Question

d) if 40 is added to each of the data values above, what is the new mean? what is the new standard deviation? what is the new variance? e) if each data value above is multiplied by 15, what is the new mean? what is the new standard deviation? what is the new variance?

Explanation:

Step1: Recall the effect of adding a constant to data

Let the original data be \(x_1,x_2,\cdots,x_n\), with mean \(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i\), standard - deviation \(s=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(x_i-\bar{x})^2}\) and variance \(s^{2}=\frac{1}{n}\sum_{i = 1}^{n}(x_i - \bar{x})^2\). If we add a constant \(c\) to each data value, the new data is \(y_i=x_i + c\) for \(i = 1,2,\cdots,n\).
The new mean \(\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i=\frac{1}{n}\sum_{i = 1}^{n}(x_i + c)=\bar{x}+c\).
The new standard - deviation \(s_y=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(y_i-\bar{y})^2}=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}[(x_i + c)-(\bar{x}+c)]^2}=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(x_i-\bar{x})^2}=s\).
The new variance \(s_y^{2}=s^{2}\).
Given \(c = 40\), if the original mean is \(\bar{x}\), the new mean is \(\bar{x}+40\), the new standard - deviation is the same as the original standard - deviation, and the new variance is the same as the original variance.

Step2: Recall the effect of multiplying data by a constant

If we multiply each data value \(x_i\) by a constant \(k\), the new data is \(z_i=kx_i\) for \(i = 1,2,\cdots,n\).
The new mean \(\bar{z}=\frac{1}{n}\sum_{i = 1}^{n}z_i=\frac{1}{n}\sum_{i = 1}^{n}(kx_i)=k\bar{x}\).
The new standard - deviation \(s_z=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(z_i-\bar{z})^2}=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(kx_i - k\bar{x})^2}=|k|s\).
The new variance \(s_z^{2}=k^{2}s^{2}\).
Given \(k = 15\), if the original mean is \(\bar{x}\), the new mean is \(15\bar{x}\), the new standard - deviation is \(15s\) and the new variance is \(225s^{2}\).

Answer:

d) New mean: Original mean + 40; New standard - deviation: Same as original; New variance: Same as original.
e) New mean: 15×Original mean; New standard - deviation: 15×Original standard - deviation; New variance: 225×Original variance.