Question

here is a data set: (lesson 1 - 12) 5 10 10 10 15 100 a. after studying the data, the researcher realized that the value 100 was meant to be recorded as 15. what happens to the mean and standard deviation of the data set when the 100 is changed to a 15?

Explanation:

Step1: Calculate the original mean

The original data - set is \(5, 10, 10, 10, 15, 100\). The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 6\) and \(\sum_{i=1}^{6}x_{i}=5 + 10+10+10+15+100=150\). So the original mean \(\bar{x}_{1}=\frac{150}{6}=25\).

Step2: Calculate the new mean

The new data - set is \(5, 10, 10, 10, 15, 15\). Now \(\sum_{i = 1}^{6}x_{i}=5 + 10+10+10+15+15=65\). The new mean \(\bar{x}_{2}=\frac{65}{6}\approx10.83\).

Step3: Recall the formula for standard deviation

The formula for the standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\). The large value of 100 in the original data - set has a large squared - deviation from the mean, which contributes significantly to the standard deviation. When 100 is replaced by 15, the squared - deviations of all data points from the new mean are smaller on average. So the standard deviation decreases.

Answer:

The mean decreases from 25 to approximately 10.83 and the standard deviation decreases.