Solutions with detailed explanations to questions on mean, median and mode in sample 18.
What is the mean of the data set given below?
\[ 10.2,\; 10.5,\; 10.9,\; 10.2,\; 10.6,\; 10.0 \]The mean is the sum of all data values divided by the number of data values.
\[ \text{mean} = \frac{10.2 + 10.5 + 10.9 + 10.2 + 10.6 + 10.0}{6} = 10.4 \]Which statement about the mean, mode, and median of the data set 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 data set has two modes
E) The median is equal to 14.5
Mean: sum of values divided by count.
\[ \text{mean} = \frac{12 + 15 + 10 + 19 + 5 + 5}{6} = 11 \]Order the data: \(5, 5, 10, 12, 15, 19\).
Median (even number of values): average of the two middle numbers.
Mode: value with highest frequency → \(5\).
Comparing, the correct statement is A) the mean is equal to the median.
What is (are) the mode(s) of the data set below?
\[ 20.1,\; 30.5,\; 10.1,\; 20.1,\; 10.6,\; 30.5,\; 10.1,\; 10.5 \]Order the data: \(10.1, 10.1, 10.5, 10.6, 20.1, 20.1, 30.5, 30.5\).
Three values appear twice: \(10.1\), \(20.1\), and \(30.5\). All are modes.
What is the median of the data set below?
\[ 0,\; 12,\; 5,\; 45,\; 12,\; 8,\; 2 \]Order the values: \(0, 2, 5, 8, 12, 12, 45\).
The median is the middle value.
What is the mean of the fractions \(\frac{2}{3}\), \(\frac{5}{6}\), and \(\frac{1}{2}\)?
Mean = sum of values divided by number of values.
\[ \text{mean} = \frac{\frac{2}{3} + \frac{5}{6} + \frac{1}{2}}{3} = \frac{\frac{4}{6} + \frac{5}{6} + \frac{3}{6}}{3} = \frac{\frac{12}{6}}{3} = \frac{2}{3} \]What is the median of the data set below?
\[ 0.3,\; 0.33,\; 0.003,\; 0.31,\; 0.0003 \]Order the values: \(0.0003,\; 0.003,\; 0.3,\; 0.31,\; 0.33\).
The median is the middle value.
Which data set 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) \(5, 6, 7, 8, 9, 5\)
A data set has no mode if no value repeats. Only set B has all distinct values.
\[ \text{Set B has no mode.} \]Find the mean, median, and mode of the following data set:
\[ 101,\; 99,\; 102,\; 105,\; 100,\; 98,\; 102 \]Mean:
\[ \text{mean} = \frac{101 + 99 + 102 + 105 + 100 + 98 + 102}{7} = 101 \]Ordered: \(98, 99, 100, 101, 102, 102, 105\)
Median (middle value): \(101\)
Mode (most frequent): \(102\)
Which data set 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) \(0, 101, 100, 201, 100\)
Each set has 5 values. For mean < 100, sum < \(5 \times 100 = 500\). Compute sums:
\[ \begin{aligned} &\text{A: } 101+99+102+99+100 = 501 \\ &\text{B: } 90+110+101+100+100 = 501 \\ &\text{C: } 100+100+100+100+99 = 499 \\ &\text{D: } 50+155+101+100+98 = 504 \\ &\text{E: } 0+101+100+201+100 = 502 \end{aligned} \]Only set C has sum 499 (< 500), so its mean is less than 100.
Which data set 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) \(2010, 2004, 2012, 2020\)
All sets have 4 values. Largest mean corresponds to largest sum.
\[ \begin{aligned} &\text{A: } 2001+2002+2008+2010 = 8021 \\ &\text{B: } 2000+2010+2005+2002 = 8017 \\ &\text{C: } 2999+1000+2001+2002 = 8002 \\ &\text{D: } 2000+2000+2000+2000 = 8000 \\ &\text{E: } 2010+2004+2012+2020 = 8046 \end{aligned} \]Set E has the largest sum, therefore the largest mean.