Question

19, 17, 29, 28, 12, 15, 27, 16, 29, 28, 16, 28
find the mean, median, mode, range, and standard deviation of the new data set obtained after adding 3 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: Add 3 to each data - point

The original data set is \(19,17,29,28,12,15,27,16,29,28,16,28\). The new data set after adding 3 to each value is \(22,20,32,31,15,18,30,19,32,31,19,31\).

Step2: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 12\) and \(\sum_{i=1}^{12}x_{i}=22 + 20+32+31+15+18+30+19+32+31+19+31=300\). So, \(\bar{x}=\frac{300}{12}=25\).

Step3: Calculate the median

First, order the data set: \(15,18,19,19,20,22,30,31,31,31,32,32\). Since \(n = 12\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th ordered values. \(\frac{n}{2}=6\) and \(\frac{n}{2}+1 = 7\). The 6th value is 22 and the 7th value is 30. So, the median is \(\frac{22 + 30}{2}=26\).

Step4: Calculate the mode

The mode is the value that appears most frequently. In the data set \(15,18,19,19,20,22,30,31,31,31,32,32\), the mode is 31.

Step5: Calculate the range

The range is the difference between the maximum and minimum values. The maximum value is 32 and the minimum value is 15. So, the range is \(32-15 = 17\).

Step6: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
\(\sum_{i=1}^{12}(x_{i}-25)^{2}=(22 - 25)^{2}+(20 - 25)^{2}+(32 - 25)^{2}+(31 - 25)^{2}+(15 - 25)^{2}+(18 - 25)^{2}+(30 - 25)^{2}+(19 - 25)^{2}+(32 - 25)^{2}+(31 - 25)^{2}+(19 - 25)^{2}+(31 - 25)^{2}\)
\(=(-3)^{2}+(-5)^{2}+7^{2}+6^{2}+(-10)^{2}+(-7)^{2}+5^{2}+(-6)^{2}+7^{2}+6^{2}+(-6)^{2}+6^{2}\)
\(=9 + 25+49+36+100+49+25+36+49+36+36+36\)
\(=486\).
\(s=\sqrt{\frac{486}{11}}\approx\sqrt{44.1818}\approx6.65\).

Answer:

Mean of the new data set: 25
Median of the new data set: 26
Mode of the new data set: 31
Range of the new data set: 17
Standard deviation of the new data set: 6.65