What is statistics? – Percentiles, Quartiles, Deciles and InterQuartile Range for Grouped data

Percentile positions in Grouped data / in Ranges: Example
The following table presents some random generated data about ages that are grouped in some specified intervals. Both, the convenient as well True Upper and Lower limits are presented for each Age Range. Also, the Simple and Cumulative frequency is presented for each Age Range.

True Limits
True Limits are defined as such:

Lower True Limits: Subtract Half Point from the Lower Range Limit e.g.
Range given 20-30
Lower Limit=20
Subtract half point: 20-0.5=19.5

Upper True Limits: Add Half Point to the Upper Range Limit e.g.
Upper Limit=30
Add Galf point: 30+0.5=30.5

For example: The Age Range of 31 – 40 has True Upper and Lower limits 30.5 – 40.5. It has a Simple Frequency of 5, which means that 5 participants claimed that had an age between 31 to 40. Finally, its Cumulative Frequency is 9, that is, 9 (4+5) participants claimed that had an Age between 21 to 30 (4) and between 31 to 40 (5).

Convenient Range limitsTrue Range limits
u and Χl)
Freq. FiCumulative Frequency cf
21 - 3020.5 - 30.544
31- 4030.5 - 40.559
41- 5040.5 - 50.5 312
51- 6050.5 - 60.5820

Statistical formula
Note that in order to find what value corresponds to a specified -ile position such as the positions of Percentile, Quartile, or Decile in grouped data, the following formula must be used:

X_{t}=R_{L}+(\frac{R_{W}}{f_{t}})(n(\frac{k}{i_{m}})-cf_{b})

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Symbol explanation
The X_{t} is the value that is indicated from a specified -ile.
The t is the specified -ile e.g. 25th Percentile, 5th Decile, 2nd Quartile.
The R_{L} is the True Lower Range limit of this Range that includes the specified -ile position.
The R_{W} is the size / Width of this Range that includes the specified -ile position.
The f_{t} is the Simple Frequency of this Range that includes the specified -ile position.
The k takes the value that is denoted to the position of a specified -ile e.g. it takes the value of 2 for Q_{2}, it takes the value of 50 for P_{50}.
The i_{m} is the number of how much Arithmetic parts / pieces or “moieties” are produced from the specified -ile e.g. Quartiles produce 4 Arithmetic pieces while Percentiles produce 100 “moeties”.
The n is the size of the dataset, that is, the total Simple Frequency.
The cf_{b} is the Cumulative Frequency of the Range that is right before from the Range that includes the specified -ile position.

Note that this part of formula: “n(\frac{k}{i_{m}})}” can indicate in what observation lies a specified -ile position. Note that it is mainly based on the size of the total number of observations as well to which -ile will be used.

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Statistical Example Ι
The steps that we must follow in order to find the value that is indicated by a specified / chosen -ile -let’s say by 50th Percentile X_{t}=X_{P50}- for the given example of Grouped data are the following ones:

i) A prerequisite in order to find the value that is indicated by a specified -ile is that the given Ranges must be arranged in an ascending order e.g. from the Lowest Age Range to the Highest Age Range, which has been done.
ii) Note that the total number of observations in this example is n=20.
iii) then, we must find where lies the 50th position between these 20 observations. This 50th position can also be expressed as \frac{k}{i_{m}}=\frac{50}{100}=0.5 when using Percentile concept. Therefore, the 50th position (which is the middle position) lies in the 10th observation 0.5*20=10.
iv) Note that the Age Ranges of 21-30 and 31- 40 includes the first 9 observations while the Age Range that includes the 10th observation is the next one, which is: 40-51. It can also named as the -ile group or the Percentile Range / Group.
v) The True Lower limit of this Age Range is: R_{L}=40.5
vi) The R_{W} is the size of the “Percentile Range” which can be found by subtracting the True Lower Range limit from the True Upper Range limit. Therefore: R_{W}=R_{U}-R_{L}=50.5-40.5=10, and thus R_{W}=10.
vii) The f_{t} is the Simple Frequency that has the “Percentile Group”, which is 3: f_{t}=3.
viii) The cf_{b} is the Cumulative Frequency that cumulate to those Age Ranges before the “Percentile Group”. The Age Ranges 21- 30 and 31- 40 includes the first 9 observations which are before the “Percentile Group”, therefore: cf_{b}=9.

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Statistical Example Ι: Calculations
Now, we must ready to enter these values into the Statistical Formula:

X_{t}=40.5+(\frac{10}{3})(20(\frac{50}{100})-9)
X_{t}=40.5+(3.33)*(20*(0.5)-9)=40.5+(3.33)*(10-9)
X_{t}=40.5+(3.33)*(1)=40.5-9=43.83

Therefore, the Age that is indicated by the position of P_{50} is the age of 43.83. Note that the position of P_{50} “always” coincide with the position of Q_{2} (both refer to the “Median position”. Therefore, this Age value is also the Median value. Therefore, 50% of the total Age observations will be before this value and 50% of the total Age observations will be after this value.

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Statistical Example IΙ: Interquartile Range
What is the value that is indicated by the Q_{1} and Q_{3} positions, respectively ?

i) The first step is to find in which observation lies the 1st and 3rd Quartile positions. This can be found by using this part of formula which indicates the 5th and 15th age observation, respectively:
—–Q_{1}=n(\frac{k}{i_{m}})}=20*(\frac{1}{4}})}=5 and
—–Q_{3}=n(\frac{k}{i_{m}})}=20*(\frac{3}{4}})}=15

ii) The next step is to find in which Age Range these observations lies. According to the Cumulative Frequencies:
—–The 5th observation is included in Age Range of: 31-40 and
—–The 15th observation is included in Age Range of: 51-60

iii) The True Lower limit of these Age Ranges is:
—–R_{L}=30.5 and
—–R_{L}=50.5

iv) The size of both these Age Ranges is 10:
—–R_{W}=R_{U}-R_{L}=40.5-30.5=10
—–R_{W}=R_{U}-R_{L}=60.5-50.5=10

v) The Simple Frequency for each Age Range is:
—–f_{t}=5
—–f_{t}=8

viii) Finally, the Cumulative Frequency that exists before these given Age Ranges, respectively is:
—–cf_{b}=4
—–cf_{b}=12

Example IΙ: Results
X_{Q1}=30.5+(\frac{10}{5})(5-4)=30.5+2*1=32.5
X_{Q3}=50.5+(\frac{10}{8})(15-12)=50.5+1.25*3=50.50+3.75=54.25

Therefore the Interquartile Range of Age groups is “32.5 – 54.25″. This Age “Range” includes the 50% of the total Age Observations, those that are placed in the “middle”.

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Example IΙ: The value of the Interquartile Range

The value of Interquartile Range is the result that we will get when the value that exists in the First Quartile position will be subtracted from the value that exists in the Third Quartile position, which is 21.75:

IQR=Q_{3}-Q_{1}=54.25-32.5=21.75

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Statistical Example ΙII: Decile positions: D_{1} and D_{9}

i) The first step is to find in which observation lies the 1st and 9th Decile positions. This can be found by using this part of formula which indicates the 2nd and 18th age observation, respectively:
—–D_{1}=n(\frac{k}{i_{m}})}=20*(\frac{1}{10}})}=2 and
—–D_{9}=n(\frac{k}{i_{m}})}=20*(\frac{9}{10}})}=18

ii) The next step is to find in which Age Range these observations lies. According to the Cumulative Frequencies:
—–The 2nd observation is included in Age Range of: 21-30 and
—–The 18th observation is included in Age Range of: 51-60

iii) The True Lower limit of these Age Ranges is:
—–R_{L}=20.5 and
—–R_{L}=50.5

iv) The size of both these Age Ranges is 10:
—–R_{W}=R_{U}-R_{L}=30.5-20.5=10 and
—–R_{W}=R_{U}-R_{L}=60.5-50.5=10

v) The Simple Frequency for each Age Range is:
—–f_{t}=4 and
—–f_{t}=8

viii) Finally, the Cumulative Frequency that exists before these given Age Ranges, respectively is:
—–cf_{b}=0 and
—–cf_{b}=12

Example IIΙ: Results
X_{D1}=20.5+(\frac{10}{4})(2-0)=20.5+5=25.5
X_{D9}=50.5+(\frac{10}{8})(18-12)=50.5+1.25*6=50.5+7.5=58

This Decile “Range” of “25.5 – 58″ includes the 80% of the total Age observations. The “middle” ones.

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