Mean, Median, Mode, Range

Mean, Median, Mode, Range exercise

• Determine the information that the table shows.

  Hints

  Member $3$ from the German team ate $31$ bratwursts and Member $4$ from the Canadian team ate $43$ bratwursts.

  The mean of the German team is $\frac{25+40+31+29+90}{5}=\frac{215}{5}=43$.

  The mode is the number that occurs the most in a data set. The German and US teams have no mode because each data value shows up the same amount of times.

  The range of the Canadian team is $43-3=40$.

  Solution

  1. We are not given the names of the team members.
  2. We can see the number of bratwursts eaten by each member of the team. For example, Member $3$ from the German team has eaten $31$ bratwursts, and Member $4$ from the Canadian team has eaten $43$ bratwursts. We can also see that Team Canada only has $4$ members.
  3. We can use the information from the table to find the mean. However, it takes some calculating. We calculate the mean as the sum of all elements divided by the number of the elements.
  4. We can also determine the median as the middle of the ordered list, if the number of elements is odd, for example, the median of Team USA is given by $30$. If the number is even, as the data set of Team Canada, the median is the average of both middle data values of the ordered list: $\frac{23+23}{2}=23$.
  5. There is only one team, Canada, that we can determine the mode for, the data value that's shown most. Canada's mode is $23$. The German and US teams have no mode because each data value shows up the same amount of times.
  6. With this table we can determine the range for each team as the difference between the highest and the lowest data value.

  • Team Germany's range: $90-25=65$.
  • Team USA's range: $63-24=39$.
  • Team Canada's range: $43-3=40$.

• Explain how to find the median, mode, and range of a data set.

  Hints

  $\begin{array}{c|c|c|c|c} \text{Member 1}&\text{Member 2}&\text{Member 3}&\text{Member 4}&\text{Member 5}\\ \hline 25&40&31&29&90 \end{array}$

  The mean of this data set is $43$. As you can see from the table, four elements are lower and just one data element is higher than $43$.

  $\begin{array}{c|c|c|c|c} \text{Member 1}&\text{Member 2}&\text{Member 3}&\text{Member 4}&\text{Member 5}\\ \hline 25&40&31&29&90 \end{array}$

  The data elements can be rewritten as $25$, $29$, $31$, $40$, $90$. The median of this list is given by $31$.

  $\begin{array}{c|c|c|c|c} \text{Member 1}&\text{Member 2}&\text{Member 3}&\text{Member 4}&\text{Member 5}\\ \hline 25&40&31&29&90 \end{array}$

  This data set doesn't contain a mode.

  $\begin{array}{c|c|c|c|c} \text{Member 1}&\text{Member 2}&\text{Member 3}&\text{Member 4}&\text{Member 5}\\ \hline 25&40&31&29&90 \end{array}$

  The range of this data set is $90-25=65$.

  Solution

  When examining data sets you can use measures of central tendency (mean, median, and mode) or range, as a measure of dispersion, to describe the set.

  Measures of Central Tendency

  • The mean, also known as the average, is the sum off all elements of the set divided by the number of elements.
  • The median is the middle of the ordered set. So you first have to order a given data set and then take the middle of an odd numbered set. If the number of elements is even, you take the average of the both elements in the middle.
  • The mode of a data set is the data value that's repeated most. If all numbers in the set occur the same amount of times then there is no mode.

  Measure of Dispersion

  • The range of a data set is defined as the difference between the highest and lowest data value.

• Determine the mean and the median of the data set.

  Hints

  The mean of the following list $2$, $4$, $4$, $3$, $7$ is calculated by $\dfrac{2+4+4+3+7}{5}=\dfrac{20}{5}=4$.

  The average of two numbers is the sum of those numbers divided by $2$.

  For example, the average of $12$ and $36$ is $\frac{12+36}2=\frac{48}2=24$.

  To get the median of any data list you first have to order it, starting with the lowest value.

  The median separates the list in two list, both with the same number of elements.

  Solution

  From this table you see the ordered list of Aaron and Blake's number of goals. It's only important to order the list for the median, but you can also calculate the mean of an ordered list.

  Aaron:

  • The mean is the sum of all elements divided by the number of the elements: $\frac{0+3+4+4}{4}=\frac{11}{4}=2.75$. Note, that the mean doesn't have to belong to the data set.
  • The median is the middle number of an ordered data set. If the ordered list is even, the median is the average of the both middle data values. Here, the median is $\frac{3+4}{2}=\frac{7}{2}=3.5$. The median also doesn't have to belong to the data set.

  Blake

  • The mean is given by $\frac{1+2+3+5}{4}=\frac{11}{4}=2.75$.
  • We calculate the median as follows: $\frac{2+3}{2}=\frac{5}{2}=2.5$.

  Comparing Aaron and Blake

  • Aaron and Blake have the same mean of $2.75$.
  • Aaron has a greater median than Blake.
  • Because Aaron's median is greater we can consider Aaron the winner.

  It may be worth noting that Aaron's mode is $4$. Also Aaron and Blake have the same range of $4$.

• Calculate the mean, median, mode, and range of the given data set.

  Hints

  The mean is the sum of all elements divided by the number of elements.

  The data values have been ordered for you: $14$, $16$, $16$, $17$, $21$, $21$, $21$.

  The median is the middle of the ordered list.

  The mode is the data value that's repeated most.

  The range is the difference between the highest and the lowest value.

  Solution

  First we put the data set in order: $14$, $16$, $16$, $17$, $21$, $21$, $21$.

  • We can determine the range as the difference between the highest, $21$, and the lowest, $14$, data value; thus, $21-14=7$.
  • The mean is the sum of all elements divided by the number of the elements: $\dfrac{14+16+16+17+21+21+21}7=\frac{126}7=18$.
  • The median is the middle of the ordered list. Here it is $17$.
  • The mode is the data value that's repeated most: $21$.

• Explain how to calculate the mean.

  Hints

  The mean is also known as the average.

  For example the average, also known as mean, of $10$ and $12$ is given by $\frac{10+12}2=\frac{22}2=11$.

  If you multiply the mean by the number of all elements you get the total sum of all the bratwursts eaten.

  Solution

  The mean of a data set is the sum of all elements of the set divided by the total number of the elements. It's also known as the average.

  Using the results of the German team:

  $\begin{array}{c|c|c|c|c} \text{Member 1}&\text{Member 2}&\text{Member 3}&\text{Member 4}&\text{Member 5}\\ \hline 25&40&31&29&90 \end{array}$

  • We first add all numbers of bratwursts: $25+40+31+29+90=215$.
  • Next, we divide this sum by $5$, the number of all elements. This gives us the mean $\frac{215}5=43$.

• Determine the mean, median, and range of the following data sets.

  Hints

  The range is the difference between the highest and the lowest data value.

  The mean is the sum of all elements divided by the number of the elements.

  The median is the middle of the ordered list.

  If the number of elements is even, the median is the average of the both elements in the middle.

  Let's have a look at this example: $2$, $3$, $7$, $9$

  • The mean: $\frac{2+3+7+9}4=\frac{21}4=5.25$
  • The median: $\frac{3+7}2=\frac{10}2=5$
  • The range: $9-2=7$

  Solution

  Mean is the sum of all the data values divided by the total number of values. Median is the middle number in the ordered data set. Range is the difference between the highest and lowest data values.

  Carlos' information: $3$, $7$, $9$, $5$

  • The mean: $\frac{3+7+5+9}4=\frac{24}4=6$
  • The median: First order the set $3$, $5$, $7$, $9$. Then find the number in the middle $\frac{5+7}2=\frac{12}2=6$
  • The range: $9-3=6$

  It is possible for all three values to be the same.

  Alexis' information: $3$, $7$, $8$, $9$, $5$

  • The mean: $\frac{3+7+8+9+5}5=\frac{32}5=6.4$
  • The median: $3$, $7$, $7$, $8$, $9$ the middle of the ordered list is $7$.
  • The range: $9-3=6$

  Jordan's information: $5$, $7$, $9$, $5$, $12$, $4$

  • The mean: $\frac{5+7+9+5+12+4}6=\frac{42}6=7$
  • The median: $4$, $5$, $5$, $7$, $12$, the middle number of the ordered list is $\frac{5+7}2=\frac{12}2=6$
  • The range: $12-4=8$