Reading Data Displays

Definition The frequency of a value, or of an interval of values, is the number of observations in a data set that equal that value or fall in that interval. The frequencies of all the distinct values add up to the total number of observations. A relative frequency is a frequency divided by that total, often written as a percent.

Definition A dot plot places each possible value on a number line and draws one dot above that value for each observation equal to it, stacking the dots in a column. The height of a column, counted in dots, is the frequency of that value.

Definition A histogram divides the number line into intervals called bins, usually of equal width, and draws a bar over each bin whose height is the frequency (or relative frequency) of observations in that bin. The display must make clear which endpoint belongs to which bin. A common convention puts a value that falls on a boundary into the bin that starts there, so a bin from $30$ to $40$ contains $30$ but not $40$. Always check the convention a question states. Data shown only as counts per interval are called grouped data.

Definition The scale of a display is the labeling of its axes, the value assigned to each gridline and what the vertical axis counts (frequency, relative frequency or percent). Every bar height or dot count must be read against these labels. When the bins of a histogram have equal width, bar heights compare frequencies directly. When the bins have unequal widths, the display must state whether the height or the area of a bar represents frequency.

Definition The median of a data set is the middle value after the values are sorted from least to greatest. With $n$ values, it is the value in position $\frac{n+1}{2}$ when $n$ is odd and the mean of the two middle values when $n$ is even. The quartiles split the sorted data into four parts with roughly equal numbers of values. The second quartile $Q_2$ is the median. One common convention takes the first quartile $Q_1$ as the median of the values below the median position and the third quartile $Q_3$ as the median of the values above it, leaving out the median itself when $n$ is odd. Other conventions exist and can give slightly different quartiles, so when a question supplies quartiles or shows them on a box plot, use those values.

Definition A box plot is drawn over a number line from five numbers, the minimum, $Q_1$, the median, $Q_3$ and the maximum. A box runs from $Q_1$ to $Q_3$ with a line inside at the median, and whiskers run out to the minimum and the maximum. Some box plots mark outliers as separate points, and then the whiskers stop at the most extreme values that are not outliers.

Definition The interquartile range of a data set is

It is the length of the box in a box plot and measures the spread of the middle half of the data. The range, by contrast, is the maximum minus the minimum.

Definition Recall that the mean is the sum of the values divided by how many values there are. A display determines a quantity only when every data set consistent with the display gives that quantity the same value. A dot plot determines every value, so it determines the mean, the median and the range. A histogram determines the count in each bin and the total, but not individual values, so in general it does not determine the exact mean, median, minimum or maximum. A box plot determines its five numbers, the IQR and the range, but not the number of observations or the mean.