"Or as Dr Gödel put it through his equations: “Ax. 1. {P(φ)∧◻∀x[φ(x)→ψ(x)]} →P(ψ)Ax. 2.P(¬φ)↔¬P(φ)Th. 1.P(φ)→◊∃x[φ(x)]Df. 1.G(x)⟺∀φ[P(φ)→φ(x)]Ax. 3.P(G)Th. 2.◊∃xG(x)Df. 2.φ ess x⟺φ(x)∧∀ψ{ψ(x)→◻∀y[φ(y)→ψ(y)]}Ax. 4.P(φ)→◻P(φ)Th. 3.G(x)→G ess xDf. 3.E(x)⟺∀φ[φ ess x→◻∃yφ(y)]Ax. 5.P(E)Th. 4.◻∃xG(x)”."
First, let's examine what is meant by modal.
How to Find the Mode or Modal Value
The mode is simply the number which appears
most often.
Finding the Mode
To find the mode, or modal value, first put the numbers
in order, then count how many of each number. A number that appears
most often is the
mode.
Example:
3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29
In order these numbers are:
3, 5, 7, 12, 13, 14, 20,
23, 23, 23, 23, 29, 39, 40, 56
This makes it easy to see which numbers appear
most often.
In this case the mode is
23.
Another Example: {19, 8, 29, 35, 19, 28, 15}
Arrange them in order:
{8, 15, 19, 19, 28, 29, 35}
19 appears twice, all the rest appear only once, so
19 is the mode.
More Than One Mode
We can have more than one mode.
Example: {1, 3, 3, 3, 4, 4, 6, 6, 6, 9}
3 appears three times, as does 6.
So there are two modes: at
3 and
6
Having two modes is called
"bimodal".
Having more than two modes is called
"multimodal".
Grouping
When all values appear the same number of times the idea of a mode is not useful. But we could group them to see if one group has more than the others.
Example: {4, 7, 11, 16, 20, 22, 25, 26, 33}
Each value occurs once, so let us try to group them.
We can try groups of 10:
- 0-9: 2 values (4 and 7)
- 10-19: 2 values (11 and 16)
- 20-29: 4 values (20, 22, 25 and 26)
- 30-39: 1 value (33)
In groups of 10, the "20s" appear most often, so we could choose 25 as the mode.
You could use different groupings and get a different answer!
We can apply this concept to practical applications.
Grouping also helps to find what the typical values are when the real world messes things up!
Example: How long to fill a pallet?
Philip recorded how long it takes to fill a pallet in minutes:
{35, 36, 32, 42, 58, 56, 35, 39, 46, 47, 34, 37}
It takes longer if there is break time or lunch so an average is not very useful.
But grouping by 5s gives:
- 30-34: 2
- 35-39: 5
- 40-44: 1
- 45-49: 2
- 50-54: 0
- 54-59: 2
"35-39" appear most often, so we can say it normally takes
about 37 minutes to fill a pallet.
How to Calculate the Mode or Modal Value