Which two of the following statements are true?

(A) The sum of the deviations from the mean (ignoring algebraic signs) is greater than the sum of the deviations from the median (ignoring algebraic signs)

(B) Standard deviation is independent of change of origin and change of scale

(C) In a symmetrical distribution, mean deviation equals 4/5 of standard deviation

(D) In a symmetrical and bell-shaped distribution, quartile deviation is 1/3 of standard deviation

Choose the correct answer from the options given below:

Correct option is B

· Statement (A): True The sum of deviations from the mean (ignoring signs) is always greater than or equal to the sum of deviations from the median (ignoring signs). This is because the median minimizes the absolute deviations, while the mean minimizes the squared deviations.
· Statement (B): False Standard deviation is independent of the change of origin but not independent of the change of scale. When data is multiplied by a constant, the standard deviation also gets multiplied by that constant.
· Statement (C): True In a symmetrical distribution, the relationship between mean deviation (MD) and standard deviation (SD) is such that MD ≈ 4/5 SD.
· Statement (D): False In a symmetrical bell-shaped distribution, quartile deviation is approximately 2/3 of the standard deviation, not 1/3.
Information Booster:
1. Mean vs. Median for Absolute Deviations: The mean is affected more by extreme values compared to the median, which is why the absolute deviations around the median are always less than or equal to those around the mean. This is a fundamental property used in statistics.
2. Relationship Between Mean Deviation and Standard Deviation in Symmetrical Distributions:
· Mean Deviation ≈4/5​× Standard Deviation This holds true under normal or symmetrical distributions because standard deviation measures spread in squared terms, while mean deviation measures absolute differences.
Additional Knowledge:
1. Statement (B): Standard Deviation and Change of Scale
· Change of Origin: Adding or subtracting a constant to all data points does not affect the standard deviation.
· Change of Scale: Multiplying all data points by a constant also multiplies the standard deviation by the same constant.
2. Statement (D): Quartile Deviation and Standard Deviation Relationship
· Quartile deviation is defined as

· In normal distributions, Quartile Deviation ≈ 2/3× Standard Deviation.