Dot Plots, Histograms and Distribution Shape

Definition The distribution of a variable describes which values it takes and how often each value, or each range of values, occurs, as counts or as proportions. The distribution of an observed data set is its empirical distribution. It gives each distinct observed value its relative frequency, so with $n$ observations each observation contributes $1/n$. A population distribution, or a probability model for observations not yet made, is a different object. The empirical distribution of a sample is evidence about the population distribution, and how good that evidence is depends on how the sample was drawn. It does not reveal the population distribution exactly, and in a small sample its pattern can differ from the population’s considerably.

Definition A dot plot draws a number line for a numerical variable and places one dot above the value of each observation. Equal values, or values that coincide at the resolution of the plot, are stacked vertically. Since there is one dot per observation, the number of dots is $n$, and for a small data set the plot shows every value.

Definition A histogram displays a numerical variable by dividing an interval that contains all the observations into adjacent, non-overlapping subintervals called bins. Over each bin it draws a bar whose base is the bin, so the bars of neighboring bins touch. A boundary rule assigns every observation to exactly one bin. The rule used here puts each bin’s left endpoint in the bin and its right endpoint in the next bin, except that the last bin also contains its right endpoint. When all bins have the same width, the height of each bar can be the bin’s frequency or its relative frequency, and the vertical axis says which.

Definition For a bin of width $w_j$ that contains $n_j$ observations, the frequency density is

measured in observations per unit of the variable. A histogram drawn with these heights has bar areas $h_j w_j = n_j$, so area shows the count, and

Dividing by $n$ as well gives the relative frequency density $n_j/(n w_j)$, whose bar areas are the relative frequencies and add to 1. With unequal bin widths the heights should be densities, because count heights would make wide bins look larger than their share, and the vertical axis should be labeled as frequency density with its units. With equal widths, density heights are the counts divided by one common width, so the shape is the same either way. A density height is not itself a count or a probability, and a relative frequency density histogram, although its total area is 1, describes the sample, not a known population density.

Definition A data set is symmetric about a center $c$ if reflecting every value through $c$, which sends $x$ to $2c - x$, gives back the same values with the same frequencies. Data are rarely exactly symmetric, so in practice a distribution is called symmetric when the two sides of its display roughly mirror each other. A tail is a thinly populated part of a distribution that stretches away from the bulk of the values. A distribution is skewed to the right, or positively skewed, when its right tail reaches much farther from the bulk than its left tail, and skewed to the left, or negatively skewed, when the left tail does. The direction of skewness is named after the longer tail, not after the side where most observations lie.

Definition The modality of a distribution is the number of clear peaks it shows. A unimodal distribution has one peak, a bimodal distribution has two, and a multimodal distribution has more than two. A distribution whose display shows no clear peak, with bars of roughly equal height, is called approximately uniform. For data, peaks are judged from a display, so the apparent number of peaks can depend on the bins, and small bumps in a small sample can be chance variation.