A population of scores has µ = 50 and σ = 5. If every score in the population is multiplied by 3, then what are the new values for the mean and standard deviation?

The new mean = 3 × (the old mean) = 150

The new standard deviation is also = 3 × (The old standard deviation) = 15

Step-by-step explanation:

µ = 50 and σ = 5

The mean is the sum of variables divided by the number of variables.

Mean = (Σx)/N = µ = 50

x = each variable

N = number of variables

If each variable changed to 3x

Mean becomes

Mean = (Σ(3x))/N = 3 (Σx)/N = 3 × µ = 3 × 50 = 150.

The standard deviation is the square root of variance. And variance is an average of the squared deviations from the mean.

The standard deviation measures the rate of spread of the data set around the mean.

Standard deviation = σ = √[Σ(x - µ)²/N]

x = each variable

µ = mean

N = number of variables

If each variable changed to 3x

Recall µ changed to 3µ

Standard deviation = σ = √[Σ(3x - 3µ)²/N]

σ = √[Σ 3² (x - µ)²/N] = √[(3²)Σ(x - µ)²/N] = 3×√[Σ(x - µ)²/N] = 3 × σ = 3 × 5 = 15

If every score is multiplied by 3, it is logical to reason that the average of the new set of numbers also is 3× the old average.

And the new set of numbers spread out similarly around this new mean, only that the new space of spread is now 3× the old one.

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