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There are twenty classes at Northwestern Middle School and twenty classes at Southeastern Middle School. The number of students in each class at each school is shown in the dot plots below.

Number of Students in Each Class at Northeastern Middle School
A dot plot. A number line going from 20 to 29 labeled Number of Students. There are 3 dots above 20, 5 above 21, 7 above 22, 4 above 23, 1 above 24, and 0 above 25, 26, 27, 28, and 29.

Number of Students in Each Class at Southeastern Middle School
A dot plot. A number line going from 20 to 29 labeled Number of Students. There is 1 dot above 20, 2 above 21, 2 above 22, 4 above 23, 3 above 24, 2 above 25, 2 above 26, 2 above 27, 1 above 28, and 1 above 29.

Michela says that the modes of the two data are the same so the median and mean must also be the same. What is Michela’s error?
The modes are not the same.
The means and medians are not the same.
Only the modes and the medians are the same.
Only the modes and the means are the same.

1 answer

To evaluate Michela's claim, we must first determine the modes, medians, and means of the student counts at both Northwestern Middle School and Southeastern Middle School.

Northwestern Middle School

Dot Plot Summary:

  • 20: 3
  • 21: 5
  • 22: 7
  • 23: 4
  • 24: 1
  • 25: 0
  • 26: 0
  • 27: 0
  • 28: 0
  • 29: 0

Mode: The mode is the number that appears the most frequently. Here, the mode is 22, since it occurs 7 times.

Median: To find the median, we arrange the data in order and find the middle value. The total number of classes is 20 (even), so we find the average of the 10th and 11th values.

Counting the cumulative frequency:

  • 20 occurs 3 times
  • 21 occurs 5 times (3 + 5 = 8)
  • 22 occurs 7 times (8 + 7 = 15)
  • 23 occurs 4 times (15 + 4 = 19)
  • 24 occurs 1 time (19 + 1 = 20)

The 10th and 11th values (counting from the beginning) are both 22. Thus, the median is 22.

Mean: The sum of all students and then dividing by the number of classes:

  • \( (20 \cdot 3) + (21 \cdot 5) + (22 \cdot 7) + (23 \cdot 4) + (24 \cdot 1) = 60 + 105 + 154 + 92 + 24 = 435 \)
  • Number of classes = 20
  • Mean = \( \frac{435}{20} = 21.75 \)

Southeastern Middle School

Dot Plot Summary:

  • 20: 1
  • 21: 2
  • 22: 2
  • 23: 4
  • 24: 3
  • 25: 2
  • 26: 2
  • 27: 2
  • 28: 1
  • 29: 1

Mode: The mode is the most frequently occurring number. The mode here is 23, since it occurs 4 times, more than any other number.

Median: With 20 classes, we again compute the average of the 10th and 11th values: Counting the cumulative frequency:

  • 20 occurs 1 time
  • 21 occurs 2 times (1 + 2 = 3)
  • 22 occurs 2 times (3 + 2 = 5)
  • 23 occurs 4 times (5 + 4 = 9)
  • 24 occurs 3 times (9 + 3 = 12)

The 10th and 11th values in the cumulative order are both 24. Thus, the median is 24.

Mean: The total sum will be calculated similarly:

  • \( (20 \cdot 1) + (21 \cdot 2) + (22 \cdot 2) + (23 \cdot 4) + (24 \cdot 3) + (25 \cdot 2) + (26 \cdot 2) + (27 \cdot 2) + (28 \cdot 1) + (29 \cdot 1) \)
  • \( 20 + 42 + 44 + 92 + 72 + 50 + 52 + 54 + 28 + 29 = 429 \)
  • Number of classes = 20
  • Mean = \( \frac{429}{20} = 21.45 \)

Summary:

  • Northwestern Middle School:
    • Mode: 22
    • Median: 22
    • Mean: 21.75
  • Southeastern Middle School:
    • Mode: 23
    • Median: 24
    • Mean: 21.45

Conclusion:

  • The modes are not the same (22 vs. 23).
  • The medians are not the same (22 vs. 24).
  • The means are also not the same (21.75 vs. 21.45).

Michela's assertion that the modes are the same is incorrect, therefore the correct answer is: "The modes are not the same."