Question

here is a data set. 36, 41, 33, 40, 42, 36, 41, 36, 37, 36. find the mean, median, mode, range, and standard deviation of the new data set obtained after adding - 8 to each value in the data set above. round your answer for the standard deviation to the nearest hundredth. mean of the new data set: median of the new data set: mode of the new data set: range of the new data set: standard deviation of the new data set:

Explanation:

Step1: Find the original mean

The original data - set is \(36,41,33,40,42,36,41,36,37,36\). The sum of the original data - set is \(36 + 41+33 + 40+42+36+41+36+37+36=378\). There are \(n = 10\) data points. The original mean \(\bar{x}=\frac{378}{10}=37.8\). When we add \(- 8\) to each value, the new mean \(\bar{y}=\bar{x}-8=37.8 - 8=29.8\).

Step2: Find the original median

First, arrange the original data - set in ascending order: \(33,36,36,36,36,37,40,41,41,42\). Since \(n = 10\) (an even number), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. The \(\frac{10}{2}=5\)th value is \(36\) and the \((\frac{10}{2}+1)=6\)th value is \(37\). The original median \(M=\frac{36 + 37}{2}=36.5\). The new median is \(36.5-8 = 28.5\).

Step3: Find the original mode

The mode of the original data - set is \(36\) (it appears \(4\) times). The new mode is \(36-8 = 28\).

Step4: Find the original range

The range of the original data - set is \(42−33 = 9\). Adding a constant to each data point does not change the range. So the new range is still \(9\).

Step5: Find the original standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
\(\sum_{i = 1}^{10}(x_{i}-37.8)^{2}=(33 - 37.8)^{2}+4\times(36 - 37.8)^{2}+(37 - 37.8)^{2}+(40 - 37.8)^{2}+2\times(41 - 37.8)^{2}+(42 - 37.8)^{2}\)
\(=(-4.8)^{2}+4\times(-1.8)^{2}+(-0.8)^{2}+(2.2)^{2}+2\times(3.2)^{2}+(4.2)^{2}\)
\(=23.04+4\times3.24 + 0.64+4.84+2\times10.24+17.64\)
\(=23.04+12.96+0.64+4.84+20.48+17.64\)
\(=79.6\)
The original standard deviation \(s=\sqrt{\frac{79.6}{9}}\approx2.97\). Adding a constant to each data point does not change the standard deviation. So the new standard deviation is approximately \(2.97\).

Answer:

Mean of the new data set: \(29.8\)
Median of the new data set: \(28.5\)
Mode of the new data set: \(28\)
Range of the new data set: \(9\)
Standard deviation of the new data set: \(2.97\)