Question

two data sets are shown:
set a: 41, 43, 50, 41, 56, 56, 48, 45, 73, 70
set b: 42, 41, 43, 43, 56, 46, 47, 45, 49, 49
the mean of set a is 50.3. the mean of set b is 46.
how do the mean and median of the data sets change if the value 45 is removed from both sets?
move the options to the blanks to complete the sentences.
after removing 45 from both data sets:
the difference between the median of set a and set b ____ because the median of set __ increases.
the difference between the mean of set a and set b ____ because the mean of set __ increases.
options: increases, decreases, stays the same, a, b

Explanation:

Step1: Analyze Set A (Original)

Set A: 40, 40, 50, 41, 56, 60, 48, 65, 70, 70 (10 elements). Sort: 40, 40, 41, 48, 50, 56, 60, 65, 70, 70. Median (for even n) is average of 5th and 6th: $\frac{50 + 56}{2} = 53$. After removing 40 (now 9 elements): Sort: 40, 41, 48, 50, 56, 60, 65, 70, 70. Median is 5th: 56. So median of A increases from 53 to 56.

Step2: Analyze Set B (Original)

Set B: 40, 41, 43, 45, 60, 66, 67, 65, 49, 49 (10 elements). Sort: 40, 41, 43, 45, 49, 49, 60, 65, 66, 67. Median: $\frac{49 + 49}{2} = 49$. After removing 40 (9 elements): Sort: 41, 43, 45, 49, 49, 60, 65, 66, 67. Median is 5th: 49. So median of B stays same (49).

Step3: Median Difference (Original)

Original median difference: $53 - 49 = 4$. New median difference: $56 - 49 = 7$. So difference increases.

Step4: Mean Calculation (Original)

Set A mean: 60.6 (given). Sum of Set A: $60.6 \times 10 = 606$. Remove 40: new sum = $606 - 40 = 566$. New mean: $\frac{566}{9} \approx 62.89$.

Set B mean: 40? Wait, no, likely typo (original mean of B: let's recalculate. Sum of B: 40+41+43+45+60+66+67+65+49+49 = 40+41=81, +43=124, +45=169, +60=229, +66=295, +67=362, +65=427, +49=476, +49=525. Mean: $\frac{525}{10} = 52.5$ (not 40, maybe typo). After removing 40: sum = $525 - 40 = 485$. New mean: $\frac{485}{9} \approx 53.89$. Wait, original problem says mean of B is 40? Maybe data entry error, but proceed with median first.

Step5: Median Difference Conclusion

Median of A increases, median of B stays. Original median diff: 53 - 49 = 4; new: 56 - 49 = 7. So difference between median of A and B increases.

Step6: Mean Difference (Assume Original Means)

Original mean diff: 60.6 - 52.5 = 8.1 (correcting B's mean). After removal: A mean ≈62.89, B mean ≈53.89. Diff: 62.89 - 53.89 = 9. So mean difference increases? Wait, but let's check with given (maybe B's mean was 52.5, not 40). But the key for median: difference between medians increases because A's median increases. For mean: Let's recalculate Set B sum correctly. Set B: 40,41,43,45,60,66,67,65,49,49. Sum: 40+41=81, +43=124, +45=169, +60=229, +66=295, +67=362, +65=427, +49=476, +49=525. Mean: 52.5. Remove 40: sum=525-40=485, mean=485/9≈53.89. Set A sum: 60.6×10=606, remove 40: 566, mean=566/9≈62.89. Original mean diff: 60.6 - 52.5 = 8.1. New diff: 62.89 - 53.89 = 9. So mean difference increases? Wait, but the problem's first blank for median difference: "increases" because A's median increases. For mean: Let's check the options. Wait, the problem's second part: "The difference between the mean of Set A and Set B ____ because the mean of Set __ increases."

Original mean of A: 60.6, B: 52.5 (corrected). After removal: A mean ≈62.89 (increase), B mean ≈53.89 (increase). But A's mean increase is larger? Wait, original A mean: 60.6, new: ~62.89 (increase by ~2.29). B mean: 52.5 to ~53.89 (increase by ~1.39). So mean difference (A - B) was 8.1, now 9 (increase). So difference increases, and mean of A increases.

Answer:

The difference between the median of Set A and Set B increases because the median of Set A increases.
The difference between the mean of Set A and Set B increases because the mean of Set A increases.