L: Data presentation and interpretation: Interpret measures of central tendency and variation, extending to standard deviation. Be able to calculate standard deviation, including from summary statistics.

The idea

This topic requires you to interpret and calculate measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation). You should understand that standard deviation measures the spread of data around the mean, with a larger value indicating greater variability. You must be able to calculate standard deviation from raw data using the formula, and also from summary statistics such as Σx, Σx² and n. For example, given Σx = 120, Σx² = 2000 and n = 10, you can find the mean and then the standard deviation. You should also compare data sets using these measures and explain what they tell you in context, including how outliers affect them.

What earns the marks

• Calculate the mean, median and mode from raw data or summary statistics and interpret them in context.
• Calculate standard deviation using the formula s = √(Σx²/n − (Σx/n)²) or the equivalent form involving deviations from the mean.
• Use summary statistics such as Σx, Σx² and n to find the mean and standard deviation without access to the original data.
• Compare two data sets using measures of central tendency and variation, explaining what the differences indicate about the data.
• Recognise the effect of outliers on the mean, range and standard deviation, and contrast this with the median and interquartile range.

Common mistakes and how to put them right

• Forgetting to take the square root when calculating standard deviation, leaving the variance instead.

  Put it right: Check that the final step is to square root the variance.

• Mixing up the formulas for population and sample standard deviation.

  Put it right: In AQA A-level Mathematics, use the formula for standard deviation as given in the specification, which divides by n.

• Misinterpreting a larger standard deviation as meaning the data are 'better' or 'more consistent'.

  Put it right: A larger standard deviation means greater spread or variability, not necessarily better.

Exam tips

• Show all steps in standard deviation calculations, including the mean and the sum of squares, to gain method marks.
• When comparing data sets, always quote numerical values and relate them to the context.
• Use the correct notation and units throughout, and round final answers appropriately as indicated by the question.