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Heights of four fifth-grade students are 58, 56, 60, and 62 inches. What are the mean, median, and standard deviation of this dataset? Also, determine the standard deviation of the new dataset by adding 2 to each number, i.e., 60, 58, 62, 64.

A. 59, 59, 2.58 and 2.58
B. 59, 59, 2.58 and 4.58
C. 58, 58, 2.58 and 2.58
D. 58, 59, 2.58 and 4.58
E. 59, 58, 2.58 and 4.58

Answer :

To solve this problem, we need to calculate the mean, median, and standard deviation for the given dataset of heights: 58, 56, 60, and 62 inches.

Step-by-step Solution:

  1. Mean:

    The mean is calculated by adding up all the numbers and then dividing by the total count of the numbers.

    [tex]\text{Mean} = \frac{58 + 56 + 60 + 62}{4} = \frac{236}{4} = 59[/tex]

  2. Median:

    The median is the middle number when all numbers are arranged in order. For an even number of observations, it's the average of the two middle numbers.

    Arranging the data: 56, 58, 60, 62.

    Median = [tex]\frac{58 + 60}{2} = \frac{118}{2} = 59[/tex]

  3. Standard Deviation (Original Dataset):

    Standard deviation measures the amount of variation or dispersion in a set of values.

    First, calculate the variance:

    • Calculate the differences from the mean and square them:

      • [tex](58 - 59)^2 = 1[/tex]
      • [tex](56 - 59)^2 = 9[/tex]
      • [tex](60 - 59)^2 = 1[/tex]
      • [tex](62 - 59)^2 = 9[/tex]

    • Then find the average of these squared differences:

      [tex]\text{Variance} = \frac{1 + 9 + 1 + 9}{4} = \frac{20}{4} = 5[/tex]

    • The standard deviation is the square root of the variance:

      [tex]\text{Standard Deviation} = \sqrt{5} \approx 2.58[/tex]

  4. New Dataset by Adding 2 to Each Number:

    New dataset: 60, 58, 62, 64.

    • The mean will increase by 2 for the new data:

      [tex]\text{New Mean} = 59 + 2 = 61[/tex]

    • The standard deviation, however, remains the same when a constant is added to all data points because standard deviation measures spread.

      Thus, the standard deviation of the new dataset is also approximately 2.58.

Conclusion:

  • Original dataset mean is 59, median is 59, and standard deviation is approximately 2.58.
  • New dataset mean is 61, and the standard deviation is still approximately 2.58.

Therefore, the correct option for this question would be the first: O 59, 59, 2.58 and 2.58.

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