Measures of dispersion – range, inter-quartile range and standard deviation.

The idea

Measures of dispersion describe how widely data values are spread. The range is maximum minus minimum, so it is quick but strongly affected by extreme values. The interquartile range (IQR) is upper quartile minus lower quartile and describes the middle 50% of ordered values, making it more robust for skewed data or outliers. Standard deviation is the square root of the mean squared deviation from the mean (using the required population or sample convention). It uses every observation and is useful with the mean, but extreme values can influence it. In context, a low standard deviation in pebble size indicates values clustered near the mean, while a high value indicates greater variation; interpretation still depends on units, sampling and the process being studied.

What earns the marks

• Calculate the range by subtracting the minimum value from the maximum value in a dataset.
• Determine the inter-quartile range by finding the difference between the upper quartile and lower quartile.
• Interpret standard deviation as a measure of how spread out numbers are from the mean, where a lower value indicates greater consistency.
• Evaluate the inter-quartile range as a more reliable measure of spread than the range because it ignores extreme outliers at either end of the dataset.
• Apply standard deviation to geographical contexts, such as comparing variability in climate measurements or the spread of sediment sizes.

Common mistakes and how to put them right

• Using the range to compare datasets with extreme outliers.

  Put it right: Use the inter-quartile range instead to focus on the core data spread.

• Miscalculating quartiles by not ranking the data first.

  Put it right: Order the dataset sequentially before finding the percentiles.

• Assuming a high standard deviation means the data is incorrect.

  Put it right: It simply indicates a wide variation or spread in the geographical phenomena being measured.

Exam tips

• When comparing two datasets, always pair a measure of central tendency with a measure of dispersion to provide a complete analysis.
• If asked to calculate standard deviation, follow the stated population or sample formula, square each deviation, obtain the variance and then take its square root.
• Use the inter-quartile range when your data is represented on a box and whisker plot, as it directly corresponds to the box length.