Question

1. the dot plots show how long it took for a dart to hit a dartboard.

dot plot 1
time to hit dartboard in seconds
dot plot 2
time to hit dartboard in seconds
a. are the mean and the median values shown in dot plot 1 the same? why or why not?
b. are the mean and the median values shown in dot plot 2 the same? why or why not?
c. is the median greater for plot 1 or plot 2? explain your reasoning.
d. is the mean greater for plot 1 or plot 2? explain your reasoning.
e. a dart is thrown and it takes 4 seconds to hit the dartboard. a dot is added on plot 1 at 4 s. will this new data value affect the median or the mean more? explain your reasoning.

Explanation:

Step1: Recall mean - median relationship in symmetric data

In a symmetric distribution, the mean and median are equal. Dot Plot 1 appears symmetric, so the mean and median are the same.

Step2: Recall mean - median relationship in skewed data

In a non - symmetric (skewed) distribution, the mean and median are different. Dot Plot 2 is non - symmetric, so the mean and median are not the same.

Step3: Estimate medians for comparison

For Dot Plot 1, assume an equal number of data points on either side of the center value. For Dot Plot 2, the data is more spread out on the right. The median of Dot Plot 1 is around 1.3 - 1.4 seconds and for Dot Plot 2 is around 1.1 - 1.2 seconds. So the median of Plot 1 is greater.

Step4: Estimate means for comparison

The mean is affected by the values and their frequencies. In Dot Plot 1, values are centered around 1.3 - 1.4. In Dot Plot 2, there are some lower values pulling the mean down. So the mean of Plot 1 is greater.

Step5: Analyze the effect of an outlier

The value of 4 seconds in Plot 1 is an outlier. The mean is calculated as $\frac{\sum_{i = 1}^{n}x_{i}}{n}$, and an outlier will change the sum and thus the mean. The median is the middle value (or average of two middle values). Adding one outlier to a set of data with many points will not change the position of the middle value(s) as much as it will change the mean. So it will affect the mean more.

Answer:

a. Yes, because the distribution in Dot Plot 1 is symmetric.
b. No, because the distribution in Dot Plot 2 is not symmetric.
c. Plot 1. The data in Plot 1 is centered at a higher value than Plot 2.
d. Plot 1. The data in Plot 1 has less influence from lower - valued data points.
e. The mean. Since 4 seconds is an outlier and the mean is more sensitive to outliers than the median.