Experimental Design and Statistical Analysis for Pharmacology and the Biomedical Sciences
to show the relationship between the magnitude of two sets of continuous variables. Figure 1.3 is an example of an X‐Y scatter plot, where Log10 of the molar drug concentration is plotted along the X‐axis and the magnitude of the ensuing response (expressed as % maximal response) is plotted up the Y‐axis. In both cases the sets of values may take any value within the range set along each axis.
Bar charts are typically used to compare values across a few categories. Figure 1.1 is an example of a bar chart where the height of each bar represents the magnitude of the parameter measured (i.e. locomotor activity) according to the category of drug treatment combination administered to the animals used in the study. Consequently, bar charts are very similar to line charts and just convey a different visual impression of the data.
Histograms are similar to bar charts where the frequency of continuous data (Y‐axis) is plotted against the pre‐defined ranges of the values (X‐axis).
Box–whisker plots provide a representation of the key features of a univariate sample of data. The whiskers indicate the range while the box indicates the median and upper and lower quartiles.
Pie charts may be used when you wish to express and compare categories of observations as proportions of a whole. So, if you can set your total to 100%, then each category should reflect a proportion of the total and be expressed as a percentage. I have used pie charts in my explanation of the theory behind Analysis of Variance, where the total size of the chart reflects the total sample variance in the data while the size of the segments reflects the relative sizes of the Between and Within sample variances (see Chapter 15).
3 Numbers; counting and measuring, precision, and accuracy
When we obtain data in our experiments, we either count the occurrence of an event or we make measurements of a specific parameter.
A count can only be a whole number (i.e. an integer), while a measurement may have any number of decimal places depending on either the accuracy of the instrumentation of the equipment used to make the measurement or any subsequent calculation that uses such measurements. So, if we use a typical home room thermometer to measure temperature, then we should be able to observe whole degree Celsius differences in temperature. If we use a laboratory thermometer then, hopefully, we should achieve an accuracy of half a degree Celsius, and accuracy may be further improved to 0.1 of a degree Celsius by using a reasonably priced digital thermometer. However, if we have access to a high‐quality industrial thermometer for use with thermocouples or resistance thermometer probes to measure temperature, then the accuracy may be improved even further. However, no number of decimal places will yield the true temperature as we will always be unsure of what number lies beyond the last digit. Perhaps the best we can hope for, no matter what we are measuring, is to achieve data with the highest level of precision and accuracy available to us.
Precision and accuracy
The term precision reflects the consistency of a series of measurements and is therefore the ability to obtain the same value on multiple occasions. So, the measure of precision is related to the spread of the data; the higher the precision of the data, then the less the spread of the values. The measurement of precision is provided by the coefficient of variation while accuracy reflects the nearness of the measurement to the true value.
Coefficient of variation is calculated as follows; (see Chapter 5 for information regarding calculation of the mean and standard deviation).(3.1)
% Accuracy is calculated by dividing the measured value by the true value and is expressed as a percentage.(3.2)
Consider the following example.
Example 3.1
Two groups of students were asked to make 10 measurements of the pH of a solution (with a known pH of 8.0). The resulting data is shown in the table below.
The respective coefficients of variation show that Group 2 was more precise in their pH measurements compared with the data obtained by Group 1; however, their estimation of the solution's pH was quite poor with a % accuracy of 107.4%. In contrast, Group 1 was 100% accurate in their estimation of the pH, but their measurements were highly variable. Consequently, while Group 2 was very precise, in contrast Group 1 was highly accurate but did not know it due to the inherent variation in their data (see Figure 3.1)!
Table 3.1 Estimation of pH.
| Observation | Group 1 | Group 2 |
|---|---|---|
| 1 | 6.0 | 8.4 |
| 2 | 8.7 | 8.6 |
| 3 | 9.2 | 8.7 |
| 4 | 10.0 | 8.9 |
| 5 | 6.5 | 8.8 |
| 6 | 7.5 | 8.5 |
| 7 | 9.7 | 8.3 |
| 8 | 8.8 | 8.4 |
| 9 | 6.5 | 8.4 |
| 10 | 7.1 | 8.9 |
| Mean | 8.0 | 8.59 |
| St dev | 1.453 | 0.223 |
| Coefficient of variation (%) | 18.2 | 2.6 |
| % Accuracy | 100.0 | 107.4 |
Figure 3.1 Estimation of pH: precision and accuracy. Summary of data from Table 3.1. Group A (open bar) pH = 8.0 ± 1.453 (mean ± St. dev). Group B (shaded bar) pH = 8.59 ± 0.223. Closed circles indicate raw data values for each group.
So, you can have precision without accuracy, but you will never know how accurate you are without precision.