Question

In a study of a parasite in humans and animals. Researchers measured the lengths (in mm) of 90 individual parasites of certain species from the blood of a mouse. The measures are shown in the following table:

Length	19	20	21	22	23	24	25	26	27	28	29
Frequency	1	2	11	9	13	15	13	12	10	2	2

a. Find the sample median and quartiles.

b. Compute the sample mean and sample standard deviation.

c. Compute the sample range and interquartile range.

d. Find the 85^th percentile.

Answer 1

a.

sample median M is the (n+1)/2 th item of the ordered data if n is odd and if n is even the median is the average of n/2 th item and n/2+1 th item

Here we have n=90 observations. The median is the average of 90/2=45th item and 90/2+1 = 46th item.

We can see that both 45th and 46th items have a length of 24

Hence the median is

the lower quartile (Q1) is the item at 25th percentile and the upper quartile Q3 is the item at 75th percentile.

There are different methods of calculating the quartiles. We use the following.

We know that the median is between the 45th and 46th items. the lower quartile is the median of lower 45 items and upper quartile is the median of upper 45 items (46th to 90th)

Lower quartile. Since n is odd the median of these 45 items is (45+1)/2=23rd item.

The 23rd item has a length of 22. Hence Q1=

Upper quartile is the 23rd item starting from 46th item, which is the 68th item.68th item has a length of 26. Hence Q3=

b) The sample mean is calculated as

where

f- frequency or the number of observations of each length

x- is the length

n=90 is the total number of observations

The sample standard deviation is

where

f - is the frequency of each length

x - is the length

is the sample mean

n=90 is the total number of observations

or

c) sample range is calculated as

Range = Highest value - lowest value = 29-19=10

Inter quartile range (IQR) is given by

IQR=Q3-Q1

We have already calculated Q1=22 and Q3=26

IQR=26-22=4

d) pth percentile is given by

n*p/100 th item if p<50% and

n+1-(n(1-p/100)) if p>50%

We want 85th percentile. Since p=85 and is >50%

n=90

The 85th percentile is

90+1 - (90*(1-0.85)) = 77.5 or 78th item is the 85th percentile

We can see from the table that 78th item has a length of 27. that means 85th percentile is