Statistics

🔒 Log in to track

high importance~3 Q in Tier 128 formulas⚡ 10 shortcuts5 subtopics

Overview Learn (5)Formulas (28)Examples (5)Practice (68)

All subtopics·Subtopic 2 of 5

Median and mode of raw data

🔒 Log in to track

⏱ 4 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

The median is the middle value after you sort the list: half the values sit below it, half above.

The mode is the value that appears most often. A list has one middle and one most-frequent value, and exam questions mostly test how each behaves when the list changes.

Median: the middle seat

Sort the values. The median is the middle one.

Odd count: the value at position $\dfrac{n+1}{2}$ .
Even count: the average of the two middle values, positions $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$ .

List 12, 5, 9, 17, 3, 11, 7 sorts to 3, 5, 7, 9, 11, 12, 17. Seven values, so the median is the 4th: 9.

List 4, 8, 15, 16, 23, 42 has six values. Median $= \dfrac{15+16}{2} = 15.5$ .

Rule: Sort first, always. The median of an unsorted list is the most common wrong answer in this chapter.

Mode: the crowd favourite

The mode is the value with the highest count. No sorting needed, just tally.

4, 7, 9, 7, 11, 4, 7, 2: the count of 7 is three, every other value appears at most twice. Mode = 7.

A list can have two modes (7 and 4 each appearing three times) or no mode at all (every value once).

Tip: Mode is the only average that must be a value from the list. Mean and median can be numbers that are not in the list.

The empirical relation

For data that is mildly lopsided (like marks or incomes), the three averages follow one rough rule:

\text{mode} = 3 \times \text{median} - 2 \times \text{mean}

Mean 30, median 32: mode $= 3 \times 32 - 2 \times 30 = 36$ . It also rearranges: median $= \dfrac{\text{mode} + 2\,\text{mean}}{3}$ .

Watch: This is an estimate, not an identity. Use it only when two of the three averages are given and the third is asked.

The relation, rearranged

Any one of the three can be the unknown.

Mean 27, mode 45: median $= \dfrac{45 + 2 \times 27}{3} = \dfrac{99}{3} = 33$ .

Sanity check: mean 27 < median 33 < mode 45 rises steadily, the shape the formula assumes.

Tip: After solving, place the three values in order. If they do not rise or fall together, redo the arithmetic.

When every value is transformed

Add the same number to every value: mean, median and mode all shift by that number.

Multiply every value by $k$ : all three get multiplied by $k$ .

Median 15, then every value is doubled and 1 is added: new median $= 2 \times 15 + 1 = 31$ .

Rule: The median follows the same shift and scale as the values themselves.

A missing value from the median

The median pins one value of the sorted list, so it can pin an unknown.

Seven values: 10, 12, 13, $x+4$ , 20, 22, 25. With seven entries the median is the 4th, which is $x+4$ itself:

x + 4 = 15 \Rightarrow x = 11

Example: With an even count the two middle entries average to the median, so their sum is $2 \times$ median.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Median of a raw list

How to spot it:

A jumbled list of values is given; the middle value is asked.

\text{Med} = \left(\frac{n+1}{2}\right)\text{th value}

Method

Sort the values in rising order.
Find the middle position: $\frac{n+1}{2}$ .
Even count: average the two middle values.

Why it works:

Sorting places half the values on each side of the answer.

Try this

Find the median of 12, 5, 9, 17, 3, 11 and 7.

Show solution

Sorted: 3, 5, 7, 9, 11, 12, 17.
$n = 7$ , so the median is the 4th value.
4th value $= 9$ .

Answer

Type 2common2 practice Q

Mode of a raw list

How to spot it:

A list with repeats is given; the most frequent value is asked.

\text{Mode} = \text{value with maximum frequency}

Method

Tally how often each value appears.
Pick the value with the highest count.
Two ties means two modes; all equal means no mode.

Why it works:

The mode counts frequency only; size and order do not matter.

Try this

Find the mode of 4, 7, 9, 7, 11, 4, 7, 2.

Show solution

Counts: 7 appears 3 times, 4 appears twice.
All other values appear once.
Highest count is 7.

Answer

Type 3very common4 practice Q

Empirical relation

How to spot it:

Two of mean, median, mode are given; the third is asked.

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Method

Write the relation.
Substitute the two known averages.
Solve for the third.

Why it works:

The three averages keep a fixed rough spacing in mildly skewed data.

Try this

For a distribution, the mean is 30 and the median is 32. Using the empirical relation, the mode is:

Show solution

$\text{Mode} = 3 \times 32 - 2 \times 30$ .
$= 96 - 60 = 36$ .

Answer

Type 4common2 practice Q

Median under a shift or scale

How to spot it:

Every value of a list is doubled, halved or shifted; the new median is asked.

\text{Med}(kx + c) = k\,\text{Med}(x) + c

Method

Find (or note) the median of the original list.
Apply the same multiply and add to that median.
Order of operations: scale first, then shift.

Why it works:

The middle position does not move when all values move together.

Try this

The median of a list is 15. Each value is doubled and then 1 is added to it. The new median is:

Show solution

New median $= 2 \times 15 + 1$ .
$= 31$ .

Answer

Type 5occasional

A missing value fixed by the median

How to spot it:

A list contains x or an expression in x; the median is given.

\text{middle entry} = \text{median}

Method

Sort so the unknown's slot is clear.
Set the middle entry (or the middle pair's average) equal to the median.
Solve for x.

Why it works:

The median names one exact entry of the sorted list when n is odd.

Try this

The median of 10, 12, 13, $x+4$ , 20, 22, 25 is 15. Find x.

Show solution

Seven values, so the 4th value is the median.
The 4th value is $x + 4 = 15$ .
$x = 11$ .

Answer

x = 11

Formula sheet

Median (odd n)

\text{value at } \frac{n+1}{2}\text{th place}

Position in the sorted list.

Median (even n)

\frac{\text{(n/2)th} + \text{(n/2+1)th}}{2}

Empirical relation

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Rough rule for mildly skewed data; rearrange for any missing one.

Median from mode and mean

\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3}

Transform

y = kx + c \Rightarrow \text{Med}_y = k\,\text{Med}_x + c

Shortcuts that save time

⚡ Count positions, do not hunt

For odd n, the median sits at position (n+1)/2 of the sorted list. Count to that position instead of scanning for the middle by eye.

Example

Find the median of 3, 5, 7, 9, 11, 12, 17.

Show solution

$n = 7$ , position $= \dfrac{7+1}{2} = 4$ .
4th value of the sorted list is 9.

Answer

⚡ Median without sorting everything

You only need the middle order statistics. In a long list, quickly bucket values as low or high; full sorting wastes time.

Example

Find the median of 4, 8, 15, 16, 23, 42.

Show solution

Six values: average the 3rd and 4th.
Here 15 and 16 are already the middle pair: $\dfrac{15+16}{2} = 15.5$ .

Answer

15.5

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Taking the middle of the list as given, without sorting.

Sort first; the given order is usually shuffled on purpose.

Mistake 02

Reporting the lower of the two middle values when n is even.

Average the two middle values.

Mistake 03

Using the empirical relation on very lopsided data and expecting exactness.

Treat it as an estimate; it fits mildly skewed data.

Mistake 04

Forgetting that a new value can change the count from odd to even.

Recheck parity after any value is added or removed.

Mistake 05

Calling the tallest bar's value the median in a table.

That value is the mode; the median needs cumulative counting.

Quick revision

Read this the night before the exam.

Median: sort, then middle value; even $n$ averages the middle pair.
Mode: highest frequency; it must be a value from the list.
Mode $= 3$ Median $- 2$ Mean (estimate).
Median $= \dfrac{\text{Mode} + 2\,\text{Mean}}{3}$ .
Shift by $c$ : all averages shift by $c$ ; scale by $k$ : all scale by $k$ .
A missing middle entry equals the median itself (odd n).

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.

Continue learning