Median from a frequency table

The idea

To find the median from a frequency table, add the frequencies successively from the smallest value until the middle position(s) are reached; the value at those positions gives the median.

Which family is in the middle?

The 36 students stand in a line, from the smallest family to the largest. The median is the family size in the middle of the line.

What is the median family size?

• Family size | Students
• 3 | 3
• 4 | 11
• 5 | 9
• 6 | 7
• 7 | 3
• 8 | 1
• 9 | 1
• 10 | 1
• 4
• 5
• 6.5, the middle of the sizes 3 to 10

Add the frequencies

The students are numbered 1, 2, 3 … in order. The ringed places are in the middle. Add the frequencies from the smallest value, one row at a time, until you reach the ringed places. Do both lists.

Reach the middle places

To find the median from a frequency table, add the frequencies successively from the smallest value until the middle position(s) are reached; the value at those positions gives the median.

There are 36 values, so the median is the average of the 18th and 19th values. Add the frequencies from the top: 3 + 11 = 14 The values at the places 1 to 14 are 3 or 4, so the 14th value is 4. Add the next row: 3 + 11 + 9 = 23 The places 15 to 23 are all 5. The 18th and the 19th are both 5, so the median is 5.

We never wrote out the 36 numbers, and we did not need the rows for 6 to 10.

If the two middle places have different values, take their average. With 20 quiz marks the middle places are the 10th and 11th. If they are 1 and 2, the median is (1 + 2) ÷ 2 = 1.5. With an odd number of values, say 25, there is one middle place: the 13th.

Notes

To find the median from a frequency table, add the frequencies successively from the smallest value until the middle position(s) are reached; the value at those positions gives the median.

Check yourself

25 students got these marks: 1 mark: 5 students, 2 marks: 8, 3 marks: 7, 4 marks: 4, 5 marks: 1. What is the median mark?

Answer: 2

There are 25 students, so the median is the 13th value. 5 students have 1 mark and 5 + 8 = 13, so the 13th student has 2 marks. The median is 2.

A frequency table has 40 values in all. Which places hold the middle values for the median?

20 children have these shoe sizes: size 4: 3 children, size 5: 7, size 6: 6, size 7: 3, size 8: 1. What is the median shoe size?

Answer: 5.5

3 children wear size 4 and 3 + 7 = 10 wear size 4 or 5, so the 10th value is 5. The 11th value is 6. The median is (5 + 6) ÷ 2 = 5.5.

In the family table, why can we stop after adding 3 + 11 + 9 = 23?

• The 19th and 20th. The 19th and 20th have 18 values before them and 20 after them. The two middle places need the same number of values on each side: the 20th and 21st.
• The 20th only. With an even number of values there are two middle places, the 20th and the 21st.
• The 20th and 21st — correct. Yes! 19 values come before the 20th and 19 come after the 21st. The median is the average of the 20th and 21st values.
• The 21st only. There are two middle places, the 20th and the 21st. The median is their average.
• The other rows are not important.. They are part of the data, but they lie after the middle. They cannot change the values at the middle places.
• Because 23 is more than 19, so both middle values are in the row for 5. — correct. Yes! The places 15 to 23 are all 5, and the 18th and 19th are inside them.
• The median is always the third row.. It depends on the frequencies. Here the middle places happen to be in the third row.
• Because 23 is the total number of students.. The total number of students is 36. 23 is only the running total so far.