# How do I find the 50th percentile, the third quartile point and the 33rd percentile of the data set 1,2,4,7,8,9,10,20?

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Given the data set 1,2,4,7,8,9,10,20:

Percentiles tell the number of data points below a given data point. Thus the number at the 50th percentile has half of the data values below it. (This is, of course, the median of the set.) We can also describe these as quartiles or deciles where the 1st quartile is the 25th percentile, the 2nd decile is the 20th perentile, etc...

(1) The 50th percentile is the number such that one half of the data lies below that number. This is the median of the data set.

` P_(50)="median"=(7+8)/2=7.5 `

(2) The third quartile is the 75th percentile. It is easiest to compute this by finding the median of the upper half of the data set.

Considering 8,9,10,20 the median of this set is 9.5.

` Q_3=P_(75)=(9+10)/2=9.5`

(3) Find the 33rd percentile:

There is no agreed upon standard method to compute this. For a large enough data sets, the various methods will generally agree, but with such a small set you will need to compare your text/instructor's given method.

Here are a couple of ways:

(a) Nearest rank: Take `P/100*n+1/2 ` where P is the percentile and 2 the number of elements of the set.

Then `33/100*8+1/2=3.14 ` ; this rounds to 3 so the third data element is the 33rd percentile.

`P_(33)=4 `

(b) Compute ` c=(nP)/100 ` with P the percentile and n the number of elements. `c=(8*33)/100=2.64 ` . This number should be rounded up (regardless of the fractional part -- thus 2.01 rounds up to 3.) If the result is a whole number, you would take the mean of the cth and (c+1)st number in the list.

Here we take the 3rd number in the list.

`P_(33)=4 `

There are other methods that interpolate the value. Also, you can graph the cumulative frequency percentile graph -- then draw a horizontal line y=33 to meet the frequency graph and drop a vertical line to the horizontal axis.

1,2,4,7,8,9,10,20

50th percentile=?

third quartile point=?

33rd percentile=?

Percentile are the values that the data into 100 equal parts, therefore the formula is 1st Percentile= (n+1)/100

The formula for the 50th percentile will be= **50[(n+1)/100] th value**

** **= 50[(8+1)/100]

= 50(9/100)

=4.5th value

Therefore, 4th value+0.5(5th-4th value)

7+0.5(8-7)

** 7.5**

Quartiles are the values which divide the arranged data into 4 equal parts. The formula for 1st quartile is=(n+1)/4

3rd quartile point=Q3= 3[(n+1)/4]th value

= 3[(8+1)/4]

= 6.75 th value

Therefore, 6th value+ 0.75( 7th-6th value)

= 9+0.75(10-9)

= **9.75**

33rd percentile=P33= 33[(n+1)/100] th value

= 33[(8+1)/100]

= 33(9/100)

= 2.97 th value

Therefore, 2nd value+0.97(3rd-2nd value)

=2+0.97(4-2)

=2+0.97(2)

=**3.94**

50th percentile=**7.5**

third quartile point=**9.75**

33rd percentile=**3.94**