Handling data: Calculate standard deviation.

The idea

Standard deviation measures how far scores typically spread from their mean. First calculate the mean, subtract it from every score, square each deviation and add the squared deviations. Divide by N when the data are treated as the complete population, or by N−1 when estimating a population from a sample, then take the positive square root. For scores 2, 4 and 6, the mean is 4 and the squared deviations total 8: the population standard deviation is √(8÷3), while the sample standard deviation is √(8÷2)=2. State the convention used, show each stage and retain enough precision until the final answer. A smaller value means tighter clustering around the mean; a larger value means greater dispersion.

What earns the marks

• Calculate the arithmetic mean of the full data set before finding each score's deviation from that mean.
• Square every deviation and add the squared values so that positive and negative deviations do not cancel.
• Divide the squared-deviation total by the denominator required by the stated convention: N for a population or N−1 for a sample estimate.
• Take the positive square root of the variance and report the standard deviation with suitable units and precision.
• Interpret the result comparatively: a smaller standard deviation indicates scores clustered more tightly around their mean.

Common mistakes and how to put them right

• Letting positive and negative deviations cancel.

  Put it right: Square each deviation before adding them.

• Using N and N−1 interchangeably.

  Put it right: Identify whether the requested convention is population or sample standard deviation and label the result.

• Stopping at the variance.

  Put it right: Take its positive square root to obtain the standard deviation.

• Rounding intermediate values too early.

  Put it right: Retain calculator precision and round only the final result.

Exam tips

• Write the mean and the squared-deviation total in your working so the calculation method remains clear even if the final arithmetic slips.
• Check which standard-deviation convention the question, supplied formula or calculator mode requires, and state the one used.
• Give the final value to sensible precision and in the same units as the original scores.
• When comparing groups, discuss the standard deviations alongside their means because spread and central tendency answer different questions.