The heights (in inches) of 10

Question

graders were compared to the heights of 10
graders. These heights are shown below.
Graders: 42, 44, 45, 42, 36, 48, 39, 48, 49, 37
Graders: 64, 65, 64, 66, 63, 62, 67, 64, 65, 65

Which of the following statistical values is smaller for the
grader's heights when compared to the
grader's heights?
A.
interquartile range

B.
standard deviation

C.
mean

D.
range

Answer 1

To compare the statistical values of the heights of the two groups of graders, we need to calculate the relevant statistics for both sets of heights.

Heights of the first graders:

Data: 42, 44, 45, 42, 36, 48, 39, 48, 49, 37

1. Range:
   Range = Maximum - Minimum = 49 - 36 = 13

2. Mean:
   Mean = (42 + 44 + 45 + 42 + 36 + 48 + 39 + 48 + 49 + 37) / 10 = 44.0

3. Standard Deviation:
   First, calculate the variance:

   • Find the mean (already found to be 44).
   • Compute each squared deviation from the mean:
     (42 - 44)² = 4, (44 - 44)² = 0, (45 - 44)² = 1, (42 - 44)² = 4, (36 - 44)² = 64, (48 - 44)² = 16, (39 - 44)² = 25, (48 - 44)² = 16, (49 - 44)² = 25, (37 - 44)² = 49
   • Sum of squared deviations = 4 + 0 + 1 + 4 + 64 + 16 + 25 + 16 + 25 + 49 = 204
   • Variance = 204 / 10 = 20.4
   • Standard Deviation = √20.4 ≈ 4.52

4. Interquartile Range (IQR):
   Ordered data: 36, 37, 39, 42, 42, 44, 45, 48, 48, 49
   Q1 (1st Quartile) = Average of 3rd and 4th values = (39 + 42) / 2 = 40.5
   Q3 (3rd Quartile) = Average of 7th and 8th values = (45 + 48) / 2 = 46.5
   IQR = Q3 - Q1 = 46.5 - 40.5 = 6.0

Heights of the second graders:

Data: 64, 65, 64, 66, 63, 62, 67, 64, 65, 65

1. Range:
   Range = Maximum - Minimum = 67 - 62 = 5

2. Mean:
   Mean = (64 + 65 + 64 + 66 + 63 + 62 + 67 + 64 + 65 + 65) / 10 = 64.5

3. Standard Deviation:
   Mean (already found to be 64.5):

   • Compute each squared deviation from the mean:
     (64 - 64.5)² = 0.25, (65 - 64.5)² = 0.25, (64 - 64.5)² = 0.25, (66 - 64.5)² = 2.25, (63 - 64.5)² = 2.25, (62 - 64.5)² = 6.25, (67 - 64.5)² = 6.25, (64 - 64.5)² = 0.25, (65 - 64.5)² = 0.25, (65 - 64.5)² = 0.25
   • Sum of squared deviations = 0.25 + 0.25 + 0.25 + 2.25 + 2.25 + 6.25 + 6.25 + 0.25 + 0.25 + 0.25 = 18.5
   • Variance = 18.5 / 10 = 1.85
   • Standard Deviation = √1.85 ≈ 1.36

4. Interquartile Range (IQR):
   Ordered data: 62, 63, 64, 64, 64, 65, 65, 65, 66, 67
   Q1 = (64 + 64) / 2 = 64
   Q3 = (65 + 65) / 2 = 65
   IQR = 65 - 64 = 1.0

Summary of Results:

• Grader Heights:
  • Range = 13
  • Mean = 44
  • Standard Deviation ≈ 4.52
  • IQR = 6.0
• Second Grader Heights:
  • Range = 5
  • Mean = 64.5
  • Standard Deviation ≈ 1.36
  • IQR = 1.0

Comparison:

Now we compare the values:

• Range: 13 (graders) vs. 5 (second graders) → Grad grads are larger
• Mean: 44 (graders) vs. 64.5 (second graders) → Grad grads are smaller
• Standard Deviation: 4.52 (graders) vs. 1.36 (second graders) → Grad grads are larger
• IQR: 6.0 (graders) vs. 1.0 (second graders) → Grad grads are larger

Thus, when comparing the heights of the two groups, the only statistical value that is consistently smaller for the graders compared to the second graders is the mean.

The answer is: C. mean