Residual Plot vs Histogram

A residual plot puts $y - \hat{y}$ on the vertical axis against either $x$ or $\hat{y}$ on the horizontal axis, with a reference line at 0. Both horizontal choices are standard: plotting against $\hat{y}$ rescales the horizontal axis, and reverses it when the slope is negative. You read the plot for shape, not for size: a bend says a straight line is the wrong model for the trend, and a fan says the scatter is not constant across the data.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. Their least-squares line is $\hat{y} = 58 + 4x$, so the residuals plotted above $x = 1$ through $x = 6$ are -2, 6, -5, 4, -7 and 4. They land on both sides of zero, they do not bend, and they do not widen from left to right. With six points that is about as much as such a plot can support, so read it beside the scatterplot rather than on its own.

Here is the sentence to retire: "the residual plot has no upward trend, so hours and score are not associated." A residual plot from a least-squares fit can never show a linear trend. The correlation between the residuals and $x$ is exactly 0 by construction, for these six points and for every other data set, because the line already absorbed the straight-line part of the pattern. The plot answers whether the line has the right shape, not whether the variables are related.

A flat, patternless residual plot is also not proof that the model is correct. It only means nothing obvious is left over, and a small sample can hide a real curve. Curvature is still evidence that a linear model does not belong on the data, and the Fall 2026 course has no topic on transforming data to achieve linearity, so the expected response to a bend is to say a line is not appropriate rather than to re-express the variables.

A histogram cuts the number line into bins of equal width and draws a bar over each bin whose height is the count of values inside it, or the relative frequency. The bars touch because the axis is a continuous scale rather than a list of labels. Every histogram needs a boundary rule, and the usual one puts a value in the bin whose lower bound it meets or exceeds and whose upper bound it falls below.

Thirty exam scores binned by tens give 9 scores in 70 to 80 and 12 in 80 to 90, and three of those students scored exactly 80. Switch to the other convention, where a value belongs to the bin whose upper bound it equals, and those three move down: the counts become 12 and 9, the tallest bar changes, and so does the shape you would report. Nothing in the drawn picture tells you which rule was used.

The misreading is "twelve students scored 85." The bar over 80 to 90 says twelve students scored somewhere in that interval and nothing more. The individual values have been thrown away, which is also why you cannot read a median off a histogram, only the bin the median lands in.

Equal width is doing real work in the definition. When bins differ in width, height stops encoding frequency fairly: a bin of width 20 holding 12 values and a bin of width 10 holding 9 draw bars of height 12 and 9, so the wide one looks bigger, while the densities are $12 / 20 = 0.6$ and $9 / 10 = 0.9$ values per unit and the narrow bin is the more crowded one. With unequal bins the vertical axis has to carry density and you read area, not height.

A histogram also destroys the order the values arrived in, which is what a time plot keeps, and its shape depends on the bin width you choose. Histograms are topic 1.5.