Measures of Location
A measure of location provides us information on the percentage of observation in the collection whose values are less than or equal to it. We also refer to these measures of location as quantiles or fractiles. The percentile, decile and quartile are the common measure of location.
Percentile
Percentile are measures of location or position used in educational and health-related fields to indicate the position of an individual in a group. Further, this type of fractiles divide the ordered observations into 100 equal parts. The formula of percentile is: Pk=(k(n+1))/100 – Weighted average estimate method
If Pk=(k(n+1))/100 is not an integer then the weighted average estimate makes use of simple interpolation between the two observed values, using the formula below: Pk=(1-m) Xi+mX((i+1)) Where; m – is the fractional part i – is the integer part k – the desired location
Decile
The deciles divide the ordered observation into ten equal parts. Basically, the first decile, D1 is the number that divides the bottom 10% of the data from the top 90%. To obtain the deciles, divide the data set into tenths and then determine the number dividing the tenths. The formula is:
Dk=(k(n+1))/10 – Weighted average estimate method
If Dk=(k(n+1))/10 is not an integer then the weighted average estimate makes use of simple interpolation between the two observed values, using the formula below:
Dk=(1-m) Xi+mX((i+1))
Where;
m – is the fractional part
i – is the integer part
k – the desired location
Quartile
The quartile divides the ordered observations into four (4) equal parts. The formula to compute quartile is:
Qk=(k(n+1))/4 – Weighted average estimate method
If Qk=(k(n+1))/10 is not an integer then the weighted average estimate makes use of simple interpolation between the two observed values, using the formula below:
Qk=(1-m) Xi+mX((i+1))
Where;
m – is the fractional part
i – is the integer part
k – the desired location
Example 1: The following data sets are the number of years of operation of 20 mining companies: 4, 6, 7, 5, 6, 30, 23, 25, 20, 21, 17, 18, 17, 19, 11, 10, 10, 8, 20, 16. Determine the 95th percentile, D6 , and Q1.
Solution: Arrange the data set in order.
4, 5, 6, 6, 7, 8, 10, 10, 11, 16, 17, 17, 18, 19, 20, 20, 21, 23, 25, 30
Compute for P95=95(20+1)/100=19.95. Since, P95 is not an integer, then the P95 is computed by P95=(1-.95) X19+0.95X((19+1))=0.05(25) +0.95(30) = 1.25 + 28.5 = 29.75 years.
Therefore, we can say that 95 percent of the 20 mining companies have been operating for less than 29.75 years.
The solution to the D6 and Q1 locations are have been left as exercises.
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Happy Reading!
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