Distributions
High-Yield Summary
- A distribution's shape reveals how data is spread — the key shapes are normal, skewed, and bimodal.
- In a normal distribution, mean = median = mode, all at the center; the 68-95-99.7 rule describes spread by standard deviation.
- In a skewed distribution, the mean is pulled toward the tail — left-skewed lowers the mean below the median, right-skewed raises it above.
- A bimodal distribution has two peaks and often signals two underlying subgroups.
The 68-95-99.7 Rule (Empirical Rule)
±1σ ≈ 68% of data, ±2σ ≈ 95%, ±3σ ≈ 99.7%
- σ = Standard deviation from the mean
- Applies only to a normal (bell-shaped) distribution.
- Example: mean height 68 in, σ = 3 in → 65–71 in covers 68% of adults, 62–74 in covers 95%.
Negative vs. Positive Skew
| Skew direction | Tail / clustering / mean vs. median |
|---|---|
| Negative (left-skewed) | Tail extends left; most data on higher end; mean is lower than median |
| Positive (right-skewed) | Tail extends right; most data on lower end; mean is higher than median |
Key Terms
- Normal distribution
- Symmetrical, bell-shaped; mean, median, and mode coincide at the center.
- Skewed distribution
- Asymmetrical; mean, median, and mode differ, named for the direction of the tail.
- Bimodal distribution
- A distribution with two distinct peaks, often signaling two underlying subgroups.
Common MCAT Trap
- "Left-skewed" and "right-skewed" are named for the tail, not the peak — a left-skewed distribution has most of its data bunched on the right, with a long tail trailing left.
- A bimodal distribution can still have only one true mode if the two peaks differ slightly in height — it's still called bimodal by shape, not by strict mode count.
Quick Recall
A distribution's tail extends to the right. Is the mean higher or lower than the median?
In a normal distribution with mean 68 and σ = 3, what range covers ~95% of the data?