10.2 Scatter Plots, Histograms & Distribution Curves

Read a scatter plot as paired observations

A scatter plot shows pairs of measurements. Each point has an x-coordinate and a y-coordinate belonging to the same observation. For example, an exercise might pair journey length with elapsed time for several fictional trips. Before interpreting a pattern, read both axis labels, scales, units and any legend identifying groups.

A point at $(40,60)$ means x is 40 and y is 60 in the stated units. It does not mean that 60% of observations belong to a 40-unit category. Unlike a bar chart, the horizontal coordinate often represents a measured numerical variable, and point spacing can carry meaning. Do not count an apparent gap as missing observations unless the question explains what was sampled.

Direction, strength and unusual points

A rising cloud of points suggests a positive association: larger x-values tend to occur with larger y-values. A falling cloud suggests a negative association. A widely scattered cloud may show weak association or no clear pattern. A curved relationship can be strong even when it is not well described by a straight line.

Association does not establish causation. A relationship between journey distance and cost is evidence about those observed pairs, not proof that distance alone explains every cost difference. A third variable, such as shipment weight, may influence the measurements. In a CBT question, restrict the conclusion to what the data and accompanying text justify.

An outlier lies away from the main pattern. First check whether the point was read correctly; then decide what the question permits you to infer. An outlier is not automatically a recording error, and it should not be deleted merely because it weakens a convenient conclusion. It can strongly affect a calculated mean or fitted trend.

Worked example: trend and prediction

Consider these fictional pairs, with x measured in kilometers and y in minutes:

Each 10-kilometer increase accompanies a 15-minute increase, so the rate of change is $15/10=1.5$ minutes per kilometer. These points fit $y=1.5x+10$. For x = 25, the rule gives 47.5 minutes. This is interpolation because 25 lies within the observed x-range. A prediction at x = 100 is extrapolation; the rule gives 160, but the observed range alone does not prove that the relationship continues that far.

The intercept 10 is the y-value of the mathematical line at x = 0. It could represent a fixed delay only if the problem supplies that interpretation. Do not turn a fitted parameter into an asserted operational fact. Likewise, the slope's unit is minutes per kilometer; reversing it changes the meaning.

The accompanying frequency profile plots the four delay-class midpoints used below. Its connecting curve is illustrative: the data give class totals, not a measured continuous density or exact within-class counts.

Understand a frequency distribution

A frequency distribution tells how observations are spread across values or intervals. A histogram uses adjacent numerical intervals, while a frequency polygon connects points representing class frequencies. A smooth distribution curve summarizes the overall shape. Read whether the vertical scale represents frequency, relative frequency or density before treating height as a count.

For a histogram with equal class widths and a frequency axis, a higher bar indicates a more frequent interval. With unequal class widths and a density axis, area represents frequency: height must be multiplied by width. For example, a class of width 10 and density 3 represents frequency 30, while a class of width 20 and density 2 represents frequency 40. The shorter second bar contains more observations.

A symmetric distribution has balanced tails around its center. A right-skewed distribution has a longer right tail; a left-skewed distribution has a longer left tail. Do not identify skewness from the side containing the tallest point alone. In many familiar right-skewed examples the mean exceeds the median, but this is a useful pattern rather than a rule that determines every possible dataset.

Cumulative information and percentages

Suppose a fictional delay distribution has 20 observations in 0–10 minutes, 30 in 10–20, 25 in 20–30 and 25 in 30–40, with intervals defined as including their lower endpoint and excluding their upper endpoint. There are 100 observations. The number below 20 minutes is $20+30=50$. The number from 20 up to 40 is also 50. The count below 30 is 75.

A cumulative graph at a class boundary gives a running total. A count between two boundaries is obtained by subtracting their cumulative totals. It is not the sum of the two plotted heights. If only class totals are given, the exact number below a value inside a class, such as 17 minutes, is not known without an additional assumption about the within-class distribution.

Interpret curve areas carefully

For a probability-density curve, total area is one and an interval's area is its probability. The height at one point is not that point's probability. For basic normal-distribution exercises explicitly assuming a normal model, symmetry places half the area on either side of the mean. The familiar approximate 68%, 95% and 99.7% intervals refer to one, two and three standard deviations under that model; they must not be applied to every distribution.

When answering, identify the representation first, then choose the correct operation: read coordinates, compare a trend, add frequencies, subtract cumulative totals or compare areas. This prevents a single visual habit from being misapplied across scatter plots, histograms and distribution curves.