Basic Statistics
A news report says the average household income in a city is $120,000. You visit and notice most people seem to be earning around $50,000. How can both be true? Because a handful of millionaires are pulling the average (mean) way up, while most people cluster near $50,000. The median income - the value where half earn more and half earn less - would be much closer to $50,000 and would tell a more honest story. Understanding why mean and median can tell different stories is exactly what the MCAT tests.
Measures of Central Tendency
Mean (Average)
Add all values and divide by the number of values. The mean is sensitive to outliers - a single extreme value can shift it substantially.
Formula: mean = (sum of all values) / n
When to use: Symmetric distributions with no extreme outliers.
Median
The middle value when all data points are arranged in order. If there is an even number of values, the median is the average of the two middle values. The median is resistant to outliers.
When to use: Skewed distributions or data with outliers. Household income, hospital stay duration, and reaction times are typically reported as medians.
Mode
The most frequently occurring value. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal).
When to use: Categorical data (most common blood type, most popular answer choice) or identifying peaks in a distribution.
Measures of Spread
Range
The difference between the largest and smallest values. Simple but heavily influenced by outliers.
Standard Deviation (SD)
Standard deviation measures how spread out the data points are from the mean. A small SD means data points cluster tightly around the mean. A large SD means they are widely scattered.
You do not need to calculate SD on the MCAT (the formula involves square roots of summed squared deviations). You need to interpret it.
Variance
Variance = SD. It is the standard deviation before taking the square root. The MCAT occasionally references variance, but standard deviation is far more common.
The Normal Distribution
The normal distribution (bell curve, Gaussian distribution) is symmetric around the mean. It is defined by two parameters: the mean (center) and the standard deviation (width).
Example: MCAT scores have a mean of 500 and an SD of about 10. So:
- 68% of test-takers score between 490 and 510
- 95% score between 480 and 520
- 99.7% score between 470 and 530
Skewed Distributions
Right-skewed (positive skew): The tail extends to the right. Most values cluster on the left. Mean is pulled right. Examples: income, hospital length of stay, reaction times.
Left-skewed (negative skew): The tail extends to the left. Most values cluster on the right. Mean is pulled left. Examples: age at retirement in a healthy population, easy exam scores.
Key relationship for skewed data: The mean is always pulled toward the tail. The median resists the pull. So the median is a better measure of “typical” for skewed data.
Standard Error and Confidence Intervals
Standard error of the mean (SEM) = SD / sqrt(n). It measures how much the sample mean is expected to vary from the true population mean. As sample size increases, SEM decreases - larger samples give more precise estimates.
95% confidence interval = mean +/- 1.96 x SEM (about mean +/- 2 SEM). This means: if you repeated the experiment many times, 95% of the calculated confidence intervals would contain the true population mean.