SPI Mathematics · Statistics

Empirical Relation and Graphs

Mode is about 3 times the median minus 2 times the mean, and each kind of graph shows data in its own way.

Why this matters

Seven questions in the bank, five of them EASY. Four rest on the one relation between mean, median and mode, and three ask which graph shows what: the axes of a histogram, the graph for parts of a whole, and the average a graph cannot give.

Concept 1 of 2: Mode equals 3 median minus 2 mean

For data that is only a little lopsided, 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 pattern gives one equation linking all three, so any two give the third.

Definition

Mode $= 3$ Median $- 2$ Mean.
The same fact written another way: Mean $-$ Mode $= 3$ (Mean $-$ Median).
It is an approximate (empirical) rule for moderately skewed data, found by observation, not proved.
To find the median from the other two: $3\,\text{Median} = \text{Mode} + 2\,\text{Mean}$ .

Empirical relation

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

Worked example

A distribution has mean 24 and mode 21. Use the empirical relation to find its median.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Statistics · Empirical Relation and Graphs

Using empirical relationship, the mode of a distribution whose mean is 7.2 and the median 7.1, is:

Swapping mean and median

The 3 goes with the median and the 2 with the mean: Mode

= 3

Median

- 2

Mean. 'Mode

= 3

Mean

- 2

Median' is the printed wrong option.

Losing the minus sign when matching a pattern

Written as x Median

+

y Mean

= z

Mode, the relation has

x = 3

y = -2

and

z = 1

. The minus sign belongs to y.

Drill 3 more on mode equals 3 median minus 2 mean

Concept 2 of 2: Which graph shows what

Each graph answers one kind of question. A pie chart shows shares of a whole, a histogram shows how a frequency table is spread out, and an ogive tracks a running total. Knowing which is which settles the definition questions at once.

Definition

Three facts settle most graph questions:

In a histogram the class intervals go on the x-axis and the frequencies on the y-axis, as bars with no gaps.
A pie chart shows the parts of a whole: each sector angle is the part's share of 360°.
The median can be read from an ogive and the mode from a histogram, but the mean cannot be found from a graph.

Graph	What it shows	What you can read from it
Histogram	Class intervals on the x-axis, frequencies on the y-axis, bars touching	The mode, from the tallest bar
Bar graph	Separate items as bars with gaps between them	Which item is largest or smallest
Pie chart	A whole divided into sectors for its parts	The share of each part: sector angle = part ÷ whole × 360°
Pictograph	Quantities drawn as rows of pictures	Counts, from the number of pictures
Frequency polygon	The mid-points of the histogram bar tops joined by lines	The shape of the distribution
Ogive	Cumulative frequencies plotted against class limits	The median, where the less-than and more-than ogives cross No graph gives the mean: it must be calculated.

The mean is the one average that cannot be found from a graph.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Statistics · Empirical Relation and Graphs

A geometric representation showing the relationship between a whole and its parts, is a

Putting the classes on the y-axis

In a histogram the class intervals run along the x-axis and the frequencies up the y-axis. A histogram's bars also touch, unlike a bar graph's.

Thinking every average has a graph

The median comes from an ogive and the mode from a histogram, but the mean has no graphical method. It is the answer when a question asks which cannot be found graphically.

Drill 2 more on which graph shows what

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)

Mode equals 3 median minus 2 mean
Empirical relation
$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

Reference tables (1)

Which graph shows what6 rows

Graph	What it shows	What you can read from it
Histogram	Class intervals on the x-axis, frequencies on the y-axis, bars touching	The mode, from the tallest bar
Bar graph	Separate items as bars with gaps between them	Which item is largest or smallest
Pie chart	A whole divided into sectors for its parts	The share of each part: sector angle = part ÷ whole × 360°
Pictograph	Quantities drawn as rows of pictures	Counts, from the number of pictures
Frequency polygon	The mid-points of the histogram bar tops joined by lines	The shape of the distribution
Ogive	Cumulative frequencies plotted against class limits	The median, where the less-than and more-than ogives cross No graph gives the mean: it must be calculated.

The mean is the one average that cannot be found from a graph.

Watch out for (4)

Swapping mean and median
→ Mode equals 3 median minus 2 mean
Losing the minus sign when matching a pattern
→ Mode equals 3 median minus 2 mean
Putting the classes on the y-axis
→ Which graph shows what
Thinking every average has a graph
→ Which graph shows what

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