Question

The following data are the ages (in years) at diagnosis for 20 patients under treatment for meningitis: 18 18 25 19 23 20 69 18 21 18

20 18 18 20 18 19 28 17 18 18

(a) . Calculate and interpret the values of the sample mean, variance, and standard deviation.

(b) . Compute the sample median. Why might you recommend it as a measure of centre rather than the sample mean? 2

(c) . Compute the upper fourth, the lower fourth, and the fourth spread. (d) . Illustrate the center, spread, and symmetry or skewness of this data using a horizontal modified boxplot.

Answer 1

(a)

Following table shows the calculations:

	X	(X-mean)^2
	18	17.2225
	18	17.2225
	25	8.1225
	19	9.9225
	23	0.7225
	20	4.6225
	69	2194.9225
	18	17.2225
	21	1.3225
	18	17.2225
	20	4.6225
	18	17.2225
	18	17.2225
	20	4.6225
	18	17.2225
	19	9.9225
	28	34.2225
	17	26.5225
	18	17.2225
	18	17.2225
Total	443	2454.55

Sample mean shows average age (in years) at diagnosis for 20 patients under treatment for meningitis while standard deviation shows the measure of dispersion.

(b)

Following table shows the ordered data set:

S.NO.	X
1	17
2	18
3	18
4	18
5	18
6	18
7	18
8	18
9	18
10	18
11	19
12	19
13	20
14	20
15	20
16	21
17	23
18	25
19	28
20	69

Since there are 20 data values so median will be average of 10th and 11th data values. That is median is

Since it seems that there is an outlier namely 69 so median should be used as a measure of center rather than the sample mean.

(c)

The lower fourth or first quartile:

There are 10 data values in the first half of data set so first quartile will be average of 5th and 6th data values. So

The upper fourth or third quartile:

There are 10 data values in the second half of data set so third quartile will be average of 15th and 16th data values. So

The fourth spread is

(d)

Lower fence of box plot:

Upper fence of box plot:

Since data values 25, 28 and 69 are not between fences so these are outliers.

Following is the box plot;