The ages of 100 persons are tabulated below. Find Q1, Q3, D2, D8, P20 and P60. Above what age we have the oldest 10% persons? Age(in years) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of persons 16 20 21 28 10 3 2 Note: the oldest 10% are above P90. Ans.: P20 = 22, P60=41.071, the oldest 10% persons are of the age 55 and above.
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The frequency table for the data given above is as shown below.
class boundary f cf
10-20 16 16
20-30 20 36
30-40 21 57
40-50 28 85
50-60 10 95
60-70 3 98
70-80 2 100
Here,
is the quartile defined by the formula,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class in which lies
is the frequency of the class containing
To determine the 1st quartile, we first find its position in order to find its class boundary as follows,
Therefore, the class boundary for the 1st quartile is 20-30
Thus,
is the third quartile defined by the formula,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class containing
is the frequency of the class containing
To determine the 3rd quartile, we first its position in order to find its class boundary as follows,
Therefore, the class boundary for the 3rd quartile is 40-50
Thus,
Therefore,
To find the 2nd decile; , we use the formula stated below,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class in which lies
is the frequency of the class containing
We first determine the class containing ,
Therefore, the class where can be found is 20-30
Hence,
Therefore,
To find we apply the formula below,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class in which lies
is the frequency of the class containing
The position of is,
The class in which lies is 40-50
Thus,
Therefore,
To find the 20th percentile given as , we apply the following formula,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class in which lies
is the frequency of the class containing
Let us determine the class where lies by first finding its position,
Now, lies in the class 20-30
Thus,
Therefore,
To find the percentile given as, , the following the formula is used,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class in which lies
is the frequency of the class containing
To find the class where lies, we first determine its position as follows,
Thus, is in the class 40-50
Therefore,
Therefore,
vii)
To find the age where the oldest 10% persons are above, we determine the 90th percentile given as, . We shall apply the formula below,
where,
is the lower class boundary of the class containing
is the cumulative frequency of the class preceding the class containing
is the width of the class containing
is the frequency of the class containing
To find the class where lies, we first determine its position as follows,
Thus, is in the class 50-60
Therefore,
Hence, the oldest 10% persons are of the age 55 and above.