• Connection to weighted mean: the weights are probabilities (p_i), which sum to 1.
  Thus the weighted arithmetic mean is a special case of expectation.

3.2 Relation to Variance

Variance ( \sigma^2 = E[(X-\mu)^2] ) measures dispersion.
The mean appears as the point that minimizes squared error