Certainly! Let's break down the process of finding the mean, median, and mode for the given frequency distribution in a clear, step-by-step manner.
### Frequency Distribution Table
| Data | Frequency |
|------|-----------|
| 11 | 1 |
| 12 | 4 |
| 13 | 2 |
| 14 | 4 |
| 15 | 9 |
### Steps to Find the Central Tendencies:
1. Mean:
- The mean is calculated using the formula:
[tex]\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\][/tex]
- Where [tex]\(x\)[/tex] is the data value and [tex]\(f\)[/tex] is the frequency.
- First, calculate the weighted sum of the data values:
[tex]\[
(11 \times 1) + (12 \times 4) + (13 \times 2) + (14 \times 4) + (15 \times 9)
= 11 + 48 + 26 + 56 + 135 = 276
\][/tex]
- Next, calculate the total number of data points (sum of the frequencies):
[tex]\[
1 + 4 + 2 + 4 + 9 = 20
\][/tex]
- Finally, calculate the mean:
[tex]\[
\text{Mean} = \frac{276}{20} = 13.8
\][/tex]
2. Median:
- The median is the middle value of the data set when it is ordered.
- To find the median, we first need to arrange the data points in order based on frequency. The ordered list based on frequency would be:
[tex]\[
11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15
\][/tex]
- Since there are 20 data points (an even number), the median is the average of the 10th and 11th data points.
- The 10th and 11th data points are both 14.
[tex]\[
\text{Median} = \frac{14 + 14}{2} = 14.0
\][/tex]
3. Mode:
- The mode is the data value(s) that appear most frequently.
- From the frequency distribution, we can see that the number 15 appears 9 times, which is more frequent than any other data value.
[tex]\[
\text{Mode} = 15
\][/tex]
### Final Measures of Central Tendency:
- Mean [tex]\( = 13.8 \)[/tex]
- Median [tex]\( = 14.0 \)[/tex]
- Mode [tex]\( = 15 \)[/tex]
I hope this provides a clear solution! If you have any further questions, please feel free to ask.
