To find the mean, median, and mode for the given frequency distribution table (FDT), let's proceed step by step:

Data:

Data (x)	Frequency (f)
43	1
44	3
45	2
46	3
47	11

1. Mean:

The formula for the mean is: $\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}$ Where:

• $x$ is the data value
• $f$ is the frequency of the data value

Step 1: Calculate the sum of $x \cdot f$ for each data point.

$(43 \cdot 1) + (44 \cdot 3) + (45 \cdot 2) + (46 \cdot 3) + (47 \cdot 11) = 43 + 132 + 90 + 138 + 517 = 920$

Step 2: Calculate the total frequency $\sum f$. $1 + 3 + 2 + 3 + 11 = 20$

Step 3: Compute the mean. $\text{Mean} = \frac{920}{20} = 46$

2. Median:

The median is the middle value of the data set when it is arranged in ascending order. For frequency distributions, we locate the median based on cumulative frequencies.

Step 1: Find the total frequency ($N = 20$) and determine the median position: $\text{Median Position} = \frac{N + 1}{2} = \frac{20 + 1}{2} = 10.5$ So, the median is the 10.5th value in the ordered data set.

Step 2: Use the cumulative frequency to locate the median.

• Cumulative frequency for 43: 1
• Cumulative frequency for 44: 4
• Cumulative frequency for 45: 6
• Cumulative frequency for 46: 9
• Cumulative frequency for 47: 20

The 10.5th value falls within the data value 47 (cumulative frequency from 10 to 20).

Thus, the median is 47.

3. Mode:

The mode is the value that occurs most frequently.

From the table, the highest frequency is 11, which corresponds to the data value 47.

So, the mode is 47.

Final Answers:

• Mean = 46
• Median = 47
• Mode = 47

Would you like further details or explanations on any of the calculations?

Here are 5 related questions you can explore:

1. How would the mean change if the frequency of 47 increased by 2?
2. How do you calculate the mean if given grouped data instead of a frequency table?
3. Can a data set have more than one mode?
4. How does the median compare to the mean in a skewed distribution?
5. What is the impact of outliers on the mean and median?

Tip: The mode is the only measure of central tendency that can be used with categorical data.