Statistics: Appendix C: Formulas for different types of frequencies

Basic input & symbols

The following basic input will be used:
k = number of classes
F_i => (absolute) frequency of class i
h_i=> Class width of class i
UB_i => Upper bound of class i
LB_i => Lower bound of class i
N => Total Absolute Frequency

Simple calculation:
$h_{i}=UB_{i}-LB_{i}$
$N=\sum_{i=1}^{k}F_{i}$

The following mathematical symbols are used:

Symbol	Description
=	Equal sign
>	Greater than
<	Less than
≥	Greater than or equal to
≤	Less than or equal to
∑	Summation
{	Set of equations
$\frac{a}{b}$	Division

Frequency Density (FD)

$FD_{i}=\frac{F_{i}}{h_{i}}$

Example

Class	Width (h)	Frequency (F)	Frequency Density (FD)
0 < 10	10	4	4/10 = 0.4
10 < 20	10	8	8/10 = 0.8
20 < 50	30	10	10/30 ≈ 0.33
50 < 100	50	3	3/50 = 0.06
TOTAL		25

Relative Frequency

$RF_{i}=\frac{F_{i}}{N}$

Example

Class	Frequency (F)	Relative Frequency (RF)
0 < 10	4	4/25 = 0.16
10 < 20	8	8/25 = 0.32
20 < 50	10	10/25 = 0.4
50 < 100	3	3/25 = 0.12
TOTAL	25	1

Relative Frequency Density

$RFD_{i}=\frac{RF_{i}}{h_{i}}=\frac{FD_{i}}{N}$
(for a proof of equivalence of these two formulas click here)

Example

Class	Width (h)	Frequency (F)	Relative Frequency (RF)	Frequency Density (FD)	Relative Frequency Density (RFD)
0 < 10	10	4	0.16	0.4	0.16 / 10 = 0.4 / 25 = 0.016
10 < 20	10	8	0.32	0.8	0.32 / 10 = 0.8 / 25 = 0.032
20 < 50	30	10	0.4	0.33	0.4 / 30 ≈ 0.33 / 25 ≈ 0.0133
50 < 100	50	3	0.12	0.06	0.12 / 50 = 0.06 / 25 = 0.0024
TOTAL		25	1

Cumulative Frequency

$CF_{i}=\sum_{j=1}^{i}F_{j}$
or as an iterative formula:
$CF_{i}=\left\{\begin{matrix}F_{i} &,i=1 \\ F_{i-1} &, i\geq 2 \end{matrix}\right.$

Example

Class	Frequency (F)	Cumulative Frequency (CF)
0 < 10	4	4 = 4 = 4
10 < 20	8	4 + 8 = 4 + 8 = 12
20 < 50	10	4 + 8 + 10 = 12 + 10 = 22
50 < 100	3	4 + 8 + 10 + 3 = 22 + 3 = 25
TOTAL	25	1

Cumulative Frequency Density

$CFD_{i}=\frac{CF_{i}}{h_{i}}$

Example

Class	Width (h)	Frequency (F)	Cumulative Frequency (CF)	Cumulative Frequency Density (CFD)
0 < 10	10	4	4	4 / 10 = 0.4
10 < 20	10	8	12	12 / 10 = 1.2
20 < 50	30	10	22	22 / 30 ≈ 0.73
50 < 100	50	3	25	25 / 50 = 0.5
TOTAL		25

Cumulative Relative Frequency

$CRF_{i}=\sum_{j=1}^{i}RF_{j}=\frac{CF_{i}}{N}$
or as an iterative formula
$CRF_{i}=\left\{\begin{matrix}RF_{i} &, i=1 \\ CRF_{i-1}+RF_{i} &, i\geq2 \end{matrix}\right.$

Example

Class	Frequency (F)	Relative Frequency (RF)	Cumulative Frequency (CF)	Cumulative Relative Frequency (CRF)
0 < 10	4	0.16	4	0.16 = 4 / 25 = 0.16 = 0.16
10 < 20	8	0.32	12	0.16+0.32 = 12/25 = 0.16+0.32 = 0.48
20 < 50	10	0.4	22	0.16+0.32+0.4 = 22/25 = 0.48+0.4 = 0.88
50 < 100	3	0.12	25	0.16+0.32+0.4+0.12 = 25/25 = 0.88+0.12 = 1
TOTAL	25	1

Cumulative Relative Frequency Density

$CRFD_{i}=\frac{CRF_{i}}{h_{i}}$

Example

Class	Width (h)	Frequency (F)	Cumulative Relative Frequency (CRF)	Cumulative Relative Frequency Density (CRFD)
0 < 10	10	4	0.16	0.16 / 10 = 0.016
10 < 20	10	8	0.48	0.48 / 10 = 0.048
20 < 50	30	10	0.88	0.88 / 30 ≈ 0.0293
50 < 100	50	3	1.00	1 / 50 = 0.02
TOTAL		25

Inverse Cumulative Frequency

$ICF_{i}=\sum_{j=i}^{k}F_{j}$
or as an iterative formula
$ICF_{i}=\left\{\begin{matrix}ICF_{i+1}+F_{i} & ,i<k\\ F_{i} & ,i=k \end{matrix}\right.$
or as an iterative formula based on the cumulative frequencies:
$ICF_{i}=\left\{\begin{matrix}N & ,i=1\\ N-CF_{i-1} & ,i>1 \end{matrix}\right.$

Example

Class	Frequency (F)	Cumulative Frequency (CF)	Inverse Cumulative Frequency (ICF)
0 < 10	4	4	4+8+10+3 = 21 + 4 = 25 = 25
10 < 20	8	12	8+10+3 = 13 + 8 = 25 - 4 = 21
20 < 50	10	22	10+3 = 3 + 10 = 25 - 12 = 13
50 < 100	3	25	3 = 3 = 25 - 22 = 3
TOTAL	25

Inverse Cumulative Frequency Density

$ICFD_{i}=\frac{ICF_{i}}{h_{i}}$

Example

Class	Width (h)	Frequency (F)	Inverse Cumulative Frequency (ICF)	Inverse Cumulative Frequency Density (ICFD)
0 < 10	10	4	25	25 / 10 = 2.5
10 < 20	10	8	21	21 / 10 = 2.1
20 < 50	30	10	13	13 / 30 ≈ 0.43
50 < 100	50	3	3	3 / 50 = 0.06
TOTAL		25

Inverse Cumulative Relative Frequency

$ICRF_{i}=\sum_{j=i}^{k}RF_{j}=\frac{ICF_{i}}{N}$
or as an iterative formula
$ICRF_{i}=\left\{\begin{matrix}RF_{i} & i=k\\ICRF_{i+1}+RF_{i} & i<k \end{matrix}\right.$
or as an iterative formula
$ICRF_{i}=\left\{\begin{matrix}1 & i=1\\1-CRF_{i-1} & i>1 \end{matrix}\right.$

Example

Class	Frequency (F)	Relative Frequency (RF)	Cumulative Relative Frequency (CRF)	Inverse Cumulative Frequency (ICF)	Inverse Cumulative Relative Frequency (ICRF)
0 < 10	4	0.16	0.16	25	0.16+0.32+0.4+0.12 = 25 / 25 = 0.84 + 0.16 = 1 = 1
10 < 20	8	0.32	0.48	21	0.32+0.4+0.12 = 21 / 25 = 0.52+0.32 = 1 - 0.16 = 0.84
20 < 50	10	0.4	0.88	13	0.4+0.12 = 13 / 25 = 0.12+0.4 = 1 - 0.48 = 0.52
50 < 100	3	0.12	1	3	0.12 = 3 / 25 = 0.12 = 1 - 0.88 = 0.12
TOTAL	25	1

Inverse Cumulative Relative Frequency Density

$ICRFD_{i}=\frac{ICRF_{i}}{h_{i}}$

Example

Class	Width (h)	Frequency (F)	Inverse Cumulative Relative Frequency (ICRF)	Inverse Cumulative Relative Frequency Density (ICRFD)
0 < 10	10	4	1	1 / 10 = 0.1
10 < 20	10	8	0.84	0.84 / 10 = 0.084
20 < 50	30	10	0.52	0.52 / 30 ≈ 0.0173
50 < 100	50	3	0.12	0.12 / 50 = 0.0024
TOTAL		25

Statistics

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donderdag 1 mei 2014

Appendix C: Formulas for different types of frequencies

Basic input & symbols

Frequency Density (FD)

Relative Frequency

Relative Frequency Density

Cumulative Frequency

Cumulative Frequency Density

Cumulative Relative Frequency

Cumulative Relative Frequency Density

Inverse Cumulative Frequency

Inverse Cumulative Frequency Density

Inverse Cumulative Relative Frequency

Inverse Cumulative Relative Frequency Density

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