3.4.1, 3.4.2.4 - Central tendency and dispersion in field data

Choosing a central measure

A central measure provides a typical value, but different measures answer different questions and respond differently to unusual observations.

For five sediment widths of 2, 4, 4, 6 and 14 mm, the mean is the sum divided by the number: $\bar{x}=(2+4+4+6+14)/5=6$ mm. It uses every value, so the large width of 14 pulls it upwards. The median is the middle value after sorting, here 4 mm. With an even number, average the two middle values. The mode is the most frequent value, here 4 mm. A dataset can have no repeated mode or more than one equally frequent mode; do not force a unique answer.

The mean suits numerical measurements when a total-based average is useful. The median is less affected by extremes and can be suitable for skewed numerical data or ordered categories whose spacings are not equal. The mode can describe unordered categories such as the most common land use, where adding category labels to calculate a mean has no meaning. A coded satisfaction scale from 1 to 5 is ordered, but its labels do not automatically establish equal differences in experience.

Sorting is essential for a median and should preserve all valid observations. Investigate an unusual value rather than deleting it because it changes the mean; it may reflect a real local process or a recording error. Report units, sample size and the chosen measure so that a reader knows what 'average' means. The central value alone cannot show whether the observations are clustered or highly variable.

Range and interquartile range

Dispersion measures reveal how widely the observations vary, helping us judge what a central value conceals.

The range is maximum minus minimum. The interquartile range (IQR) is upper quartile minus lower quartile, describing the spread of the central portion and giving less weight to extremes. Quartile algorithms differ, so state or follow the supplied convention rather than silently changing it.

In this lesson, sort the data, split into lower and upper halves, excluding the single middle observation when the number is odd, then take the median of each half. For the ordered widths 2, 4, 4, 6, 8, 10, 12, 14 mm, the median is $(6+8)/2=7$ mm. The lower half 2, 4, 4, 6 has lower quartile $Q_1=(4+4)/2=4$. The upper half 8, 10, 12, 14 has $Q_3=(10+12)/2=11$. Thus IQR is $11-4=7$ mm and range is $14-2=12$ mm.

If the last value were 40 instead of 14, the range would rise to 38 mm but these middle-half quartiles would remain 4 and 11. The IQR would still be 7 mm, showing the central spread without concealing the need to report the extreme separately. Range is simple but depends on only two readings and may increase as more observations are sampled. Neither measure describes the shape completely, so inspect the original values or a dispersion graph.

A software quartile function may interpolate differently and yield a different legitimate IQR. For a question prescribing the method above, use that method. For a report, name the convention and use the same one for both sites. Range and IQR retain the measurement's units; a width spread is in mm, not mm².