4. A given data set has a mean and standard deviatio questions. a) What are the

Question

4. A given data set has a mean and standard deviatio questions. a) What are the new values of the mean and standard deviation if the same constant k is added to n . Answer each of the following each data value in the given set? Explain. (6 pts.) b) What a is multiplied by the same constant k? Explain. (6 pts.) re the new values of the mean and standard deviation if each data value of the data set o) If the standard deviation of a given data set is zero, what can you say about the data values in the dataset? (4 pts.) 4 mean s all v.bes mulJheyr ar«W

Explanation / Answer

4) a) If we add the same constant k to all data values included in a dataset, we obtain a new data set whose mean is the mean of the original data set PLUS k. The standard deviation does not change.

b) If we multiply all data values included in a data set by a constant k, we obtain a new data set whose mean is the mean of the original data set TIMES k and the standard deviation of the original data set TIMES the absolute value of k.

' = (kx + ky + kz) / 3 = k

' = [ ((kx - k)2 +(ky - k)2+(kz - k)2)/3 ] = |k|

c) We limit the discussion to a data set with 4 values for simplicity, but the conclusions are true for any data set with quantitative data.

Let x, y, z and w be the data values making a data set with mean .

The standard deviation = [ ((x - )2 + (y - )2 + (z - )2 + (w - )2)/3 ]

Let = 0, hence

[ ((x - )2 + (y - )2 + (z - )2 + (w - )2)/3 ] = 0

Which gives

(x - )2 + (y - )2 + (z - )2 + (w - )2 = 0

All terms in the equation are positive and therefore, the above equation is equivalent to

(x - )2 = 0, (y - )2 = 0, (z - )2 = 0 and (w - )2 = 0.

Which gives

x = y = z = w = : all data values in the set with = 0 are equal.

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