Charts for Discrete and Time Series Data

Ungrouped discrete numerical data

Some numerical data arise by counting, so only separate values are possible. For example, the number of pets in a household might be 0, 1, 2, 3 or 4; a household cannot have 2.6 pets. These are discrete values.

The data are ungrouped when each value is kept separate rather than placed into intervals. A frequency table organises the results by listing each possible value alongside the number of times it occurs. If the frequencies for 0, 1, 2, 3 and 4 pets are 2, 5, 6, 4 and 1, the table represents $2+5+6+4+1=18$ households.

Constructing a vertical line chart

A vertical line chart represents each discrete value with its own vertical line. Put the values of the variable on the horizontal axis and frequency on the vertical axis. Above each value, draw a thin line from the horizontal axis to the correct frequency.

The lines remain separate because the values between them are not possible or have not been recorded as individual categories. Do not join their tops: joining successive observations is a feature of a line graph, not a vertical line chart.

Choose an even vertical scale that includes the greatest frequency, label both axes and give the chart a clear title. The value labels on the horizontal axis must also be correctly spaced when they are numerical: the distance from 0 to 1 should represent the same numerical change as the distance from 1 to 2.

Interpreting a vertical line chart

To read a frequency, start at a discrete value on the horizontal axis and move vertically to the top of its line. Then read across to the frequency scale. In the example, the line above 2 reaches 6, so 6 households have 2 pets.

The tallest line identifies the most frequent value, or mode. You can also combine frequencies: the number of households with at least 3 pets is $4+1=5$. Be precise about inequalities—“at least 3” includes both 3 and 4, whereas “more than 3” includes only 4.

As a check, adding all the frequencies should give the total number of observations.

Time series tables

A time series records a variable at successive times, such as a shop's weekly sales or a town's monthly rainfall. A time series table should place the times in chronological order and give the corresponding value, with its unit, for each time.

Before drawing, check what each entry means. A value might be a total over an interval, such as rainfall during a month, or a measurement at a particular instant, such as temperature at midday. The label and unit should make this clear.

Constructing a time series line graph

Put time on the horizontal axis and the measured variable on the vertical axis. Plot each ordered pair $(\text{time},\text{value})$, then join consecutive points with straight line segments. Joining them makes changes from one observation to the next easy to see.

Time must progress from left to right. If observations are equally spaced in time, their points should be equally spaced horizontally. If the time intervals differ, their horizontal positions should represent the actual elapsed time. Choose a suitable vertical scale, plot accurately, and include a title, axis labels and units.

Data for Weekly sales over six weeks

The connecting segments show the movement between recorded observations; they do not prove that every intermediate value was measured. For example, a segment between weekly totals does not turn those totals into daily measurements.

Interpreting change over time

Read an individual value by locating its time and reading across from the plotted point to the vertical scale. To find a change, subtract the earlier value from the later value. A positive result is an increase and a negative result is a decrease.

Look for the overall trend as well as short-term fluctuations. A generally rising series can still contain individual falls. A peak is a local or overall high point, while a trough is a low point. Repeated rises and falls at similar times may suggest a recurring pattern, but the graph alone does not explain its cause.

When comparing periods, use both direction and size: “sales rose” is less informative than “sales rose from 18 to 27, an increase of 9”. Any continuation of a trend beyond the plotted period is only an estimate because the pattern may change.

Vertical line charts

Use for ungrouped discrete numerical data.

• Horizontal axis: separate numerical values.
• Vertical axis: frequency.
• Draw one separate vertical line above each value.
• Do not join the tops of the lines.
• Add frequencies to check the total number of observations.

Read the height above a value to find its frequency. The tallest line gives the mode.

Time series

Record values in chronological order. For a line graph:

1. put time on the horizontal axis;
2. put the variable and its unit on the vertical axis;
3. plot each observation;
4. join consecutive points;
5. label both axes and use a suitable scale.

Interpretation

$\text{change}=\text{later value}-\text{earlier value}$

Identify individual values, increases, decreases, peaks, troughs and the overall trend. Do not confuse a short-term fluctuation with the overall direction, and treat predictions beyond the data as estimates.