Frequency Tables and Histograms

A frequency table pairs each value, or each interval of values, with a count of how many times it shows up. A histogram is the picture version: bars sitting over number-line intervals, where the height of each bar is the count of data points inside that interval. Both tools trade the raw list of numbers for a shape you can actually see — where the data piles up, where it thins out, whether it's lopsided.

The payoff shows up fast when you need a mean. Instead of writing out every repeated value, you multiply each value by its frequency, add those products, and divide by the total count. That's the same arithmetic as adding up a long list, just compressed. The same compression works in reverse: if you know the total count and the heights of all but one bar, the missing bar's height is just the total minus the rest, since every bar's count has to add up to the total number of data points.

The trap is assuming the tallest bar marks the median. It doesn't have to. The median is the point where, if you lined up every data point in order, half would sit below it and half above. That depends on counts across the whole histogram, not just which single interval has the most values. A tall bar on the left can still leave more than half the data sitting in the bars to its right, which pushes the median past that bar entirely. This is why histograms connect to the bigger idea of distributions: shape, center, and spread are related but separate questions, and reading one off a picture takes care, not a glance at the tallest bar.