Answer :
The correct statement regarding the measure of dispersion of ungrouped data is:
C. If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change
Here is a detailed explanation step-by-step:
1. Understanding Variance and Standard Deviation:
- The variance of a dataset is a measure of how spread out the numbers in the dataset are. It is calculated as the average of the squared differences from the mean.
- The standard deviation is the positive square root of the variance.
2. Statement A:
- Variance is the mean deviation of each value from the arithmetic mean
- This statement is incorrect because the variance is actually the mean of the squared deviations from the arithmetic mean, not just the mean deviation.
3. Statement B:
- Variance is a positive square root of the standard deviation
- This statement is incorrect because it states exactly the opposite of what is true. The standard deviation is the positive square root of the variance, not the other way around.
4. Statement C:
- If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change
- This statement is correct. When a constant [tex]\( c \)[/tex] is added to every value in the dataset, the mean of the dataset will increase by [tex]\( c \)[/tex], but the individual deviations from the mean do not change. Hence, the squared deviations remain the same, and the variance does not change.
5. Statement D:
- If each value is multiplied by a constant [tex]\( c \)[/tex], then the variance is changed by a factor of [tex]\( c^2 \)[/tex]
- This statement is incorrect because it should be: If each value is multiplied by a constant [tex]\( c \)[/tex], then the variance is changed by a factor of [tex]\( c^2 \)[/tex]. For example, if each value is multiplied by 3, the variance is multiplied by [tex]\( 3^2 = 9 \)[/tex].
Given these explanations, the true statement about the measure of dispersion of ungrouped data is Statement C, which is:
C. If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change.
C. If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change
Here is a detailed explanation step-by-step:
1. Understanding Variance and Standard Deviation:
- The variance of a dataset is a measure of how spread out the numbers in the dataset are. It is calculated as the average of the squared differences from the mean.
- The standard deviation is the positive square root of the variance.
2. Statement A:
- Variance is the mean deviation of each value from the arithmetic mean
- This statement is incorrect because the variance is actually the mean of the squared deviations from the arithmetic mean, not just the mean deviation.
3. Statement B:
- Variance is a positive square root of the standard deviation
- This statement is incorrect because it states exactly the opposite of what is true. The standard deviation is the positive square root of the variance, not the other way around.
4. Statement C:
- If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change
- This statement is correct. When a constant [tex]\( c \)[/tex] is added to every value in the dataset, the mean of the dataset will increase by [tex]\( c \)[/tex], but the individual deviations from the mean do not change. Hence, the squared deviations remain the same, and the variance does not change.
5. Statement D:
- If each value is multiplied by a constant [tex]\( c \)[/tex], then the variance is changed by a factor of [tex]\( c^2 \)[/tex]
- This statement is incorrect because it should be: If each value is multiplied by a constant [tex]\( c \)[/tex], then the variance is changed by a factor of [tex]\( c^2 \)[/tex]. For example, if each value is multiplied by 3, the variance is multiplied by [tex]\( 3^2 = 9 \)[/tex].
Given these explanations, the true statement about the measure of dispersion of ungrouped data is Statement C, which is:
C. If a constant [tex]\( c \)[/tex] is added to each value then, the variance does not change.
