Video Transcript
Malcolm measured the height of 40 plants planted in his school garden. If the mean height of the plants is 140 centimeters and the median height is 110 centimeters, which of the following sentences would explain the difference between the mean and the median of the plants’ heights? A) Malcolm must have made an error in the calculation of the mean. B) A small number of plants have a very small height, which decreased the median. C) A small number of plants have a very big height, which increased the mean. Or D) Malcolm must have made an error in the calculation of the median.
Let’s start by defining what the mean and median values measure. The median is the middle value in a sorted list. If we order the values from least to greatest or greatest to least, the median is the value in the middle. That means there’ll be an equal number of data points below and above the median. And what about the mean? The mean is the sum of all values divided by the number of values you have.
Two of the answer choices rely on us saying that Malcolm made some kind of mistake. And we haven’t been given any information about how Malcolm made these calculations. Before we choose A or D, we would need to know that option B and C are not possible. That is to say, we need to know for sure that it is impossible for a set of 40 plants to have a mean of 140 and a median of 110 before we claim that Malcolm has made an error.
First, let’s think about the median. If we ordered the plant heights from least to greatest, 110 centimeters is the middle. There are 20 plant heights above 110 and 20 plant heights below 110. This is where our median lies. Where would our mean fall? The mean is slightly higher at 140 centimeters.
Option B says a small number of plants have a very small height, which decreased the median. If out of the plants, two or three had very small heights, we could still have 17 heights below 110 but very close to 110. Having a few very small heights would not necessarily decrease the median.
Option C says a small number of plants have a very big height, which increased the mean. If, for example, we had two very large plants, these two really large values would increase the numerator when you’re calculating the mean. It would increase the sum of the values. But the denominator, the number of values, would stay the same. A larger numerator with the same denominator creates a higher value and would make the mean larger. A few very large plants would have a direct impact on increasing the mean. This means that option C could explain the numbers that Malcolm found.
