SAT Math: One-variable data: center and spread

One-variable data: centre and spread.

Practise one-variable data: center and spread →Adaptive SAT module

What the test covers

Key ideas

Common mistakes

Do it in Desmos

Enter the list as L = [ ... ] and use mean(L), median(L), stdev(L).

IB link: IB SL 4.2-4.3.

Worked example

Worked example

The data set 4, 7, 7, 9, 13 has one value, 25, added to it. Which statement about the new data set is true?

  1. The mean and median increase by the same amount.
  2. The mean and median do not change.
  3. The median increases by more than the mean does.
  4. The mean increases by more than the median does.

Answer: D: The mean increases by more than the median does.

Old mean \(40/5=8\), old median 7. New mean \(65/6\approx10.83\) (up about 2.83); new median \(\tfrac{7+9}{2}=8\) (up 1). The mean increases by more.

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.

Question 1

The mean of six numbers is 12. When one of the numbers is removed, the mean of the remaining five numbers is 11. What number was removed?

Student-produced response: type your answer.

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Answer: 17

Total before: \(6\times12=72\). Total after: \(5\times11=55\). Removed: \(72-55=17\).

Question 2

The mean of 8 numbers is 15. When one of the numbers is removed, the mean of the remaining 7 numbers is 16. What number was removed?

Student-produced response: type your answer.

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Answer: 8

Total before: \(8\times15=120\). Total after: \(7\times16=112\). Removed: \(8\).

Question 3

The mean of 5 numbers is 16. When one of the numbers is removed, the mean of the remaining 4 numbers is 18. What number was removed?

Student-produced response: type your answer.

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Answer: 8

Total before: \(5\times16=80\). Total after: \(4\times18=72\). Removed: \(8\).

Question 4

The mean of 8 numbers is 20. When one of the numbers is removed, the mean of the remaining 7 numbers is 18. What number was removed?

Student-produced response: type your answer.

Show the answer and solution

Answer: 34

Total before: \(8\times20=160\). Total after: \(7\times18=126\). Removed: \(34\).

Question 5

The mean of 5 numbers is 19. When one of the numbers is removed, the mean of the remaining 4 numbers is 20. What number was removed?

Student-produced response: type your answer.

Show the answer and solution

Answer: 15

Total before: \(5\times19=95\). Total after: \(4\times20=80\). Removed: \(15\).

Question 6

The mean of 5 numbers is 20. When one of the numbers is removed, the mean of the remaining 4 numbers is 21. What number was removed?

Student-produced response: type your answer.

Show the answer and solution

Answer: 16

Total before: \(5\times20=100\). Total after: \(4\times21=84\). Removed: \(16\).

Question 7

The data set \(7, 11, 14, 16, 19\) has one value, 50, added to it. Which statement about the new data set is true?

  1. The median increases by more than the mean does.
  2. The mean and median increase by the same amount.
  3. The mean increases by more than the median does.
  4. The mean and median do not change.
Show the answer and solution

Answer: C: The mean increases by more than the median does.

Old mean \(67/5=13.4\), old median \(14\). New mean \(117/6\approx19.50\); new median \(\tfrac{14+16}{2}=15\). The mean increases by more than the median does.

Question 8

The data set \(5, 11, 13, 15, 19\) has one value, 50, added to it. Which statement about the new data set is true?

  1. The mean and median increase by the same amount.
  2. The mean increases by more than the median does.
  3. The mean and median do not change.
  4. The median increases by more than the mean does.
Show the answer and solution

Answer: B: The mean increases by more than the median does.

Old mean \(63/5=12.6\), old median \(13\). New mean \(113/6\approx18.83\); new median \(\tfrac{13+15}{2}=14\). The mean increases by more than the median does.

Question 9

The data set \(2, 7, 9, 12, 19\) has one value, 40, added to it. Which statement about the new data set is true?

  1. The mean and median increase by the same amount.
  2. The median increases by more than the mean does.
  3. The mean increases by more than the median does.
  4. The mean and median do not change.
Show the answer and solution

Answer: C: The mean increases by more than the median does.

Old mean \(49/5=9.8\), old median \(9\). New mean \(89/6\approx14.83\); new median \(\tfrac{9+12}{2}=10.5\). The mean increases by more than the median does.

Question 10

The data set \(4, 6, 13, 15, 19\) has one value, 30, added to it. Which statement about the new data set is true?

  1. The median increases by more than the mean does.
  2. The mean and median do not change.
  3. The mean increases by more than the median does.
  4. The mean and median increase by the same amount.
Show the answer and solution

Answer: C: The mean increases by more than the median does.

Old mean \(57/5=11.4\), old median \(13\). New mean \(87/6\approx14.50\); new median \(\tfrac{13+15}{2}=14\). The mean increases by more than the median does.

Question 11

The data set \(6, 8, 9, 16, 17\) has one value, 50, added to it. Which statement about the new data set is true?

  1. The median increases by more than the mean does.
  2. The mean increases by more than the median does.
  3. The mean and median do not change.
  4. The mean and median increase by the same amount.
Show the answer and solution

Answer: B: The mean increases by more than the median does.

Old mean \(56/5=11.2\), old median \(9\). New mean \(106/6\approx17.67\); new median \(\tfrac{9+16}{2}=12.5\). The mean increases by more than the median does.

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