QT – Nov 2015 – L1 – Q4 – Measures of Central Tendency

Discuss the relationships between mean, median, and mode in non-symmetric distributions and calculate percentages using a normal distribution.

Question

(a) State the relationships between the mean, median, and mode in a non-symmetric distribution. (5 Marks)

(b) A large departmental store, BOBO, has analyzed the monthly amount spent by its credit card customers and found that it is normally distributed with a mean of GHC 100 and a standard deviation of GHC 15.

Required:
Determine the percentage of people who will spend:
(i) Over GHC 130 (2 Marks)
(ii) Over GHC 120 (2 Marks)
(iii) Below GHC 70 (2 Marks)
(iv) Between GHC 100 and GHC 130 (2 Marks)
(v) Between GHC 115 and GHC 130 (2 Marks)
(vi) The minimum amount spent by the top 10% of customers (2.5 Marks)
(vii) The minimum amount spent by the top 3% of customers (2.5 Marks)

Answer

(a) Relationships in Non-Symmetric Distribution:

In a positively skewed distribution, the order is:
Mode < Median < Mean
This indicates that the mean is pulled to the right by higher values, causing the distribution to skew right.
In a negatively skewed distribution, the order is:
Mode > Median > Mean
This indicates that the mean is pulled to the left by lower values, causing the distribution to skew left.

(b) Calculations Using Normal Distribution:

Let X be amount spent on credit cards. Then X N(100, 225)

i) Over GHC 130: