Making Sense of Data Spread
An Interactive Journey from Mean to Standard Deviation
1. The Problem: Why the Average Isn’t Enough
Two datasets can have the exact same average (mean) but look completely different. The mean tells us the center, but it doesn’t describe the data’s spread or consistency. Let’s look at two sets of student test scores.
Class A Scores
Average Score
75
2. The Process: Calculating Spread
To measure spread, we see how far each data point deviates from the mean. But we can’t just average the deviations, as positives and negatives cancel out. The solution is to square them. Let’s walk through it step-by-step.
Our dataset is [60, 70, 75, 80, 90], with a mean of 75. Click below to start.
3. The “Aha!” Moment: Why We Use Standard Deviation
Our calculation gave us a **Variance** of 100. But if our scores are in “points,” the variance is in “points squared”—which is not intuitive. We take the square root of the variance to get the **Standard Deviation**, bringing our measure of spread back to the original units.
Variance
100
points²
Hard to interpret. What is a “square point”?
Standard Deviation
10
points
Easy to interpret! A typical score is 10 points away from the mean of 75.
4. Practical Application: Investment Risk
Standard deviation is crucial for assessing risk. Consider two portfolios with the same average annual return of 8%. Their standard deviation tells a very different story about their volatility and predictability.
Portfolio A: Low Risk
Mean Return: 8%
Std. Deviation: 3%
Most returns fall between 5% and 11%.
Portfolio B: High Risk
Mean Return: 8%
Std. Deviation: 15%
Most returns fall between -7% and 23%.
5. The Rule of Thumb: Normal Distribution
For many types of data that form a bell curve (a “normal distribution”), the standard deviation has a specific, powerful meaning. It tells us what percentage of the data we can expect to find within certain ranges around the mean.