SPI Mathematics · Statistics

Median and Mode of Raw Data

The median is the middle value once the data is in order, and the mode is the value that occurs most often.

Why this matters

Eight questions in the bank, seven of them EASY. Three only ask which average is which, and the rest put a short list in order for the median or count how often each value occurs for the mode, sometimes with one value left unknown.

Concept 1 of 3: Mean, median and mode side by side

All three are called measures of central tendency because each picks one number to stand for the whole list. They pick it in different ways, and the options in a definition question are built from mixing those ways up.

Definition

Three words settle most definition questions:

Mean uses every value.
Median is the middle value, so it ignores how big the end values are.
Mode is the most frequent value. A list can have two modes, or none if every value occurs once.

Measure	What it is	How to find it
Mean	The average	Add all the values and divide by how many there are
Median	The middle value	Put the values in order and take the middle one
Mode	The most frequent value	Count how often each value occurs
Smallest and largest value removed	The median stays the same	The middle of an ordered list is still the middle The mean uses every value, so it usually changes. The mode changes only if an end value was the most frequent one.

The mode is the most frequent value, the median the middle one: 'middle most frequent' is a mix of the two.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Median and Mode of Raw Data

The most frequent value in the data is known as

Calling the mode the middle value

The middle value of ordered data is the median. The mode is the most frequent value, whether or not it sits in the middle.

Drill 3 more on mean, median and mode side by side

Concept 2 of 3: Median of a list

Line the values up from smallest to largest. With an odd number of values one sits exactly in the middle; with an even number two share the middle, and the median is halfway between them.

Definition

Put the values in order first. The median of an unordered list means nothing.
n odd: the median is the $\left(\frac{n + 1}{2}\right)$ th value.
n even: the median is the mean of the $\left(\frac{n}{2}\right)$ th and $\left(\frac{n}{2} + 1\right)$ th values.
If the median and the two middle values contain an unknown, set their mean equal to the median and solve.

Position of the median

n \text{ odd: } \left(\frac{n + 1}{2}\right)\text{th value} \qquad n \text{ even: mean of the } \left(\frac{n}{2}\right)\text{th and } \left(\frac{n}{2} + 1\right)\text{th values}

Worked example

Find the median of 12, 7, 19, 4, 15, 10.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Statistics · Median and Mode of Raw Data

If the median of the given data 24, 25, 26,

x + 2

x + 3

, 30, 31, 34 is 27.5 then the value of

x

Taking the middle of an unordered list

In 59, 46, 30, 23, 27 the middle number as written is 30, but in order the list is 23, 27, 30, 46, 59. Always sort before you count to the middle.

Giving the position for the value

For 9 values,

\frac{9 + 1}{2} = 5

says the median is the 5th value. The answer is whatever number stands 5th, not 5.

Drill 1 more on median of a list

Concept 3 of 3: Mode of a list

Make a quick tally of each value. The one with the longest tally is the mode. If a value is unknown, ask what it must be to break a tie.

Definition

Tally every value, then add the tallies up: the total must equal the number of values.
The mode is the value with the largest count.
If two values tie and the mode is given as one of them, the unknown value must be that one.

Worked example

Find the mode of 3, 6, 6, 2, 9, 3, 6, 4.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Statistics · Median and Mode of Raw Data

The mode of 5, 2, 7, 3, 2, 2, 5, 5, 2, 3, 4, 5, 5, 7, 7 is

Losing count in a long list

In a list of 15 numbers, two values can look equally common at a glance. Tally them and check that the tallies add to 15 before you choose.

Drill 1 more on mode of a list

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 (1)

Median of a list
Position of the median
$n \text{ odd: } \left(\frac{n + 1}{2}\right)\text{th value} \qquad n \text{ even: mean of the } \left(\frac{n}{2}\right)\text{th and } \left(\frac{n}{2} + 1\right)\text{th values}$

Reference tables (1)

Mean, median and mode side by side4 rows

Measure	What it is	How to find it
Mean	The average	Add all the values and divide by how many there are
Median	The middle value	Put the values in order and take the middle one
Mode	The most frequent value	Count how often each value occurs
Smallest and largest value removed	The median stays the same	The middle of an ordered list is still the middle The mean uses every value, so it usually changes. The mode changes only if an end value was the most frequent one.

The mode is the most frequent value, the median the middle one: 'middle most frequent' is a mix of the two.

Watch out for (4)

Calling the mode the middle value
→ Mean, median and mode side by side
Taking the middle of an unordered list
→ Median of a list
Giving the position for the value
→ Median of a list
Losing count in a long list
→ Mode of a list

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