S.D. of a data is 6. When each observation is increased by 1, then the...
Given:
Standard deviation (SD) of a data = 6
To find:
SD of the new data when each observation is increased by 1
Solution:
When each observation in a data set is increased by a constant value, the mean of the data set also increases by that constant value. However, the standard deviation remains unchanged.
Explanation:
Let's consider a data set with n observations: {x1, x2, x3, ..., xn}
The mean of this data set is given by:
Mean = (x1 + x2 + x3 + ... + xn) / n
When each observation is increased by 1, the new data set becomes: {x1 + 1, x2 + 1, x3 + 1, ..., xn + 1}
The mean of the new data set is given by:
New Mean = [(x1 + 1) + (x2 + 1) + (x3 + 1) + ... + (xn + 1)] / n
To find the standard deviation, we need to calculate the sum of squares of the differences between each observation and the mean, and then take the square root of the average of these squared differences.
Calculation of Standard Deviation:
SD = sqrt[((x1 - Mean)^2 + (x2 - Mean)^2 + (x3 - Mean)^2 + ... + (xn - Mean)^2) / n]
Similarly, for the new data set:
New SD = sqrt[((x1 + 1 - New Mean)^2 + (x2 + 1 - New Mean)^2 + (x3 + 1 - New Mean)^2 + ... + (xn + 1 - New Mean)^2) / n]
Comparison:
We know that the mean of the new data set is increased by 1 compared to the original mean.
New Mean = Mean + 1
Substituting this into the equation for New SD, we get:
New SD = sqrt[((x1 + 1 - (Mean + 1))^2 + (x2 + 1 - (Mean + 1))^2 + (x3 + 1 - (Mean + 1))^2 + ... + (xn + 1 - (Mean + 1))^2) / n]
= sqrt[((x1 - Mean)^2 + (x2 - Mean)^2 + (x3 - Mean)^2 + ... + (xn - Mean)^2) / n]
= SD
Hence, the standard deviation of the new data set remains unchanged, which is equal to the standard deviation of the original data set.
Therefore, the SD of the new data is 6, which is the same as the original SD.
Hence, option B is the correct answer.