Question

2. Suppose that the data for analysis includes the attribute age . The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

(a) What is the mean of the data? What is the median ?

(b) What is the mode of the data? Comment on the data’s modality (i.e., bimodal, trimodal, etc.).

(c) What is the midrange of the data?

(d) Can you find (roughly) the first quartile (Q 1) and the third quartile (Q 3) of the data? (e) Give the five-number summary of the data.

(f) Show a boxplot of the data.

(g) How is a quantile-quantile plot different from a quantile plot ?

Answer 1

Solution:

Part a

For the given data, we have

Total sum = ∑x = 809

Sample size = n = 27

Sample mean = ∑x / n = 809/27 = 29.96296

Median = Middle most observation when data is in increasing order = 14^th obs.

Median = 25

Part b

Mode = Most repeated observation = Observation with highest frequency

There are two observations 25 and 35 are repeated most times (4 times)

So, modes are 25 and 35

So, data is bimodal.

Part c

From given data, we have

Minimum = 13

Maximum = 70

Mid-range = (Minimum + Maximum) / 2

Mid-range = (13 + 70)/2

Mid-range = 41.5

Part d

WE are given n = 27

First quartile = Q1 = (1/4)*27 = 6.75 ≈ 7^th observation when data is in increasing order

First quartile = 20

Third quartile = Q3 = (3/4)*27 = 20.25 ≈ 20^th observation when data is in increasing order

Third quartile = 35

Part e

Five number summary is given as below:

Minimum = 13

First quartile = 20

Median = 25

Third quartile = 35

Maximum = 70

Part f

Box plot for the given data is given as below:

Part g

We know that quantile – quantile plot used for two data sets whether they come from same population or not; while quantile plot is used for single data set.