3. a. as this distribution in problem 2 has its peak near the center, you could use either mean or median as…

Explanation:

Step1: Recall median concept

To find the median, if the number of data - points $n$ is odd, the median is the $\left(\frac{n + 1}{2}\right)$-th ordered data - point. If $n$ is even, the median is the average of the $\frac{n}{2}$-th and $\left(\frac{n}{2}+1\right)$-th ordered data - points. First, we need to order the data set.

Step2: Recall inter - quartile range (IQR) concept

The first quartile $Q_1$ is the median of the lower half of the data, and the third quartile $Q_3$ is the median of the upper half of the data. The inter - quartile range $IQR=Q_3 - Q_1$.

Step3: Analyze data variation

To describe the variation in the data, we can look at the spread of the data. A large $IQR$ indicates a large spread (varied data), a small $IQR$ indicates a small spread (not very varied data).

However, since the data set is not given in the problem, we assume a general approach. If we had a data set $x_1,x_2,\cdots,x_n$:

1. Order the data: $x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}$
2. Calculate the median:
   • If $n = 2k+1$ (odd), median $=x_{(k + 1)}$ where $k=\frac{n - 1}{2}$
   • If $n = 2k$ (even), median $=\frac{x_{(k)}+x_{(k + 1)}}{2}$ where $k=\frac{n}{2}$
3. Calculate $Q_1$ and $Q_3$:
   • If $n = 2k+1$ (odd), the lower half is $x_{(1)},x_{(2)},\cdots,x_{(k)}$ and the upper half is $x_{(k + 2)},x_{(k+3)},\cdots,x_{(2k + 1)}$. Then $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
   • If $n = 2k$ (even), the lower half is $x_{(1)},x_{(2)},\cdots,x_{(k)}$ and the upper half is $x_{(k + 1)},x_{(k + 2)},\cdots,x_{(2k)}$. Then $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
   • Calculate $IQR = Q_3 - Q_1$
4. Analyze variation:
   • If $IQR$ is large (compared to the range of the data or some expected value), the data is very scattered.
   • If $IQR$ is small, the data is not very scattered.

Answer:

Since the data set is not provided, we cannot give numerical values for the median and inter - quartile range. For part b, if the inter - quartile range is large, the answer is "very scattered (varied)"; if it is moderate, "somewhat so"; if it is small, "not very much so".