Q30. - Algebra 1

Question

Find a set of numbers that satisfies each list of conditions.

 

a. The mean, median and mode are all the same number.

b. The mean is greater than the median.

c. The mode is 10 and the median is greater than the mean.

d. The mean is 6, the median is 5.5, and the mode is 9.

Step-by-Step Solution

Answer

Answer of a.

The set of a numbers whose mean, median and mode are equal is $\left\{2,5,5,6,7\right\}$.

Answer of b.

The set of a numbers whose mean is greater than the median is $\left\{2,3,4,7,9\right\}$.

Answer of c.

The set of a numbers whose mode is 10 and the median is greater than the mean is  $\left\{1,5,10,10,14\right\}$.

Answer of d.

The set of a numbers whose mean is 6, median is 5.5 and mode is 9 is $\left\{3,4,5,6,9,9\right\}$.

Part a. Step 1. State the concept of Central tendency.

The measure of central tendency is a value that describe the data by a central value. The central values are mean, median and mode.

Mean is the average of all the data points.

Median is the middle value after arranging the data in ascending order.

Mode is the value that is the most frequent value in the data.

Part a. Step 2. State the formulae to find the central tendencies.

Mean of a set of number $x_1,x_2,x_3,\mathrm{.............}{x}_n$ is defined by, 

$\frac{x_1+x_2+x_3+\mathrm{.............}+x_n}{n}$

 The Median is:

 $x_{\left(n+1\right)/2}\left[\text{if }n\text{ is odd}\right]$

$\frac{x_{\left(n/2\right)}\text{+}{x}_{\left(n/2\right)+1}}{2}\left[\text{if }n\text{ is even}\right]$

 The mode is the number in the list that occurs most often.

Part a. Step 3 . Solve with example.

Assume the set of the number is, $\left\{2,5,5,6,7\right\}$.

Mean of the number is obtained as:

$\begin{array}{c}\text{Mean}=\frac{2+5+5+6+7}{5}\\ =\frac{25}{5}\\ =5\end{array}$

Median is obtained as:

$\begin{array}{c}\text{Median}=x_{\left(n+1\right)/2}\\ =x_{\left(5+1\right)/2}\\ =x_{6/2}\\ =x_3\\ =5\end{array}$

Mode is 5 since it occurs more often. 

Hence, 

The set of a numbers whose mean, median and mode are equal is $\left\{2,5,5,6,7\right\}$.

Part b. Step 1. State the concept of Central tendency.

The measure of central tendency is a value that describe the data by a central value. The central values are mean, median and mode.

Mean is the average of all the data points.

Median is the middle value after arranging the data in ascending order.

Mode is the value that is the most frequent value in the data.

Part b. Step 2. State the formulae to find the central tendencies.

Mean of a set of number $x_1,x_2,x_3,\mathrm{.............}{x}_n$ is defined by, 

$\frac{x_1+x_2+x_3+\mathrm{.............}+x_n}{n}$

 

The Median is:

 $x_{\left(n+1\right)/2}\left[\text{if }n\text{ is odd}\right]$

$\frac{x_{\left(n/2\right)}\text{+}{x}_{\left(n/2\right)+1}}{2}\left[\text{if }n\text{ is even}\right]$

The mode is the number in the list that occurs most often.

Part b. Step 3 . Solve with example.

Assume the set of the number is, $\left\{2,3,4,7,9\right\}$.

Mean of the number is obtained as:

$\begin{array}{c}\text{Mean}=\frac{2+3+4+7+9}{5}\\ =\frac{25}{5}\\ =5\end{array}$

Median is obtained as:

$\begin{array}{c}\text{Median}=x_{\left(n+1\right)/2}\\ =x_{\left(5+1\right)/2}\\ =x_{6/2}\\ =x_3\\ =4\end{array}$

So, mean is greater than median.

Hence, 

The set of a numbers whose mean is greater than the median is $\left\{2,3,4,7,9\right\}$.

Part c. Step 1. State the concept of Central tendency.

 The measure of central tendency is a value that describe the data by a central value. The central values are mean, median and mode.

Mean is the average of all the data points.

Median is the middle value after arranging the data in ascending order.

Mode is the value that is the most frequent value in the data.

Part c. Step 2. State the formulae to find the central tendencies.

Mean of a set of number is $x_1,x_2,x_3,\mathrm{.............}{x}_n$ defined by, 

$\frac{x_1+x_2+x_3+\mathrm{.............}+x_n}{n}$

 

The Median is:

$x_{\left(n+1\right)/2}\left[\text{if }n\text{ is odd}\right]$

$\frac{x_{\left(n/2\right)}\text{+}{x}_{\left(n/2\right)+1}}{2}\left[\text{if }n\text{ is even}\right]$

The mode is the number in the list that occurs most often.

Part c. Step 3 . Solve with example.

Assume the set of the number is, $\left\{1,5,10,10,14\right\}$.

Mean of the number is obtained as:

$\begin{array}{c}\text{Mean}=\frac{1+5+10+10+14}{5}\\ =\frac{40}{5}\\ =8\end{array}$

Median is obtained as:

$\begin{array}{c}\text{Median}=x_{\left(n+1\right)/2}\\ =x_{\left(5+1\right)/2}\\ =x_{6/2}\\ =x_3\\ =10\end{array}$

So, median is greater than mean. 

Mode is 10 since it occurs more often. 

Hence, 

The set of a numbers whose mode is 10 and the median is greater than the mean is $\left\{1,5,10,10,14\right\}$.

Part d. Step 1. State the concept of Central tendency.

The measure of central tendency is a value that describe the data by a central value. The central values are mean, median and mode.

Mean is the average of all the data points.

Median is the middle value after arranging the data in ascending order.

Mode is the value that is the most frequent value in the data.

Part d. Step 2. State the formulae to find the central tendencies.

Mean of a set of number $x_1,x_2,x_3,\mathrm{.............}{x}_n$ is defined by, 

$\frac{x_1+x_2+x_3+\mathrm{.............}+x_n}{n}$

 

The Median is:

$x_{\left(n+1\right)/2}\left[\text{if }n\text{ is odd}\right]$

 $\frac{x_{\left(n/2\right)}\text{+}{x}_{\left(n/2\right)+1}}{2}\left[\text{if }n\text{ is even}\right]$

The mode is the number in the list that occurs most often.

Part d. Step 3. Solve with example.

Assume the set of the number is, $\left\{3,4,5,6,9,9\right\}$.

Mean of the number is obtained as:

$\begin{array}{c}\text{Mean}=\frac{3+4+5+6+9+9}{6}\\ =\frac{36}{6}\\ =6\end{array}$

Median is obtained as:

$\begin{array}{c}\text{Median}=\frac{x_{n/2}+x_{\left(n/2\right)+1}}{2}\\ =\frac{x_{6/2}+x_{\left(6/2\right)+1}}{2}\\ =\frac{x_3+x_4}{2}\\ =\frac{5+6}{2}\\ =\frac{11}{2}\\ =5.5\end{array}$

So, median is greater than mean. 

Mode is 9 since it occurs more often. 

Hence, 

The set of a numbers whose mean is 6, median is 5.5 and mode is 9 is $\left\{3,4,5,6,9,9\right\}$