Question

a classroom of students has their heights measured (in inches) for statistics investigation. the tallest student in the class is 71 inches tall. one student is assigned to record all the values. that student made a mistake when recording one of the values. when they meant to record the 71, they accidentally entered 711. what is the affect of this mistake on the mean and sd?
the mean and sd will be increased by the mistake.
the mean and sd will be decreased by the mistake.
the mean and sd will be unaffected by the mistake.

Explanation:

Step1: Recall the formula for the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. When a value $x$ is replaced by a larger value $y$ (here $x = 71$ and $y=711$), the sum $\sum_{i = 1}^{n}x_{i}$ increases. Since $n$ (number of data - points) remains the same, the mean will increase.

Step2: Recall the formula for the standard - deviation

The standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$. The new value 711 is much further from the original mean (compared to 71) and also changes the new mean. The squared - differences $(x_{i}-\bar{x})^{2}$ for all data points will be affected in a way that increases the overall sum $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}$, so the standard deviation will increase.

Answer:

The mean and SD will be increased by the mistake.