The row ‘value 5, frequency 6’ means six observations have value five. It does not mean the fifth observation is six, and it does not make six the data value. In the main example, the expanded list is 0, 2, 2, 5, 5, 5, 5, 5, 5, 9.

That expansion gives a direct check for a small table. For a large table, use counts to locate positions without writing every observation. The essential interpretation is the same: the value column tells what was observed, and the frequency column tells how often.

Add the frequencies: 1 + 2 + 6 + 1 = 10. This total is N, the number of observations. It is not the number of rows, which is four. Median positions depend on N because the median describes the centre of the full ordered collection.

For even N, locate observations N/2 and N/2 + 1, then average their values. For odd N, locate observation (N + 1)/2. These are positions, not final data values. A tenth-observation calculation does not mean that the median must be the number ten.

Order the value rows from smallest to largest, then form running frequencies. In the example, cumulative counts are 1, 3, 9 and 10. Value zero occupies position one; value two occupies positions two and three; value five occupies positions four through nine; value nine occupies position ten.

The fifth and sixth positions both lie in the block for value five. Their average is five. Reading a cumulative count of nine does not mean the median is nine: it tells us where the block ends. Translate the position back to the corresponding value.

Yes. Keep values 0, 2, 5 and 9 but change frequencies to 1, 4, 3 and 2. Total frequency is still ten. The value two now occupies positions two through five, and value five occupies positions six through eight. The median is therefore (2 + 5)/2 = 3.5.

The median need not be an observed value when the total is even. There is no recorded 3.5 in this example, yet it correctly summarises the two central observations. Do not round to two or five merely because those are the values available in the table.

If values 1, 4 and 8 have frequencies 2, 5 and 2, there are nine observations. The middle position is (9 + 1)/2 = 5. Value one occupies positions one and two, value four occupies positions three through seven, and value eight occupies positions eight and nine.

The fifth observation is four, so median is four. No averaging of two central values is required. The total frequency controls the odd-or-even decision. The number of distinct value labels can be odd while the observation count is even, or vice versa.

An exact-value table identifies the value of each observation block, so the median can be found exactly under the usual definition. A grouped table such as 0 ≤ x < 10 instead hides the individual values within an interval. Locating a median class does not automatically reveal an exact median.

Do not use an interval midpoint and present it as the recovered exact median without an appropriate method and assumptions. This guide’s examples deliberately use exact values. Read whether the first column contains observations or class intervals before transferring a calculation from one kind of table to another.

Question 1: values 1, 3, 8 have frequencies 1, 4, 1; find median. Question 2: values 2, 6, 10 have frequencies 2, 1, 2; find median. Question 3: values 0 and 7 have frequencies 3 and 3; find median. Question 4: can the median of the frequencies answer these questions?

Answers: 1) N = 6, central positions three and four both give three. 2) N = 5, position three gives six. 3) Central observations are zero and seven, so median 3.5. 4) No; those frequencies count observations of the data values. Ask for the position-to-value reasoning as well as the result.

Bring the table and the first line of your child’s working. A note such as ‘takes the median of counts instead of locating observations’ identifies the representation issue. In a small-group 3-pax lesson, students can expand one small table, then use running totals on a larger table without writing every value.

For parents considering Secondary 2 Mathematics tuition in Punggol, ask ‘Which observation positions are in the middle, and what values occupy them?’ Keep those as two separate questions. Once your child can answer both, change the frequencies while leaving the value labels unchanged to test genuine understanding.