Statistics Basics

The MCAT does not require you to calculate statistics. It requires you to read a result and know what it does and doesn’t say. This section covers every statistical concept that has appeared on the exam.

Correlation

Correlation measures the strength and direction of a linear relationship between two continuous variables. It makes no assumption about which variable causes which.

• Correlation coefficient (r) ranges from -1 to +1.
  • r = +1. Perfect positive correlation: as one variable rises, the other rises proportionally.
  • r = -1. Perfect negative correlation: as one rises, the other falls proportionally.
  • r = 0. No linear relationship.
• Strength. The closer |r| is to 1, the stronger. |r| = 0.9 is strong; |r| = 0.2 is weak.
• A coefficient outside [-1, +1] is impossible. If you see r = -1.4, it is a computational error.

Example. Height and weight in adults correlate around r = 0.7 - a strong positive correlation, but not perfect.

Regression

Regression models a directional relationship: one variable is assumed to depend on (or be predicted by) another.

• Linear regression. Fits a straight line through data, expressing the dependent variable as a function of the independent variable.
• Unlike correlation, regression assumes a one-way influence of the predictor on the outcome.

Example. Predicting blood pressure from sodium intake. Sodium is the independent variable; blood pressure depends on it.

Contrast with correlation. Correlation is symmetric (r between X and Y is the same as r between Y and X). Regression is directional.

Chi-Square Test

Chi-square (χ²) test compares two or more distributions of categorical data. Asks whether observed frequencies in each category differ from what would be expected by chance.

Example. Do men and women smoke at different rates? Count smokers and non-smokers in each group. Chi-square tests whether the two 2 × 2 distributions differ.

Use chi-square when both variables are categorical.

t-Test

t-test compares mean values of a continuous dependent variable between two groups or a group and a specific value.

• One-sample t-test. Compare a group’s mean to a known value.
• Two-sample (independent) t-test. Compare means of two different groups.
• Paired t-test. Compare two measurements on the same subjects (before/after).
• One-tailed t-test. Tests for a difference in a specific direction.
• Two-tailed t-test. Tests for a difference in either direction.

Example. Do patients on drug A have lower cholesterol than patients on drug B? Two-sample t-test.

ANOVA

ANOVA (analysis of variance) extends the t-test to three or more groups. Like a t-test, it compares means of a continuous variable across categorical groups.

Example. Compare mean test scores across four teaching methods.

If ANOVA finds a significant difference, post-hoc tests determine which groups differ.

The Null and Alternative Hypotheses

• Null hypothesis (H₀). The default assumption: there is no effect or no relationship between variables.
• Alternative hypothesis (H₁ or Hₐ). The proposal: there is an effect.

Statistical tests try to reject the null. You never “prove” the alternative directly; you only accumulate enough evidence against H₀ to make it implausible.

Type I and Type II Errors

• Alpha (α). Probability of a Type I error; conventionally set at 0.05. You are willing to accept a 5% chance of claiming an effect when none exists.
• Beta (β). Probability of a Type II error. Typically you want β ≤ 0.20.
• Power = 1 - β. The probability of detecting a real effect when one exists. Larger samples increase power.

Confidence Intervals and Confidence Level

Confidence interval (CI). A range that is likely to contain the true population parameter.

Confidence level. The percentage of such intervals, computed across repeated samples, that would contain the true parameter. A 95% confidence level implies that 95% of similarly constructed intervals would cover the true value.

Example. A study reports a mean weight loss of 5 kg with a 95% CI of [3, 7]. If we repeated the study many times, 95% of resulting intervals would contain the true population mean.

Variance

Variance. The average squared deviation of values from their mean. A measure of how spread out data is. Its square root is the standard deviation.

Low variance → values cluster near the mean. High variance → values are spread out.

The 68-95-99.7 Rule

For a normal distribution:

• 68% of values fall within 1 standard deviation of the mean.
• 95% fall within 2 standard deviations.
• 99.7% fall within 3 standard deviations.

Example. IQ has a mean of 100 and SD of 15. Then 68% of people score 85–115, 95% score 70–130, and 99.7% score 55–145.

Regression to the Mean

Regression to the mean. When an initial measurement is extreme, the next measurement on the same subjects tends to be closer to the population average - simply because extreme values often include random noise that won’t repeat.

Example. Athletes featured on the cover of Sports Illustrated often have a poor next season. Not because the magazine is cursed, but because making the cover requires an extreme recent performance, and subsequent performances regress toward the athlete’s true mean.

This is a frequent MCAT trap in passages about treatment effects: if the sickest patients were chosen for treatment and they later felt “better,” some of that improvement may just be regression to the mean, not the treatment.

Percent Change

• Doubling = 100% increase.
• Halving = 50% decrease.
• Increase from 20 to 30 = 50% increase (the base is 20, the increase is 10, $\frac{10}{20}$ = 50%).
• Decrease from 30 to 20 = 33% decrease (the base is 30, the decrease is 10, $\frac{10}{30}$ ≈ 33%).

Percent change is asymmetric - going up 50% and then down 50% does not return you to the start. (100 → 150 → 75.)