A new value moves the median

The idea

Including a new value greater than the median increases the median, and including a value less than the median decreases it.

A new score joins the line

Five quiz scores, in order: 3, 5, 8, 11, 14. The median is the middle one, 8. A new score, 11, joins the line.

What will the new median be?

• Still 8
• Between 8 and 11
• 11, the new score

Include a new card

The cards are in order. The middle card is the median. Choose a new value and press Include. The card takes its place in the line and the middle moves. Try a value more than, less than and equal to the median.

The middle moves towards the new card

Including a new value greater than the median increases the median, and including a value less than the median decreases it.

The data 3, 5, 8, 11, 14 has 5 values. The median is the 3rd: 8. Include 11: now 6 values, 3, 5, 8, 11, 11, 14. The median is the average of the 3rd and 4th: (8 + 11) ÷ 2 = 9.5 The median increased, 8 → 9.5.

Include 2 instead: 2, 3, 5, 8, 11, 14. (5 + 8) ÷ 2 = 6.5 The median decreased, 8 → 6.5. Include 8, the median itself, and it stays 8.

Each time the middle moves half a card towards the new card. What matters is where the new card lands in the line, not how big it is: include 11 or 100 and the median is 9.5 both times. (The mean would not do this: it becomes about 8.67 with 11 but 23.5 with 100.)

Notes

Including a new value greater than the median increases the median, and including a value less than the median decreases it.

Check yourself

The data 2, 5, 7, 9, 12 has median 7. A new value 20 is included. What is the new median?

Answer: 8

The line is now 2, 5, 7, 9, 12, 20: 6 cards. The middle two are the 3rd and 4th: 7 and 9. (7 + 9) ÷ 2 = 8.

Five students of different heights have a median of 150 cm. A new student, 140 cm, joins the line. What happens to the median?

To the data 3, 5, 8, 11, 14 you include 100 once, or you include 11 once. Which gives the bigger median?

The data 4, 6, 9, 15 has median 7.5. A new value 20 is included. What is the new median?

Answer: 9

The line is now 4, 6, 9, 15, 20: 5 cards. The middle one is the 3rd: 9. The median increased from 7.5 to 9.

• It goes up, because there is one more student.. A student shorter than the median stands on the left of the middle. The middle moves left, not right.
• It goes down, but stays above 140 cm. — correct. Yes! The median becomes the average of 150 and the height just below it, so it is less than 150 but still more than 140.
• It stays 150 cm, because the new student is not in the middle.. The new student stands on the left of the middle, so the middle card changes. The median moves half a place to the left.
• It becomes 140 cm, the new height.. One new value does not become the median. The middle only moves half a card towards it.
• Including 100, because 100 is much bigger than 11.. The median does not care how big the new value is, only which side of the middle it lands. With 100 the middle two cards are 8 and 11: median 9.5.
• Including 11, because it is closer to the middle.. With 11 the line is 3, 5, 8, 11, 11, 14: the middle two are 8 and 11, median 9.5. The same as with 100.
• Both give the same median, 9.5. — correct. Yes! Both land on the right of the middle and the middle two cards are 8 and 11 in both lines. The mean would not be the same.
• Neither changes the median.. Both are more than the median 8, so the median goes up, from 8 to 9.5.