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Calculate the mean, median, and mode from the following data of heights in centimeters of a group of students:

161, 162, 163, 161, 163, 164, 164, 160, 165, 163, 164, 165, 166, 164.

Now suppose that a group of students with heights of 160, 166, 159, 168, 167, and 170 centimeters is added to the original group. Find the mean, median, and mode of the combined group.

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

  1. 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]

  2. 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]

  3. 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

  1. 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]

  2. 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]

  3. 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.

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