Standard deviationIB MYP Maths Extended: Subtopic test

1. 1
   Five students in Singapore record how many minutes it takes them to solve a puzzle: $4,\ 6,\ 7,\ 9,\ 14$. Use technology and the population standard deviation $\sigma$ where needed.

   (a)

   What is the mean time?

   [1 mark]

   • A$7$ minutes
   • B$10$ minutes
   • C$40$ minutes
   • D$8$ minutes

   (b)

   What is the standard deviation of the times, to 3 significant figures?

   [1 mark]

   • A$3.41$ minutes
   • B$3.81$ minutes
   • C$11.6$ minutes
   • D$2.80$ minutes

   (c)

   A sixth student takes exactly $8$ minutes. State the new mean and find the new standard deviation. Say whether the spread has increased or decreased.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher investigates how the standard deviation changes as values spread out. Each list below has five values and mean $10$. List P is $8,9,10,11,12$ with standard deviation $1.41$. List Q is $6,8,10,12,14$ with standard deviation $2.83$. List R is $4,7,10,13,16$ with standard deviation $4.24$. (All standard deviations are given to 3 significant figures.)

   (a)

   List S is $2,6,10,14,18$. Each value in S is exactly 4 times as far from the mean as the matching value in P. What is the standard deviation of S?

   [1 mark]

   • A$2.83$
   • B$4.24$
   • C$5.66$
   • D$11.3$

   (b)

   Which statement describes the pattern in the data?

   [1 mark]

   • AAdding $k$ to every distance from the mean adds $k$ to the standard deviation
   • BMultiplying every distance from the mean by $k$ multiplies the standard deviation by $k$
   • CThe standard deviation stays the same when the values spread out
   • DMultiplying every distance from the mean by $k$ multiplies the standard deviation by $k^2$

   (c)

   List T is $5,7.5,10,12.5,15$. Use the pattern to predict its standard deviation, then verify your prediction using technology.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two courier companies in Nairobi each make five deliveries to the same shop. The delivery times in minutes are: Company X: $20,\ 24,\ 28,\ 32,\ 36$. Company Y: $26,\ 27,\ 28,\ 29,\ 30$.

   (a)

   Find the mean and the standard deviation of the delivery times for each company.

   [3 marks]

   (b)

   The shop must tell customers a delivery time. Compare the two companies and justify which one the shop should choose.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school in Dubai asks the 20 students in Class A how many books they read over the summer. The results are: $0$ books, 2 students; $1$ book, 5 students; $2$ books, 6 students; $3$ books, 4 students; $4$ books, 3 students. Class B (also 20 students) had a mean of $2.4$ books and a standard deviation of $0.7$ books.

   (a)

   (i) Find the mean number of books read by Class A.
   (ii) Use technology to find the standard deviation for Class A.
   (iii) Find how many students read a number of books within one standard deviation of the mean.

   [6 marks]

   (b)

   The librarian says that Class B has better reading habits than Class A. Use the mean and standard deviation of both classes to discuss and evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions