CDS Mathematics · Statistics

The Median of a List

Put the values in order; the median is the middle one, or the mean of the middle two when there is an even number of values.

All Statistics notes CDS Mathematics strategy

Why this matters

Fifteen PYQs, the largest page in the chapter. Half are straight computations; the rest ask what happens to the median when values change or are added, or use a given median to find an unknown value. In every case the key is the POSITION of the middle, not the size of the values.

Concept 1 of 2: Finding the median

The median splits the ordered data in half. Only the middle position matters, so sorting and counting is all the work.

Definition

Sort the values. With $n$ odd, the median is the $\dfrac{n + 1}{2}$ th value.
With $n$ even, it is the mean of the $\dfrac n2$ th and $\left(\dfrac n2 + 1\right)$ th values.
For an arithmetic sequence the median is the middle term (or the mean of the middle two), which equals the mean.
If every value is $x$ plus a constant, sort the constants.

Median position

n \text{ odd: } \left(\tfrac{n + 1}{2}\right)\text{th}; \quad n \text{ even: mean of the } \tfrac n2\text{th and } \left(\tfrac n2 + 1\right)\text{th}

Diagram · median = the middle of sorted data

Sort first, then locate the middle position. With an odd count there is one middle value; with an even count the median is the average of the two middle values. It ignores how far the extremes lie — which is why it resists outliers better than the mean.

Worked example

Find the median of

14, 3, 22, 9, 17, 6

, and then of the same list with

30

added.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (II) 2023 — Elementary Mathematics · Q10Easy

Example 1 · Statistics · Median of Ungrouped Data

If the median of observations 12, 1, 8, 54, 61, 28, 45, 35, 21, 17 is M, then what is the value of

2M + 5

Sort before you pick the middle

The middle of the list as printed is not the median. With

n

even, average the two middle values of the SORTED list.

Drill 7 more on finding the median

Concept 2 of 2: How the median responds to changes

The median only looks at the middle position. Changing values far from the middle, or adding one value on each side, leaves it where it was. That same idea lets you place an unknown value using a given median.

Definition

Increasing the largest few values (or decreasing the smallest few) does not change the median, as long as their order relative to the middle is kept.
Adding one value below and one above the median leaves it unchanged.
Missing values that are known to lie below every listed value fill the lowest positions.
Given the median, decide which positions the unknowns must occupy, then solve; reject any solution that breaks the stated order.

Middle of ten values

\text{median} = \tfrac12\left(x_{(5)} + x_{(6)}\right)

Worked example

The sorted data

4, 9, 11, x, x + 2, 20, 25, 31

has median

15

. Find

x

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q57Moderate

Example 2 · Statistics · Median of Ungrouped Data

The median of 19 observations is 30. Two more observations are made and the values of these are 8 and 32. What is the median of the 21 observations ?

Half the change when two values are averaged

If one of the two middle values rises by

1

, the median rises by

\dfrac12

, not

1

. The mean of the whole list rises by only

\dfrac1n

Drill 6 more on how the median responds to changes

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Finding the median
Median position
$n \text{ odd: } \left(\tfrac{n + 1}{2}\right)\text{th}; \quad n \text{ even: mean of the } \tfrac n2\text{th and } \left(\tfrac n2 + 1\right)\text{th}$
How the median responds to changes
Middle of ten values
$\text{median} = \tfrac12\left(x_{(5)} + x_{(6)}\right)$

Watch out for (2)

Sort before you pick the middle
→ Finding the median
Half the change when two values are averaged
→ How the median responds to changes

Test yourself on Statistics

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Statistics chapter test Drill all 15 questions