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. show data
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.
show calculator

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 of value $39.99$, the new sum is $\sum_{i = 1}^{12}x_{i}\approx48.84 + 39.99=88.83$, and the new mean is $\bar{y}=\frac{88.83}{12}\approx7.40$. So the mean increases.

Step2: Recall median concept

For $n = 11$ (an odd - numbered data set), the median is the 6 - th ordered value. When $n = 12$ (an 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 are still values from the original 11 non - olive - oil items (or at least the 6 - th value is from the original 11), and the new median is still based on the middle values of the original relatively low - priced items (with the large value of the olive oil not affecting it as much as the mean). So the median increases, but less than the mean.

Answer:

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