What is statistics? – Range

Theoretical Definition – Range
Statistical Range can be defined as any two values that can imply that intermediate numbers can exist between these two values.

Example
The Age Range of 21-30 can suggest that these integrer (whole number) age values exist 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30 between these Range limits. Generally, it can suggest that all intermediate values between 21 and 30 can be described by these two numbers, Integers and non-integer values.

Range_numbers

Useful Math Symbols For Ranges Limits

“[” or “]” : Are the symbols for Closed Range Limits
“(” or “)” : Are the symbols for Open Range Limits

For Example: The Age Range of [30-40+) shows that:
The Age Range starts from the number 30 (included) based on the bracket “[”
While the “)” shows that the Upper Range Age Limit is limitless:
All Age numbers are equal to or are above 40 can be included in your sample e.g.
60, 100, 120 years old people.

The same is true for the Age Range of (12-16]. The Age Range shows that all ages from 0 to 16 can be included in that Age Range. However, based on the Lower Age Limit “12”, someone e.g. the Survey-ist can expect that he / she will have very few participants below e.g. age of 10 or 8. However, he / she cannot exclude baby participants based on how the Age Range is given “(“.

Upper and Lower Range limits
However, some problems can arise.

1st Example: Problem I
Let’s suppose that you have defined some Age Ranges for a psychological / Social survey: 21-30, 31-40, 41-50, 51, 60 and 60+. Then you would like to formulate some guidelines how you will categorize the Upper Age Range Limit of the Last age Range that is given as open [60, 60+) or [60-60+). Such problems may arise when someone would like to analyze some data, and he/she would like to Define these (Undefined) limitless Limits after Data Collection.

This problem have the following “Rule of thumb” solutions based on your survey needs:
Example use: Age Range [60-60+)

i) You may do not define a given Upper or Lower Range Limit for Limitless Ranges.
ii) You may define as Upper Age Limit the longest living e.g. Human organism e.g. 120 years old.
–> Therefore, the Age Range [60-60+) is transformed as such: [60-120]
iii) You may define as Upper Age Limit the mean age of Death Age (Mortality Age) of a given population e.g. 81
–> Therefore, the Age Range [60-60+) is transformed as such: [60-81]
iv) You may define as Upper Age Limit the theoritical Longest Human Living Age based on the related literature e.g. 150
–> Therefore, the Age Range [60-60+) is transformed as such: [60-150]
iiv) You may define as Upper Age Limit the last Age Data point in a dataset e.g. 70
–> Therefore, the Age Range [60-60+) is transformed as such: [60-70]
vi) You may define as Upper Age Limit the last Age Data point that it cannot be considered as an outlier e.g. 67
–> Therefore, the Age Range [60-60+) is transformed as such: [60-67]

2nd Example: Problem II
Let’s suppose that you have conducted a survey and in that survey, you made two Age groups: 21-30 and 31 to 40. However, a participant told you that his age was 30 years and 6 months, that is, he was 30.5 years old. What will you decide? You will include him in the first or in the latter Age Range / Group? This problem exists in all Lower and Upper Range Limits.

Convienient Lower and Upper Range Limits
The convenient Lower and Upper limits are the two Declared values.

For Example, The Convenient Lower and Upper Range Limits for the age groups of 21-30, and 31-40 are the values of 21 and 30, and the values of 31 and 40, respectively.

True Lower and Upper Range Limits
The Real True Lower and Upper Limits can be calculated if:

i) We substcact “5” points from the most previous “Decimal Depth” of the Convenient Lower Range Limit Value.

ii) We add “5” points to the most previous “Decimal Depth” of the Convenient Upper Range Limit Value.

True Lower and Upper Range Limits: Examples
The Whole numbers / Integer values have zero “0” “Decimal depth”. We add “1” point in order to find the most previous “Decimal depth” for an Integer value.

True Lower Range Limits
A “Convenient” Lower Range Limit value of “1” has a “True” Lower Range Limit value of: 1-0.5=0.5
and a “Convenient” Lower Range Limit value of “0.04” has a “True” Lower Range Limit value of: 0.04-0.005=0.035

Here, the Decimal Depth of the “Convenient” Lower Range Limit value is 2. The most previous Decimal Depth is found by adding a point to this value: 2+1=3. From this “Decimal depth”, the value of “5” is substracted from the “Convenient” Lower Range Limit value.

True_Range_Limits

True Upper Range Limit
A “Convenient” Upper Range Limit value of “10” has a “True” Upper Range Limit value of:
10+0.5=10.5
and a “Convenient” Upper Range Limit value of “0.001” has a “True” Upper Range Limit value of:
0.001+0.0005=0.0015

Here, the Decimal Depth of the “Convenient” Upper Range Limit value is 3. The most previous Decimal Depth is found by adding a point to this value: 3+1=4. From this “Decimal depth”, the value of “5” is added to the “Convenient” Upper Range Limit value.

Table that presents the “Convenient” and True Upper and Lower Range Limtis in various examples


Convenient Range Limits
True Range Limits
21 --- 3020.5 --- 30.5
1 --- 100.5 --- 10.5
0.04 --- 0.0010.035 --- 0.0015
-0.33 --- +0.33
-0.325 --- +0.335
0 --- 1
-0.5 --- 1.5
-1--- 0
-1.5 --- 0.5

The Size of the Range – Whole numbers / integer case
Statistically, the size of the Range for integer numbers (Range Represantation) can be found by this way: “Upper Range Limit” – “Lower Range Limit” + 1.

Therefore, according to the previous example: (30-21)+1=9+1=10

This result shows that there are ten integer values that can be represented by the Range of 21-30.

Size of the Range – General Formula
In the previous example, we suggested that there were zero “0” decimals, in order to find the quantity of the integer values that can be represented by a Range with Integer Upper and Lower Limits. The Statistical Formula for such calculation is the below one, which is the simplified edition of the next one, because D_{D}=0:

R_{W}=(R_{U}-R_{L})+1

Now, if we would like to increase the “depth”, that is, to increase the decimal “depth” in a given Range, then the General Range Formula can be defined as:

R_{W}=[(R_{U}-R_{L})*D_{D}]+1

i) We subtract the “Upper” Range Limit from the “Lower” Range Limit

ii) For Decimal depth of zero “0”, we multiply the result by “1”: D_{D}=1 (default value)

iii) For Decimal depth greater than zero (>0) , we add so many zeros to the default value of “1”, as the number of decimals we are interested in to find Range Value Representation, and then:

iv) We add to this final result “one point” in order to find the Range Value Representation or Range Width R_{W}.

Applying the Formula

1) For the Range of 1.0 to 3.0, the Decimal Depth is “1”, and thus:
D_{D}=10 and then:
[(3.0-1.0)*10]+1=(2*10)+1=20+1=21

Therefore, the Range of 1.0 to 3.0 represents 21 values in a Decimal Depth of “0.1”.

2) For the Range of -0.2 to 0.5, the Decimal Depth is “1”, and thus:
D_{D}=10 and then:
[(0.5-(-0.2))*10]+1=((0.5+0.2)*10)+1=0.7*10+1=71

Therefore, the Range of -0.2 to 0.5 represents 71 values in a decimal Decimal Depth of “0.1”.

3) For the Range of 0.001 to 0.002, the Decimal Depth is “3”, and thus:
D_{D}=1000 and then:
[(0.002-0.001)*1000]+1=((0.001)*1000)+1=1+1=2

Therefore, the Range of 0.001 to 0.002 represents 2 values in a Decimal Depth of “0.001”: 0.001 and 0.002!

4) For the Range of 0.10 to 0.02, the Decimal Depth is “2”, and thus:
D_{D}=100 and then:
[(0.1-0.02)*100]+1=((0.08)*100)+1=8+1=9

Therefore, the Range of 0.10 to 0.02 represents 9 values in a Decimal Depth of “0.01”:
0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10!

5) For the Range of 2 to 1.001 –> the Decimal Depth is “3”, and thus:
D_{D}=1000 and then:
[(2-1.001)*1000]+1=((0.999)*1000)+1=999+1=1000

Therefore, the Range of 2 to 1.001 represents 1000 values.

Table that presents the “Range Value Representation” in relation to the Decimal Depth of the Range “0 to 1″

Decimal DepthRange Value Representation
FOR the range of 0-1
0 2
111
2101
3
1001
4
10001
5
10001
6100001
71000001
810000001
9100000001
101000000001

Range Value Represantation