Variability

Knowing the center of your data is only half the story. Two classes can both have a mean exam score of 75%, yet tell completely different stories. In one class, every student scored between 70% and 80%. In the other, scores ranged from 30% to 100%. The averages are identical, but the experiences are worlds apart. Variability - how spread out the data is - gives you the other half of the picture.

Range

The range is the simplest measure of spread: the difference between the maximum and minimum values.

Range = Maximum - Minimum

It is easy to calculate but very crude. A single outlier can blow up the range. If 99 students score between 70 and 80 and one student scores 15, the range jumps from 10 to 65 - even though almost all the data is tightly clustered.

Variance

Variance measures how far each data point is from the mean, on average. The calculation involves three steps:

1. Find the mean of the dataset.
2. Subtract the mean from each data point to get the deviation. Square each deviation.
3. Average the squared deviations.

Variance (s²) = Σ$(x - \bar{x})^2$ / (n - 1) for a sample

Why square the deviations? Because raw deviations (positive and negative) cancel each other out and always sum to zero. Squaring ensures all deviations are positive.

The catch: because we squared the deviations, the units of variance are squared too. If your data is in centimeters, the variance is in cm². That makes variance hard to interpret directly, which is why we usually take one more step.

Standard Deviation

Standard deviation (SD or s) is simply the square root of the variance. This brings the units back to match the original data.

Standard deviation = sqrt(variance)

What Standard Deviation Tells You

Think of SD as a “typical distance from the mean.” If the mean height in a class is 170 cm with SD = 5 cm, most students are within about 5 cm of 170 cm. If SD = 15 cm, heights are much more spread out.

Why n - 1?

You may notice the sample variance formula divides by (n - 1) instead of n. This is called Bessel’s correction. A sample tends to underestimate the true population variance because sample data points cluster closer to the sample mean than to the true population mean. Dividing by (n - 1) slightly inflates the estimate to correct for this bias. For the MCAT, just know that sample variance uses (n - 1) and population variance uses N.

Interquartile Range (IQR)

The interquartile range is the range of the middle 50% of the data. It is calculated by finding the first quartile (Q1, the 25th percentile) and third quartile (Q3, the 75th percentile), then subtracting.

IQR = Q3 - Q1

Like the median, the IQR is resistant to outliers. It ignores the top 25% and bottom 25% of the data entirely, focusing only on the central bulk. This makes it the go-to measure of spread for skewed distributions.

Comparing Measures of Variability