Question

A survey was conducted among 20 adults. The following shows the age of the respondents.

44 47 47 47 47 52 53 53 54 54

55 56 57 58 58 64 66 66 69 83

(1) Please calculate the mean, median, 1st quartile (i.e. 25th percentile), 3rd quartile (i.e. 75th percentile), and IQR for their age. (2 points for each question, 10 points in total)

(2) For the above 20 adults, are there outliers (i.e. are there people with extreme age)? Please show your calculations for identifying the outliers. (2 points)

Answer 1

1) Mean = ; ∑x is total of all data points

Mean = 1130/20 = 56.5

Median :

There are total 20 data points , so 2nd quartile or median of the data set is average of the 10th and 11th observation in the ordered ( ascending order ) data set . so 10th observation is 54 and 11th observation is 55

So Median = (54+55) /2 = 54.5

1st quartile (Q₁) :

The median divides the data set into two equal parts of 10 observations.

So the 1st quartile is the middle most number of first 10 observations.

1st quartile is average of 5th and 6th observation in the ordered data set, 5th observation is 47 and 6th observation is 52

Therefore 1st quartile = ( 47+52)/2 = 49.5

3rd quartile (Q₃):

So the 3rd quartile is the middle most number of last 10 observations.

3rd quartile is average of 15th and 16th observation ,in the ordered data set 15th observation is 58 and 16th observation is also 64

Therefore 3rd quartile = ( 58+64)/2 = 61

IQR = Q₃- Q₁   = 61 - 49.5 = 11.5

2) Lower bound = Q₁ - 1.5*IQR = 49.5 - 11.5 = 32.25

Upper bound = Q₃ - 1.5*IQR = 61 + 11.5 = 78.25

The observation in the data set that lies outside these bounds ( 32.25 , 78.25)  is considered as an outlier.

So 83 is an outlier.