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The following data was collected:

38 32 27 37 33 28 27 38 35 28 34 21
34 29 37 36 25 20 28 41 28 35 36 40
31 25 31 33 34 39 20 29 24 30 31 26
62 23 33 35 27 22 36 39 42 38 29 36
What is the 5 number summary for this data set?

1 answer

To find the five-number summary (minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum) for the data set, we first need to organize the data in ascending order.

Step 1: Organize the data.

The data set is:

38, 32, 27, 37, 33, 28, 27, 38, 35, 28, 34, 21,
34, 29, 37, 36, 25, 20, 28, 41, 28, 35, 36, 40,
31, 25, 31, 33, 34, 39, 20, 29, 24, 30, 31, 26,
62, 23, 33, 35, 27, 22, 36, 39, 42, 38, 29, 36

Ordering the data gives us:

20, 20, 21, 22, 23, 24, 25, 25, 26, 27,
27, 27, 28, 28, 28, 28, 29, 29, 29, 30,
31, 31, 31, 31, 32, 33, 33, 33, 34, 34,
34, 34, 35, 35, 35, 36, 36, 36, 36, 37,
37, 38, 38, 38, 38, 39, 39, 40, 41, 42,
62

Step 2: Identify the five-number summary.

  1. Minimum:

    • The smallest number in the dataset is 20.

  2. Maximum:

    • The largest number in the dataset is 62.

  3. Median (Q2):

    • To find the median, we need the middle number.
    • There are 48 values in the dataset (even number), so the median will be the average of the 24th and 25th values.
    • The 24th and 25th values are both 31.
    • Median (Q2) = (31 + 31) / 2 = 31.

  4. First Quartile (Q1):

    • Q1 is the median of the first half of the data set (the first 24 values).
    • The first half of the data set (sorted) is: 
      20, 20, 21, 22, 23, 24, 25, 25, 26, 27,
27, 27, 28, 28, 28, 28, 29, 29, 29, 30,
31, 31, 31, 31
    • There are 24 values, so Q1 is the average of the 12th and 13th values (both 27).
    • Q1 = (27 + 27) / 2 = 27.

  5. Third Quartile (Q3):

    • Q3 is the median of the second half of the data set (the last 24 values).
    • The second half of the data set (sorted) is: 
      31, 31, 32, 33, 33, 33, 34, 34, 34, 34,
35, 35, 35, 36, 36, 36, 36, 37, 37, 38,
38, 38, 38, 39, 39, 40, 41, 42, 62
    • Again, there are 24 values, so Q3 is the average of the 12th and 13th values (both 36).
    • Q3 = (36 + 36) / 2 = 36.

Final Five-Number Summary:

  • Minimum: 20
  • Q1: 27
  • Median (Q2): 31
  • Q3: 36
  • Maximum: 62

Thus, the five-number summary is: 20, 27, 31, 36, 62.