Question

the scores for 21 students on an exam are summarized in the stemplot below. exam scores 4|9 5 6|9 7|1 2 5 5 6 7 8|0 1 2 4 5 5 6 9 9 9|0 1 4 7 key: 9|0 = 90% on the exam the teacher realizes that a mistake was made and the student whose score was recorded as 49% should have been 69%. if the mistake is corrected, what effect will this have on the mean and median scores? the mean and median will both increase. neither the mean nor the median will change. the mean will increase but the median will decrease. the mean will increase but the median will be unchanged.

Explanation:

Step1: Recall the formula for the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here $n = 21$. The sum of the original scores is $S_{1}$. When we change a score from $49$ to $69$, the new sum of scores $S_{2}=S_{1}- 49 + 69=S_{1}+20$. Since the number of data - points $n$ remains the same and the sum of the data increases, the mean $\bar{x}=\frac{S}{n}$ will increase because $\frac{S_{2}}{n}=\frac{S_{1}+20}{n}=\frac{S_{1}}{n}+\frac{20}{n}$, and $\frac{20}{n}>0$ ($n = 21$).

Step2: Recall the concept of the median

The median is the middle - value when the data is arranged in ascending order. Since $n = 21$ (an odd number), the median is the $(\frac{21 + 1}{2})=11$th value when the data is arranged in ascending order. The score of $49$ is the lowest score. Changing the lowest score from $49$ to $69$ does not change the position of the $11$th value in the ordered list of scores. So the median remains unchanged.

Answer:

The mean will increase but the median will be unchanged.