Question

For the data set {1, -3, 5.5, 8.1, 9, -2, 0, 0.4, 21, -6.8, -5}, what are the mean, variance, Q1, Q2, Q3, IQR and mode, all accurate to one decimal point and in that order? Select one:

a. 2.6, 63.1, -3, 0.4, 8.1, 7.7, 0.

b. 2.6, 63.1, -3.4, 0.4, 8.9, 7.7, no mode.

c. 2.6, 7.9, -3, 0.4, 8.1, 7.7, no mode.

d. 2.6, 7.9, -3.4, 0.4, 8.9, 7.7, no mode.

e. 2.6, 7.9, -3, 0.4, 8.1, 7.7, 0.

f. 2.6, 63.1, -3, 0.4, 8.1, 7.7, no mode.

g. 2.6, 63.1, -3.4, 0.4, 8.9, 7.7, 0.

h. 2.6, 7.9, -3.4, 0.4, 8.9, 7.7, 0.

Answer 1

Given data set is : 1, -3, 5.5, 8.1, 9, -2, 0, 0.4, 21, -6.8, -5. Now, we will arrange all these numbers in ascending order. Thus, arranging in ascending order we have : -6.8, -5, -3, -2, 0, 0.4, 1, 5.5, 8.1, 9, 21.

Now, we have to find the mean. We know,

Mean() = (Sum of all values)/(Total number of values). Here, total number of values = 11.

Thus, Mean, = {(-6.8)+(-5)+(-3)+(-2)+0+0.4+1+5.5+8.1+9+21}/11 = 28.2/11 = 2.6(rounded up to one decimal place).

Thus, Mean, = 2.6 .

Now, we will find the variance. We know,

Variance = , where x is each data value, is the mean and n is the total number of values. First, we will find .

= 88.36 + 57.76 + 31.36 + 21.16 + 6.76 + 4.84 + 2.56 + 8.41 + 30.25 + 40.96 + 338.56 = 630.98

Thus, = 630.98. Here, n = 11.

Thus, Variance = 630.98/(11-1) = 630.98/10 = 63.098 = 63.1(rounded up to one decimal place).

Thus, Variance = 63.1 .

Now, we have to find the first quartile, Q1. We know Q1 is the median of the lower half. In a particular arranged data set, the values to the left side of the median is called the lower half. The values to the right side of the median is called the upper half. The median of the upper half is called the third quartile, Q3. The median is called the second quartile, Q2.

Now, in our arranged data set, there are 11 values (i.e., odd numbered). Thus, the median is the {(n+1)/2}th term. Here, n = 11. Thus, the median is the {(11+1)/2}th term. Thus, the median is the 6th term. The 6th term in our arranged data set is 0.4 . Thus, Median, Q2 = 0.4 .

Now, the numbers to the left side of the median (i.e, to the left side of 0.4) are -6.8, -5, -3, -2, 0. Thus, Q1 is the median of these numbers. Here, there are 5 values (i.e., odd numbered). Thus, the median is the {(n+1)/2}th term. Thus, the median is {(5+1)/2}th term. Thus, the median is 3rd term. Here, the 3rd term is -3. Thus, Q1 = -3.

Now, the numbers to the right side of the median (i.e., to the right side of 0.4) are 1, 5.5, 8.1, 9, 21. Thus, Q3 is the median of these numbers. Here, there are 5 values (i.e., odd numbered). Thus, the median is the {(n+1)/2}th term. Thus, the median is the {(5+1)/2}th term. Thus, the median is 3rd term. Here, the 3rd term is 8.1. Thus, Q3 = 8.1 .

Thus, Q1 = -3, Q2 = 0.4, Q3 = 8.1 .

Now, we have to find IQR (i.e., inter-quartile range). We know, IQR = Q3 - Q1 = 8.1 - (-3) = 8.1 + 3 = 11.1 .

Thus, IQR = 11.1 .

Now, we have to find the mode. We know, the value which occurs maximum number of times is called the mode. But, here, in our case, each number has occurred only once. Thus, there is no mode.

Thus, there is no mode.

Thus, Mean = 2.6, Variance = 63.1, Q1 = -3, Q2 = 0.4, Q3 = 8.1, IQR = 11.1, No mode.

All the answers match with option (f) except for the IQR. No option has the correct answer for IQR.