Variation: Normal Distribution, Range, and Standard Deviation

During a data collection project, a student is interested in gathering the heights of adult men in their city.

Upon returning to the classroom, the pupils graph the frequency of heights in the sample population. The resulting curve is bell-shaped, with a single peak at the center of which lies the mean.

While a single data point, such as the mean, is crucial to the analysis of these results, so too is the variation. Defined as the dispersion of measurements within a data set, this quantity describes the spread of results, giving a sense of the distance between points.

Additionally, the graph is symmetric, with half of the individuals demonstrating a stature taller than and half shorter than the average. This is referred to as a normal distribution or curve.

To evaluate the variation, they first calculate the range of the results, which is the difference between the highest and lowest heights.

Although the range describes the spread of data, it can be dramatically affected by outliers—like the school’s tallest basketball player—and doesn’t elucidate how measurements are positioned around the mean.

To address this, the student uses an equation to compute a second gauge of variation,