The value of the standard deviation does not change and remains the same.
Given, a retired statistic professor has recorded final exam results for decades. the mean final exam score for the population of a student is 82.4 with a standard deviation of 6.5.
The mean μ = 82.4
The standard deviation σ = √[ ((x - μ)2 + (y - μ)2 + (z - μ)2)/3 ]
we have to find the variance,
We now add a constant k to each data value and calculate the new mean μ'.
μ' = ((x + k) + (y + k) + (z + k)) / 3 = (x + y + z) / 3 + 3k/3 = μ + k
We now calculate the new mean standard deviation σ'.
σ' = √[ ((x + k - μ')2 +(y + k - μ')2+(z + k - μ')2)/3 ]
Note that x + k - μ' = x + k - μ - k = x - μ
also y + k - μ' = y + k - μ - k = y - μ and z + k - μ' = z + k - μ - k = z - μ
Therefore σ' = √[ ((x - μ)2 +(y - μ)2+(z - μ)2)/3 ] = σ
If we add the same constant k to all data values included in a data set, we obtain a new data set whose mean is the mean of the original data set PLUS k. The standard deviation does not change.
Hence, the standard deviation does not change.
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