Exploring distance time graphs
Exploring distance time graphs

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Exploring distance time graphs

1.3.2 Time-series graphs: an example

Figure 4
Figure 4 Temperature over the monthly menstrual cycle

Figure 4 shows a time-series graph of a woman’s temperature over her menstrual cycle, published in a pregnancy guide. Each point on the graph represents the temperature taken first thing in the morning. The points are joined by straight lines to give an overall visual indication of the temperature variation over the month. These lines contain no extra information about temperature, however. Since temperature is measured only once each day, the lines do not represent the woman’s actual temperature during the intervening 24 hours, but instead reflect an assumption that the temperature does not vary wildly between measurements.

The important function of this graph is to show the way a woman’s temperature changes over her monthly cycle. You can see that the temperature does not stay at the ‘normal’ body temperature of 37°C, but varies from a minimum of about 36.6°C at the beginning of the cycle to a peak of about 37.1°C near the end, a range of about 0.5°C. Since the variation is small compared with the average temperature, the vertical axis of the graph does not start from zero but covers only the temperature range that normally occurs.

Plotting the graph at this scale shows up variations as small as 0.1°C. which would be lost if the scale covered a wider range. At this level of detail, the time-series graph shows quite clearly the rapid rise (relatively) in temperature that signals that ovulation has taken place. Since a causal link between ovulation and temperature rise has been established by medical research, a woman plotting her daily temperature as a time-series graph can use it to determine when ovulation has occurred. Thus, she can read out the temperature rise from the graph, and bring other knowledge to bear to read in the interpretation that an egg has been released.

This interpretation of the graph also relies on the accuracy with which the temperature readings have been taken and plotted. In this case, the thermometer must be a special ‘fertility’ thermometer which will give readings to an accuracy and resolution sufficient to respond to the relatively small changes taking place. If the measurements were accurate only to within say, 0.5 degrees, or if the graph scale itself did not allow you to plot points any more accurately than to within 0.5 degrees, then the fine detail of the temperature variations would be lost and a relatively reliable indication of ovulation would be less likely.

Time-series graphs give information only at the points that have been explicitly plotted. Even if a line has been drawn joining the plotted points, it can, at best, only represent an informed guess of what is going on between the points. Trends are expressions of confidence that a set of data conforms to some recognisable pattern, rather than being just a set of random and unrelated numbers. A lot of mathematical effort has gone into developing techniques both for interpolation – estimating the values that lie between known points – and extrapolation – estimating values that lie beyond the range of plotted data. But whatever assumptions are made about long-and short-term trends, the measured data are all there is to go on. This point is brought out by Figure 5.

Figure 5
Figure 5 Temperature variation over a day

The daily, or diurnal, temperature variations are similar in men and women.

Temperature data can be plotted as a time series, but this time the graph characterises the daily rather than the monthly rhythm of a woman’s temperature variation. After waking, a woman’s temperature rises rapidly, followed by a dip in the afternoon, a gradual climb to a peak in the evening and then a sharp drop during sleep. The variation over a day is about 1.5°C, significantly greater than the monthly variation. In between the adjacent points on the monthly graph, therefore, there is a lot of hidden activity. But notice that the notion of ‘rapid’ and the visual perception of steepness is dramatically altered by the scale chosen for the vertical axis.

So there is a trade-off. The time-series graph based on a single daily measurement gives no indication of a daily cycle, particularly as it was deliberately taken at the same time each day. On the other hand, readings taken throughout the day will give a more detailed record, but the extent of the daily fluctuations may obscure the single small temperature rise that signals ovulation. Because of the variation of temperature over the course of a day, temperature measurements must be taken at the same time each day for the monthly series to be meaningful. Taking a morning reading one day and an afternoon reading the next could indicate a misleading temperature rise. Similarly, temperatures quoted on their own can be a misleading guide to a woman’s general state of health; a temperature higher than 37°C may indicate a slight fever if it is recorded in the morning, but not if it is recorded in the evening or after ovulation. This is, of course, one of the difficulties – body temperature rises for a variety of reasons.

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