Question

chucky grabbed 11 items in the grocery store. each item had a different price, and the mean was about $4.44. on his way to the register, he added a 12th item: a jug of olive oil for $39.99. how will adding the jug of olive oil affect the mean and median? choose 1 answer: a both the mean and median will increase, but the median will increase by more than the mean. b both the mean and median will increase, but the mean will increase by more than the median. c the mean will increase, and the median will decrease. d the median will increase, and the mean will decrease.

Explanation:

Step1: Recall mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Initially, $n = 11$ and $\bar{x}\approx4.44$, so $\sum_{i=1}^{11}x_{i}\approx4.44\times11 = 48.84$. After adding the 12 - th item with value $x_{12}=39.99$, the new sum is $\sum_{i = 1}^{12}x_{i}\approx48.84 + 39.99=88.83$, and the new mean $\bar{y}=\frac{88.83}{12}\approx7.40$. So the mean increases.

Step2: Analyze median

For $n = 11$ (odd - numbered data set), the median is the 6 - th ordered value. For $n = 12$ (even - numbered data set), the median is the average of the 6 - th and 7 - th ordered values. Since the new value $39.99$ is likely to be the largest value, the 6 - th and 7 - th ordered values in the new data set will be larger than the 6 - th value in the old data set, but the increase in the median is not as large as the increase in the mean because the mean is affected by the extreme value $39.99$ more.

Answer:

B. Both the mean and median will increase, but the mean will increase by more than the median.