Distributions

A distribution describes how data points are spread out or arranged, and its shape reveals important characteristics about the dataset. Distributions are classified based on measures of central tendency and measures of variability — the two most common shapes to recognize are normal and skewed, plus the special case of a bimodal distribution.

Key Takeaways

• In a normal distribution, the mean, median, and mode coincide at the center, and the 68-95-99.7 rule describes how data spreads across standard deviations.

• In a skewed distribution, the mean, median, and mode diverge — the skew is named for the direction of the tail, and the mean shifts toward that tail.

• A negatively (left-) skewed distribution has a mean lower than its median; a positively (right-) skewed distribution has a mean higher than its median.

• A bimodal distribution has two peaks and often signals two underlying subgroups within the data.

The Normal Distribution

The normal distribution is one of the most common and important distributions in statistics. It is symmetrical and bell-shaped, with the highest point at the center. In a perfectly normal distribution, the mean, median, and mode are all equal and located at the center — most data points cluster around the average, with fewer data points as you move farther away from it.

The 68-95-99.7 Rule

The 68-95-99.7 rule (the empirical rule) describes how data is distributed within standard deviations of the mean in a normal distribution:

The empirical rule, applied to adult height

Mean height = 68 inches (5'8"), standard deviation = 3 inches

This property makes the normal distribution incredibly useful for predicting probabilities and understanding variability. For example, someone who is 74 inches tall falls within two standard deviations above the mean — taller than approximately 95% of adults.

Skewed Distributions

In a skewed distribution, the data is not symmetrically distributed: the mean, median, and mode differ, and the direction of the skew is determined by the tail of the distribution.

Negative (Left) Skew

In a negatively skewed (left-skewed) distribution, the tail extends to the left. Most data points cluster on the higher end of the scale, and the mean is pulled to the left of the median and mode. Because the mean is more sensitive to outliers than the median, the mean of a negatively skewed distribution is lower than the median.

Positive (Right) Skew

In a positively skewed (right-skewed) distribution, the tail extends to the right. Most data points are on the lower end, while a few extreme values pull the mean to the right — so the mean of a positively skewed distribution is higher than the median.

Negative vs. positive skew

Bimodal Distributions

A bimodal distribution has two distinct peaks separated by a valley. A bimodal distribution can technically have only one true mode if one peak is slightly higher than the other — but even when the peaks differ in height, the distribution is still called bimodal. If there is sufficient separation between the two peaks, or relatively little data between them, a bimodal distribution can sometimes be analyzed as two distinct distributions, though this isn't always necessary.

For example, measuring the heights of all adults might produce a distribution with two peaks — one corresponding to male heights and the other to female heights. Recognizing bimodal distributions is important, since they often signal underlying differences in the data that may warrant separate analysis.

Common MCAT Mistakes

• Assuming a symmetric-looking histogram is always normal. A distribution needs the mean, median, and mode to coincide at the center and follow the bell-shaped 68-95-99.7 spread — a roughly symmetric shape alone isn't enough to call it normal.

• Mixing up which direction a skew is named for. Skew is named for the direction the tail extends, not where most of the data sits. A left-skewed (negative) distribution has most data on the higher end, with a long tail stretching left.

• Forgetting which way the mean shifts relative to the median in a skew. The mean is more sensitive to outliers than the median, so it gets pulled toward the tail — lower than the median in a negative skew, higher than the median in a positive skew.

• Treating every two-peaked dataset as two separate distributions. A bimodal distribution is still one distribution with two peaks; splitting it into two distinct distributions is only sometimes useful, not automatic, and depends on how separated the peaks are.

MCAT-Style Concept Check

Question: A dataset of exam scores has a mean of 62, a median of 75, and a mode of 80. Which best describes this distribution?

• A) Normal distribution, because the values are close together

• B) Negatively (left-) skewed, because the mean is lower than the median

• C) Positively (right-) skewed, because the mean is lower than the median

• D) Bimodal distribution, because there are three different central values

Answer: B

Explanation: The mean (62) is lower than the median (75), which is lower than the mode (80). Since the mean is more sensitive to outliers than the median, a small number of very low scores pull the mean down and to the left, producing a long left tail — this is the signature of a negatively (left-) skewed distribution. A normal distribution requires the mean, median, and mode to be equal, which rules out (A). Option (C) reverses the relationship: a positive (right-) skew has the mean higher than the median. Option (D) confuses three different central-tendency measures with two distinct peaks, which is not what bimodal describes.

FAQ

What makes a distribution "normal"?

A normal distribution is symmetrical and bell-shaped, with the mean, median, and mode all equal and located at the center. Data spreads out from that center following the 68-95-99.7 rule: about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

How can I tell if a distribution is negatively or positively skewed?

Look at which direction the tail extends. A negatively (left-) skewed distribution has a tail extending left, with most data clustered on the higher end and the mean pulled below the median. A positively (right-) skewed distribution has a tail extending right, with most data on the lower end and the mean pulled above the median.

Why does skew affect the mean more than the median?

The mean incorporates the magnitude of every value, so extreme values in the tail pull it toward them. The median only depends on position in sorted order, so it's far less affected by how extreme the tail values are — which is why the mean and median diverge in a skewed distribution.

What does a bimodal distribution tell you about a dataset?

A bimodal distribution has two distinct peaks, which often signals that the dataset actually contains two underlying subgroups (for example, male and female heights combined into one dataset). Recognizing this can prompt analyzing the subgroups separately, though it isn't always necessary.