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Topics · Measures of Dispersion

Measures of dispersion in DSE Maths Paper 2

Dispersion is about how spread out a set of data is. You need to find the range, the inter-quartile range, the standard deviation and the variance, read them off diagrams, and know what happens to each when every datum is changed in the same way.

What this topic covers

  • Range and inter-quartile range, including from box-and-whisker diagrams
  • Standard deviation and variance of a data set
  • The effect of adding to or multiplying every datum
  • Comparing the spread of two data sets

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Two sample questions

Sample 1

A set of data consists of the 5 numbers 60, 33, 15, 28 and 24. Find the standard deviation of the data, correct to the nearest 0.01.

  1. A6.79
  2. B15.19
  3. C16.99
  4. D33.97
Show the answer and solution

Answer: B

Worked solution

Mean=60+33+15+28+245=32\displaystyle = \dfrac{60 + 33 + 15 + 28 + 24}{5} = 32Variance=∑(x−mean)25\displaystyle {}= \dfrac{\sum(x - \text{mean})^2}{5}≈230.80\displaystyle {}\approx 230.80Standard deviation=Variance≈15.19\displaystyle = \sqrt{\text{Variance}} \approx 15.19

Sample 2

The standard deviation of a set of numbers is 8. Each number in the set is multiplied by negative 6 and then 13 is added to each resulting number. The variance of the new set of numbers is

  1. A2304.
  2. B2473.
  3. C3721.
  4. D15876.
Show the answer and solution

Answer: A

Worked solution

For y=−6x+13:\displaystyle y = -6x + 13: \quadmultiplying by −6\displaystyle -6 scales the spread by 6\displaystyle 6, and adding 13\displaystyle 13 does not change itσy=6 σx=6×8=48\displaystyle \sigma_y = 6\,\sigma_x = 6 \times 8 = 48Variance=(σy)2=482=2304\displaystyle = (\sigma_y)^2 = 48^2 = 2304

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Original material by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.