The Median | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

Order the data before finding the middle

\text{median position}=\dfrac{n+1}{2}

Symbols

$n$ = number of ordered values (none)
$(n+1)/2$ = middle position (position)

For odd $n$ , there is one middle value.
For even $n$ , the position ends in 0.5; take the mean of the two surrounding values.
In a frequency table, use cumulative counts to locate the middle position. Do not choose the middle printed $x$ -value.
If the table contains grouped intervals, the exact observations are hidden. Cumulative counts can identify the median class, but not an exact median value.

Worked example

Median from cumulative counts

Values 10, 20, 30 and 40 have frequencies 2, 5, 4 and 3. There are 14 values, so use the 7th and 8th positions. The cumulative counts are 2, 7, 11 and 14.

Answer

\text{median}=25

A cumulative-frequency graph uses half the total

The final cumulative frequency is the total number of observations, $n$ . Enter at $n/2$ , move across to the smooth curve and then down to the data axis. This is a continuous estimate, so use $n/2$ , not the discrete-list position $(n+1)/2$ .

Examiner note

One-variable statistics mode is a quick check for the median of a raw list or a discrete frequency table. Enter the values and their frequencies correctly. It cannot recover an exact median from grouped intervals, and unsupported calculator output does not show method.

List or frequency table: locate ordered positions. Cumulative-frequency graph: read at half of the final cumulative frequency.

Common mistake

Sorting is essential. The median is not the middle value as originally written, and it is not automatically the middle value printed in a frequency table.

Now you try

Your turnIndependent check [3 marks]

Values 1, 2, 3, 4 and 5 have frequencies 2, 5, 1, 6 and 2. Find the median.

Show worked answer

There are 16 values, so use the 8th and 9th positions. The cumulative frequencies are 2, 7, 8, 14 and 16. The 8th value is 3 and the 9th is 4.

Answer

\text{median}=(3+4)/2=3.5

Order the data before finding the middle

\text{median position}=\dfrac{n+1}{2}

Symbols

$n$ = number of ordered values (none)
$(n+1)/2$ = middle position (position)

For odd $n$ , there is one middle value.
For even $n$ , the position ends in 0.5; take the mean of the two surrounding values.
In a frequency table, use cumulative counts to locate the middle position. Do not choose the middle printed $x$ -value.
If the table contains grouped intervals, the exact observations are hidden. Cumulative counts can identify the median class, but not an exact median value.

Worked example

Median from cumulative counts

Values 10, 20, 30 and 40 have frequencies 2, 5, 4 and 3. There are 14 values, so use the 7th and 8th positions. The cumulative counts are 2, 7, 11 and 14.

Answer

\text{median}=25

A cumulative-frequency graph uses half the total

Examiner note

List or frequency table: locate ordered positions. Cumulative-frequency graph: read at half of the final cumulative frequency.

Common mistake

Sorting is essential. The median is not the middle value as originally written, and it is not automatically the middle value printed in a frequency table.

Now you try

Your turnIndependent check [3 marks]

Values 1, 2, 3, 4 and 5 have frequencies 2, 5, 1, 6 and 2. Find the median.

Show worked answer

There are 16 values, so use the 8th and 9th positions. The cumulative frequencies are 2, 7, 8, 14 and 16. The 8th value is 3 and the 9th is 4.

Answer

\text{median}=(3+4)/2=3.5