Solutions with explanations to questions on mean, median and mode in sample 18.

What is the mean of the data set given below?
10.2 , 10.5 , 10.9 , 10.2 , 10.6 , 10.0
Solution
The mean is equal to the sum of all data values divided by the number of data values.
mean = (10.2 + 10.5 + 10.9 + 10.2 + 10.6 + 10.0) / 6 = 10.4

Which of the statements, related to the mean, mode and median of the data set given below is true?
12 , 15 , 10 , 19 , 5 , 5
A) the mean is equal to the median
B) the mode is larger than the mean
C) the median is smaller than the mode
D) the above data set has two modes
E) The median of the above data is equal to 14.5
Solution
The mean is equal to the sum of all data values divided by the number of data values.
(12 + 15 + 10 + 19 + 5 + 5) / 6 = 11
Mode: First arrange data from smallest to largest
5 , 5 , 10 , 12 , 15 , 19
Median is the data value in the middle of the ordered data set. Since the number of data values is even there is no single data value in the middle of the set and therefore the median is equal to the average of the two data values in the middle
median = (10 + 12) / 2 = 11
Mode is the data value with highest frequency of repetition.
mode = 5
Examining the value of the mean, median and mode, the statement in A) "the mean is equal to the median" is correct.

What is(are) the mode(s) of the data set given below?
20.1 , 30.5 , 10.1 , 20.1 , 10.6 , 30.5 , 10.1 , 10.5
Solution
Order given data set.
10.1 , 10.1 , 10.5 , 10.6 , 20.1 , 20.1 , 30.5 , 30.5
This data set has 3 modes which are.
10.1 , 20.1 , 30.5 : these data values have equal frequencies of repetition.

What is the median of the data set given below?
0 , 12 , 5 , 45 , 12 , 8 , 2
Solution
We first order the data values in the given set
0 , 2 , 5 , 8 , 12 , 12 , 45
Median is the data value in the middle of the ordered data set. Hence
median = 8

What is the mean of the fractions 2/3 , 5/6 and 1/2?
Solution
The mean is equal to the sum of all data values divided by the number of data values.
mean = (2/3 + 5/6 + 1/2) / 2 = (4/6 + 5/6 + 3/6) / 3
= (12/6) / 3 = 2 / 3

What is the median of the data set given below?
0.3 , 0.33 , 0.003 , 0.31 , 0.0003
Solution
We first order the data values in the gievn set
0.0003 , 0.003 , 0.3 , 0.31 , 0.33
Median is the data value in the middle of the ordered data set. Hence
median = 0.3

Which of these data sets has no mode(s)?
A) 57 , 24 , 57 , 21 , 49
B) 23 , 24 , 56 , 21 , 43
C) 20 , 20 , 20 , 20 , 20
D) 2000 , 3000 , 4000 , 2000 , 20000
E) 5 , 6 , 7 , 8 , 9 , 5
Solution
A data set with no repetition of data values has no mode. Only the data set in B) above has no data value that is repeated and therefore has no mode.

What is the mean, median and mode of the following data set?
101 , 99 , 102 , 105 , 100 , 98 , 102
Solution
The mean is equal to the sum of all data values divided by the number of data values.
(101 + 99 + 102 + 105 + 100 + 98 + 102) / 7 = 101
Mode: First arrange data from smallest to largest
98 , 99 , 100 , 101 , 102 , 102 , 105
105
Median is the data value in the middle of the ordered data set. Hence
median = 101
Mode is the data value with highest frequency of repetition.
mode = 102
Examining the value of the mean, median and mode, the statement in A) "the mean is equal to the median" is correct.

Which of these data sets has a mean less than 100?
A) 101 , 99 , 102 , 99 , 100
B) 90 , 110 , 101 , 100 , 100
C) 100 , 100 , 100 , 100 , 99
D) 50 , 155 , 101 , 100 , 98
E) 0 , 101 , 100 , 201 , 100
Solution
All the data set given above have 5 data values. The mean is equal to the sum of all data values divided by the number of data values. For the mean to be less than 100, the sum of all data values in a given set must be less than 5*100 = 500. Let us find the sum of all data values in all 5 sets given above:
A) (101 + 99 + 102 + 99 + 100) = 501
B) (90 + 110 + 101 + 100 + 100) = 501
C) (100 + 100 + 100 + 100 + 99) = 499
D) (50 + 155 + 101 + 100 + 98) = 504
E) (0 + 101 + 100 + 201 + 100) = 502
Since the sum in set B) is less than 500, its mean is less than 100.

Which of the following data sets has the largest mean?
A) 2001 , 2002 , 2008 , 2010
B) 2000 , 2010 , 2005 , 2002
C) 2999 , 1000 , 2001 , 2002
D) 2000 , 2000 , 2000 , 2000
E) 2010 , 2004 , 2012 , 2020
Solution
All the data set given above have 4 data values. For the mean of a data set to be the largest, the sum of all its data values must be the largest. Find the sum of the data values in each data set:
A) (2001 + 2002 + 2008 + 2010) = 801
B) (2000 + 2010 + 2005 + 2002) = 8017
C) (2999 + 1000 + 2001 + 2002) = 8002
D) (2000 + 2000 + 2000 + 2000) = 8000
E) (2010 + 2004 + 2012 + 2020) = 8046
The data set in E) has the largest sum and therefore has the largest mean.
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