The numbers of students in the 9 schools in a district are given below.

Question

(Note that these are already ordered from least to greatest.)
178, 184, 267, 268, 279, 280, 316, 361, 387
Suppose that the number 178 from this list changes to 259. Answer the following.
(A) What happens to the median? If it increases or decreases then by how much
(B) What happens to the mean? If it increases or decreases then by how much

Answer 1

To analyze the changes in the median and mean when the number 178 is changed to 259 in the given list of student enrollments, we start with the original dataset:

Original data:
178, 184, 267, 268, 279, 280, 316, 361, 387

New data after change:
184, 259, 267, 268, 279, 280, 316, 361, 387

Part (A): Changes to the Median

1. Original Median Calculation:

   • The original dataset has 9 entries (odd number), so the median is the middle number.
   • The median is the 5th number: 279.

2. New Median Calculation:

   • The new dataset also has 9 entries (odd number).
   • The median is still the 5th number: 279.

Conclusion for Part (A):

• Since the median remains 279, there is no change to the median.

Part (B): Changes to the Mean

1. Original Mean Calculation:

   • First, we sum the original data: \[ 178 + 184 + 267 + 268 + 279 + 280 + 316 + 361 + 387 = 2220 \]
   • Now, we calculate the mean by dividing the total by the number of entries (9): \[ \text{Original Mean} = \frac{2220}{9} \approx 246.67 \]

2. New Mean Calculation:

   • Next, we sum the new data: \[ 184 + 259 + 267 + 268 + 279 + 280 + 316 + 361 + 387 = 2281 \]
   • Now, we calculate the mean: \[ \text{New Mean} = \frac{2281}{9} \approx 253.44 \]

3. Change in Mean Calculation:

   • The change in the mean is: \[ \text{Change} = 253.44 - 246.67 \approx 6.77 \]

Conclusion for Part (B):

• The mean increases by approximately 6.77.

Summary:

• (A) The median remains 279 (no change).
• (B) The mean increases by approximately 6.77.