Let's work through your question step by step.

(a) Finding the Mean, Median, and Mode

The data provided is in the form of a frequency distribution table. To interpret it, the column $X$ represents the score values, and the column $f$ represents the frequency of each score. Here's how we can calculate the mean, median, and mode.

Table Recap:

Step 1: The Mean

The formula for the mean is:

$\text{Mean} = \frac{\sum (X \times f)}{\sum f}$

I'll now calculate it.

Step 2: The Median

The median is the value where 50% of the scores lie below and 50% lie above. We can find it by locating the cumulative frequency that corresponds to the middle of the distribution.

Step 3: The Mode

The mode is the score with the highest frequency, which is simply the value of $X$ that corresponds to the highest $f$.

(b) Shape of the Distribution

Once we have the mean, median, and mode, we can determine the shape of the distribution based on their relative positions:

• Symmetrical distribution: Mean ≈ Median ≈ Mode
• Positively skewed distribution: Mean > Median > Mode
• Negatively skewed distribution: Mean < Median < Mode

Let me calculate these values for you now.Here are the calculated values for the distribution:

• Mean: 3.0
• Median: 4
• Mode: 4

(b) Shape of the Distribution

The distribution's shape can be inferred as negatively skewed (left-skewed) because the mean (3.0) is less than both the median and mode (which are both 4). In negatively skewed distributions, the mean tends to be pulled toward the lower end of the distribution due to lower scores having some weight.

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. How would the mean change if the frequencies for higher scores increased?
2. What would happen to the shape of the distribution if the median was lower than the mode?
3. How can outliers impact the mean versus the median in skewed distributions?
4. What is the relationship between skewness and the relative positions of the mean and mode?
5. Can a distribution have more than one mode? How would that affect its interpretation?

Tip: Always check for extreme values (outliers) when analyzing the mean, as they can distort the perception of central tendency in skewed distributions.