Which one of the following is correct if each of the values of the set of data increases by 12?

Correct option is A

When each value in a dataset is increased by a constant (here, 12), the following effects occur:

​Mean Effect: The mean of a dataset is given by:

​If we add 12 to each value in the dataset, then:

Hence, the mean increases by 12.

​Standard Deviation Effect: The standard deviation measures the dispersion of data points from the mean and is given by:​

Since each value increases by 12, the difference X_i - X̄ remains the same, so standard deviation does not change.

​Coefficient of Variation (C.V.) Effect: The coefficient of variation is defined as:

Since the standard deviation remains unchanged, but the mean increases, the C.V. actually decreases, meaning option (3) is incorrect.​

​No Change in C.V.: Since C.V. depends on the mean and standard deviation, and the mean has changed, the C.V. cannot remain the same. Thus, option (4) is incorrect.​

Information Booster:

• Mean Shift Property: When a constant is added to each data point, the mean shifts by the same constant, while dispersion-related measures remain unchanged.
• Effect on Variance & Standard Deviation: Both variance and standard deviation remain the same because the relative distances between data points do not change.
• Effect on Median & Mode: The median and mode also increase by the same constant as the mean when a constant is added to all values.
• Application in Statistics: This property is often used in data normalization and transformations, particularly in standardization techniques.

Additional Knowledge:

(2) Increase the standard deviation by 12. 

• Standard deviation measures how spread out the data is from the mean. Since the difference Xi−XˉX_i - \bar{X}Xi​−Xˉ does not change, standard deviation remains unchanged.
• Adding a constant shifts all data points by the same amount, keeping their relative differences the same.

(3) Increase the coefficient of variation by 12. 

• Coefficient of variation ($C.V.$) is dependent on the ratio of standard deviation to mean.
• Since standard deviation remains the same but the mean increases, C.V. actually decreases instead of increasing.

(4) No change in the value of the coefficient of variation. 

• Since the mean increases while the standard deviation remains constant, the C.V. must change.
• This contradicts the statement that there is no change, making this option incorrect.