Skewness and outliersEdexcel International A Level Maths: Subtopic test

1. 1
   The ages of customers at a shop have lower quartile 24 years, median 28 years and upper quartile 40 years. A value is classed as an outlier if it is more than $1.5\times IQR$ above the upper quartile or more than $1.5\times IQR$ below the lower quartile.

   (a)

   Describe the skewness of the distribution using the quartiles.

   [1 mark]

   • ANegative skew
   • BSymmetrical
   • CIt cannot be found from the quartiles
   • DPositive skew

   (b)

   Find the value above which a customer's age is classed as an outlier.

   [1 mark]

   • A$56$
   • B$0$
   • C$64$
   • D$88$

   (c)

   The two oldest customers are aged 62 and 71. Determine which of these ages are outliers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The numbers of text messages sent in one day by a sample of students have mode 8, median 10 and mean 12.

   (a)

   Describe the skewness of the distribution.

   [1 mark]

   • ANegative skew
   • BPositive skew
   • CSymmetrical
   • DNo skew, because the three measures are all close in value

   (b)

   Which statement best explains why the mean is greater than the median?

   [1 mark]

   • AA few students sent a very large number of messages, pulling the mean up.
   • BMost students sent more than 12 messages.
   • CThe mode is smaller than the median.
   • DSome students sent no messages, which lowers the median.

   (c)

   Explain why the median is a more suitable measure of location than the mean for these data.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The daily rainfall, in mm, at a weather station over 15 days has smallest value 0, lower quartile 2, median 5, upper quartile 11 and largest value 40. The next largest value is 19. A value is classed as an outlier if it is more than $1.5\times IQR$ above the upper quartile or more than $1.5\times IQR$ below the lower quartile.

   (a)

   Show that 40 mm is an outlier, and determine whether 19 mm is an outlier.

   [3 marks]

   (b)

   Describe how a box plot of these data would be drawn, giving the values at which each part is placed and showing how any outlier is marked.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The marks $x$ of 20 students in a test have $\sum x=1240$ and $\sum x^2=78\,880$, so the mean is 62 and the standard deviation is 10. The lowest mark is 38, the lower quartile is 55, the median is 64, the upper quartile is 70 and the highest mark is 85. A mark is classed as an outlier if it is more than 2 standard deviations from the mean.

   (a)

   (i) Determine whether the lowest mark and the highest mark are outliers.
   (ii) Use the quartiles to describe the skewness of the marks.

   [6 marks]

   (b)

   Both outliers are removed. Find the mean and standard deviation of the remaining marks, and comment on the effect of removing them.

   [6 marks]

   Total for question 4: 12 marks

End of questions