Which of the following statements are true? A. The mean of a data set lies…
2024
Which of the following statements are true?
A. The mean of a data set lies between its smallest and greatest observations, inclusive.
B. If all observations in a data set are equal, the mean may differ from their common value.
C. If x is added to every observation in a data set, the mean also increases by x.
D. Dividing every observation by a non-zero number x always leaves the mean unchanged.
Answer: C. A and C only — ConceptFor observations a₁, a₂, …, aₙ, the arithmetic mean is their sum divided by n. It lies in the closed interval from the minimum observation to the…
- A.
A, B and C only
- B.
B, C and D only
- C.
A and C only
- D.
D only
Attempted by 2 students.
Show answer & explanation
Correct answer: C
Concept
For observations a₁, a₂, …, aₙ, the arithmetic mean is their sum divided by n. It lies in the closed interval from the minimum observation to the maximum observation.
Adding the same number x to every observation adds x to the mean, while dividing every observation by a non-zero x divides the mean by x.
Application
Statement A follows from min ≤ mean ≤ max; equality is allowed when all observations are equal.
Statement B is false: if every observation equals k, then the mean (nk)/n = k.
Statement C is true: (sum + nx)/n = mean + x.
Statement D is false as a universal claim: after division by non-zero x, the new mean is mean/x, so it is not always unchanged.
Cross-check
For the data 2, 4 and 6, the mean is 4, which lies between 2 and 6.
Adding 3 gives 5, 7 and 9 with mean 7, an increase of 3; dividing the original data by 2 gives mean 2, not 4.
Result
Therefore, the true statements are A and C only.