Answer :
To find the mean, median, and mode of the given data, we'll first calculate these for the original group and then for the combined group.
Original Group Data:
Heights (in cms): 161, 162, 163, 161, 163, 164, 164, 160, 165, 163, 164, 165, 166, 164
Mean:
- Add all the heights together:
[tex]161 + 162 + 163 + 161 + 163 + 164 + 164 + 160 + 165 + 163 + 164 + 165 + 166 + 164 = 2225[/tex] - Divide by the number of data points (14):
[tex]\text{Mean} = \frac{2225}{14} \approx 159.64[/tex]
- Add all the heights together:
Median:
- Arrange the data in ascending order:
160, 161, 161, 162, 163, 163, 163, 164, 164, 164, 164, 165, 165, 166 - With 14 numbers, the median will be the average of the 7th and 8th numbers:
[tex]\text{Median} = \frac{163 + 164}{2} = 163.5[/tex]
- Arrange the data in ascending order:
Mode:
- The mode is the number that appears most frequently. In this case, 164 appears 4 times.
- Therefore, the mode is 164.
Combined Group Data:
Heights in additional group: 160, 166, 159, 168, 167, 170
Combined heights: 160, 161, 161, 162, 163, 163, 163, 164, 164, 164, 164, 165, 165, 166, 160, 166, 159, 168, 167, 170
Mean:
- Sum all heights:
[tex]2225 + 160 + 166 + 159 + 168 + 167 + 170 = 3215[/tex] - Divide by the total number of data points (20):
[tex]\text{Mean} = \frac{3215}{20} = 160.75[/tex]
- Sum all heights:
Median:
- Arrange all 20 heights in ascending order:
159, 160, 160, 161, 161, 162, 163, 163, 163, 164, 164, 164, 164, 165, 165, 166, 166, 167, 168, 170 - The median is the average of the 10th and 11th numbers:
[tex]\text{Median} = \frac{164 + 164}{2} = 164[/tex]
- Arrange all 20 heights in ascending order:
Mode:
- The mode is still 164, as it appears most frequently.
In summary, for the original data, the mean is approximately 159.64, the median is 163.5, and the mode is 164. For the combined group, the mean is 160.75, the median is 164, and the mode remains 164.