Let's analyze the data and calculations from your image step by step.

Given Frequency Table:

• The number of meetings per quarter and their corresponding frequencies (resident answers):
  • 1 meeting: 12 residents
  • 2 meetings: 15 residents
  • 3 meetings: 17 residents
  • 4 meetings: 16 residents

These values provide a distribution of the number of meetings attended by the residents.

Mode, Median, Mean, and Standard Deviation:

• Mode: The mode is already calculated as 3. This is correct, as 3 meetings have the highest frequency of 17.
• Median: The median is correctly given as 3, since the middle value of the sorted data is 3.

Mean Calculation:

The mean can be calculated using the formula for a weighted mean: $\text{Mean} = \frac{\sum(f_i \cdot x_i)}{\sum f_i}$ Where:

• $f_i$ are the frequencies
• $x_i$ are the number of meetings

So: $\text{Mean} = \frac{(12 \cdot 1) + (15 \cdot 2) + (17 \cdot 3) + (16 \cdot 4)}{12 + 15 + 17 + 16}$ $\text{Mean} = \frac{12 + 30 + 51 + 64}{60} = \frac{157}{60} \approx 2.62$ The mean should be 2.62, not 2.48 as shown.

Standard Deviation Calculation:

To compute the standard deviation, use the formula: $\text{Standard Deviation} = \sqrt{\frac{\sum f_i (x_i - \mu)^2}{\sum f_i}}$ Where $\mu$ is the mean (2.62).

Let's calculate the variance step by step:

1. $(x_1 - \mu)^2 = (1 - 2.62)^2 = 2.6244$
2. $(x_2 - \mu)^2 = (2 - 2.62)^2 = 0.3844$
3. $(x_3 - \mu)^2 = (3 - 2.62)^2 = 0.1444$
4. $(x_4 - \mu)^2 = (4 - 2.62)^2 = 1.9044$

Now multiply by the frequencies and sum up: $12 \cdot 2.6244 + 15 \cdot 0.3844 + 17 \cdot 0.1444 + 16 \cdot 1.9044 = 31.4928 + 5.766 + 2.4548 + 30.4704 = 70.184$

Finally, divide by the total frequency and take the square root: $\text{Variance} = \frac{70.184}{60} = 1.1697$ $\text{Standard Deviation} = \sqrt{1.1697} \approx 1.08$ The standard deviation should be 1.08, not 1.07.

Summary of Correct Results:

• Mode: 3 (correct)
• Median: 3 (correct)
• Mean: 2.62 (corrected)
• Standard Deviation: 1.08 (corrected)

Would you like more details on any step or have any specific questions?

Here are five questions to deepen your understanding:

1. How would the calculations change if the frequency of the third meeting were different?
2. Can you explain why the mode and median can sometimes be different in other distributions?
3. How do outliers affect the mean and standard deviation?
4. What is the significance of weighted mean in data analysis?
5. How would the data change if all values were multiplied by a constant?

Tip: Understanding how each measure (mean, median, mode, standard deviation) reacts to changes in the data helps in interpreting real-world data sets accurately.