AFCAT Numerical Ability · Statistics

Median, Mode and Comparing the Averages

Find the median and mode of a list and of grouped data, use the geometric mean, link the averages by the empirical relation, and choose the right average for the data.

All Statistics notes

Why this matters

This is the chapter's biggest page, more than half of it MODERATE, and it holds the chapter's only HARD question. The marks are in the grouped formulas: the median needs the cumulative frequency before the median class, the mode needs the classes on both sides of the modal class, and most slips come from using the formula on the wrong class.

Concept 1 of 5: Median and mode of a list

The median is the middle value once the data are in order; the mode is the value that occurs most often. Neither needs the size of every value: the median needs positions and the mode needs counts.

Definition

Put the data in order first.
For $n$ values with $n$ odd, the median is the $\tfrac{n + 1}{2}$ th value. With $n$ even, it is the mean of the $\tfrac n2$ th and $\left(\tfrac n2 + 1\right)$ th values.
In a frequency table, keep a running total of the frequencies (the cumulative frequency) to see which value holds the middle position.
The mode is the value with the highest frequency. If two values tie for the highest, there are two modes; if every value occurs equally often, there is no mode.
The median is not moved by extreme values, and it can be found even when the end values are unknown, as long as their number is known.

Position of the median

\text{median} = \left(\dfrac{n + 1}{2}\right)^{\text{th}} \text{ value} \quad (n \text{ odd})

Worked example

Find the median and the mode of 12, 5, 14, 5, 11, 10, 5 and 13.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Median, Mode and Comparing Averages

The average value of the median of 2, 8, 3, 7, 4, 6, 7 and the mode of 2, 9, 3, 4, 9, 6, 9 is

Order the data first

The middle of 8, 2, 5 as written is 2. In order the list is 2, 5, 8 and the median is 5.

Equal counts mean no mode

In 3, 3, 5, 5, 8, 8 every value occurs twice, so no value is more frequent than the others and the data have no mode.

Drill 5 more on median and mode of a list

Concept 2 of 5: Median of grouped data

With grouped data the single values are not listed, so find the class that holds the middle position. Then assume the values in that class are spread evenly across it and go the right fraction of the way in.

Definition

$N = \sum f$ . The median class is the first class whose cumulative frequency reaches $\tfrac N2$ .
Median $= l + \dfrac{\tfrac N2 - F}{f} \times h$ , where $l$ is the lower boundary of the median class, $F$ the cumulative frequency of the classes before it, $f$ its own frequency and $h$ its width.
A missing frequency with the median given: put the median into the formula and solve for the frequency. The total frequency then gives any second missing one.

Median of grouped data

M = l + \dfrac{\tfrac N2 - F}{f} \times h

Worked example

The classes 0-10, 10-20, 20-30, 30-40 and 40-50 have frequencies 5, 8, 12, 10 and 5. Find the median.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Statistics · Median, Mode and Comparing Averages

Consider the following frequency distribution

Class interval	1-4	4-7	7-10	10-13	13-16	16-19
Frequency	6	30	40	16	4	4

What is the median of the data?

F is the count before the class

F

is the cumulative frequency of the classes below the median class. Including the median class's own frequency puts the median below its class.

Grouped data use N over 2

The grouped formula uses

\tfrac N2

, not

\tfrac{N + 1}{2}

. The

\tfrac{n + 1}{2}

rule is for a list of single values.

Drill 2 more on median of grouped data

Concept 3 of 5: Mode of grouped data

The modal class is the class with the highest frequency. Inside it, the mode leans towards whichever neighbouring class is bigger, and the formula measures how far it leans.

Definition

Mode $= l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$ , where $f_1$ is the frequency of the modal class, $f_0$ the class before it, $f_2$ the class after it, $l$ its lower boundary and $h$ its width.
The mode always lies inside the modal class, so a question asking which interval holds the mode needs only the highest frequency.
To compare the modes of two distributions, find each modal class first. Often that alone decides.

Mode of grouped data

Z = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h

Worked example

The classes 0-10, 10-20, 20-30 and 30-40 have frequencies 6, 14, 22 and 10. Find the mode.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Statistics · Median, Mode and Comparing Averages

For the following frequency distribution

Class interval	0-5	5-10	10-15	15-20	20-25	25-30
Frequency	10	15	30	80	40	20

m

is the value of mode, then which one of the following is correct?

Both neighbours go in the denominator

The denominator is

2f_1 - f_0 - f_2

. With a modal frequency of 22 and neighbours 14 and 10 it is 20; using

f_1 - f_2 = 12

instead gives the wrong mode.

The tallest class, not the middle one

The modal class is the one with the highest frequency, wherever it sits in the table. The median class is a different class in most tables.

Drill 2 more on mode of grouped data

Concept 4 of 5: Geometric mean, and numbers from their averages

The geometric mean multiplies where the ordinary mean adds: the geometric mean of

n

positive numbers is the

n

th root of their product. When several averages of the same unknown numbers are given, each one is an equation, and together they fix the numbers.

Definition

The geometric mean of $a$ and $b$ is $\sqrt{ab}$ ; of $a$ , $b$ and $c$ it is $\sqrt[3]{abc}$ . So a geometric mean of 6 for three numbers means their product is 216.
A list with each value repeated, such as $p, p, q, q$ , has the same geometric mean as $p, q$ : $\sqrt[4]{p^2 q^2} = \sqrt{pq}$ .
The mean gives the sum, the geometric mean gives the product, and the median gives the middle value. Two numbers with a known sum and product are the roots of $t^2 - (\text{sum})\,t + \text{product} = 0$ .
For positive numbers the geometric mean is never more than the ordinary mean.

Geometric mean

G = \sqrt[n]{x_1 x_2 \cdots x_n}

Worked example

The geometric mean of 5, 25 and

x

is 10. Find

x

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Statistics · Median, Mode and Comparing Averages

The arithmetic mean, geometric mean and median of 6 positive numbers

a, a, b, b, c, c

where

a < b < c

are

\frac{7}{3}

, 2, 2 respectively. What is the value of

c

Multiply, do not add

The geometric mean of 2 and 18 is

\sqrt{36} = 6

, not their ordinary mean 10.

Adding to every value does not add to the geometric mean

1 and 9 have geometric mean 3. Add 1 to each and 2 and 10 have geometric mean

\sqrt{20} \approx 4.47

, not 4.

Drill 3 more on geometric mean, and numbers from their averages

Concept 5 of 5: Mean, median and mode compared

In a moderately lopsided distribution the three averages sit in a fixed pattern: the median lies between the mean and the mode, about a third of the way from the mean. That gives one relation linking all three, and it also tells you which average suits which kind of data.

Definition

Empirical relation: Mode $= 3$ Median $- 2$ Mean. Given any two, it estimates the third.
For symmetric data, mean, median and mode are equal.
Which average suits which data: the mode for the most common size or choice (shoe or collar sizes); the median for lopsided data such as incomes, or for qualities that can be ranked but not measured; the mean when every value should count, such as marks.
Adding $k$ to every value adds $k$ to the mean, the median, the mode and the mid-range (the mean of the largest and smallest values), but not to the geometric mean.
The mean is pulled by extreme values and the median is not. The mean is not always above the median: it is above it only when the values run further on the high side.

Empirical relation

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

Worked example

A distribution has mean 42 and median 40. Estimate its mode.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 5 · Statistics · Median, Mode and Comparing Averages

If the mean and median of a set of numbers are 8.9 and 9 respectively, then the mode will be

3 times the median, 2 times the mean

Mode

= 3

Median

- 2

Mean. Swapping the 3 and the 2 turns a mean of 42 and a median of 40 into a mode of 46 instead of 36.

The mean is not always above the median

In 2, 10, 12 the mean is 8 and the median 10. The mean is above the median only when the large values stretch further from the middle.

Drill 5 more on mean, median and mode compared

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 (5)

Median and mode of a list
Position of the median
$\text{median} = \left(\dfrac{n + 1}{2}\right)^{\text{th}} \text{ value} \quad (n \text{ odd})$
Median of grouped data
Median of grouped data
$M = l + \dfrac{\tfrac N2 - F}{f} \times h$
Mode of grouped data
Mode of grouped data
$Z = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$
Geometric mean, and numbers from their averages
Geometric mean
$G = \sqrt[n]{x_1 x_2 \cdots x_n}$
Mean, median and mode compared
Empirical relation
$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

Watch out for (10)

Order the data first
→ Median and mode of a list
Equal counts mean no mode
→ Median and mode of a list
F is the count before the class
→ Median of grouped data
Grouped data use N over 2
→ Median of grouped data
Both neighbours go in the denominator
→ Mode of grouped data
The tallest class, not the middle one
→ Mode of grouped data
Multiply, do not add
→ Geometric mean, and numbers from their averages
Adding to every value does not add to the geometric mean
→ Geometric mean, and numbers from their averages
3 times the median, 2 times the mean
→ Mean, median and mode compared
The mean is not always above the median
→ Mean, median and mode compared

Drill the questions on this topic

22 questions from the bank, with cart and Word export.

Drill all 22 questions