Question

1. the number of writing instruments in some teachers desks is displayed in the dot plot. which is greater, the mean or the median? explain your reasoning using the shape of the distribution.

Explanation:

Step1: Count the data - points

There is 1 dot at 5, 1 dot at 6, 1 dot at 7, 1 dot at 8, 4 dots at 9, 5 dots at 10, 2 dots at 11, 1 dot at 12. The total number of data - points is $1 + 1+1 + 1+4 + 5+2 + 1=16$.

Step2: Find the median

Since $n = 16$ (an even number), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered data - values. The 8th and 9th ordered data - values. Counting the dots, the 8th and 9th values are both 10. So the median $M = 10$.

Step3: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{k}x_{i}f_{i}}{n}$, where $x_{i}$ is the value and $f_{i}$ is the frequency. $\sum_{i = 1}^{k}x_{i}f_{i}=5\times1 + 6\times1+7\times1 + 8\times1+9\times4 + 10\times5+11\times2+12\times1=5 + 6+7 + 8+36 + 50+22+12 = 146$. And $n = 16$. So $\bar{x}=\frac{146}{16}=9.125$.

Step4: Analyze the distribution shape

The distribution is slightly left - skewed. In a left - skewed distribution, the mean is usually less than the median. But our calculated mean ($9.125$) is less than the median ($10$) which is consistent with our calculations.

Answer:

The median is greater. The median is 10 and the mean is 9.125.