Now that we know how to calculate the mean, median, mode and range, we will take a look at how to find missing data values if we already know some of the statistics.
Let's launch right in with some examples.
Example 1:
The range for the following set of data, arranged in ascending order, is $10$10.
Calculate the missing value, $x$x.
$2$2 $4$4 $7$7 $10$10 $x$x
Think: Since we know the numbers are in ascending order and we know the range is $10$10, we can add the range onto the lowest value of $2$2 to get the highest value, $x$x.
Do:
| $x$x | $=$= | $2+10$2+10 |
| $x$x | $=$= | $12$12 |
Example 2
The median for the following set of data, arranged in ascending order, is $5$5, and the range is $15$15.
Calculate the missing values of $x$xand $y$y.
$x$x $3$3 $y$y $7$7 $16$16
Think: First let's work with the range and find $x$x. This time $x$x is the smallest value, and the largest value is $16$16. We can subtract the range from the largest value to get the smallest value.
Do:
| $x$x | $=$= | $16-15$16−15 |
| $x$x | $=$= | $1$1 |
Think: Now let's use the median value to find $y$y. We know the median is the middle value, and in this set of $5$5 numbers the middle value is $y$y.
Do: $y=5$y=5
Example 3:
The following set of data, arranged in ascending order, has a mean of $4$4, a median of $3$3 and a mode of $1$1.
$1$1 $x$x $2$2 $y$y $z$z $10$10
Find the values of $x$x, $y$y and $z$z.
Think: We can either start by using the information about the mode or the median. Let's start with the mode.
Do: Since there needs to be a mode of $1$1, and the numbers are in ascending order, then $x=1$x=1
Think: Now we'll find the median. There are $6$6 data points, so the median will be the average of $2$2 and $y$y.
Do:
| $3$3 | $=$= | $\frac{2+y}{2}$2+y2 |
| $6$6 | $=$= | $2+y$2+y |
| $6-2$6−2 | $=$= | $y$y |
| $y$y | $=$= | $4$4 |
Think: Lastly, we need to use the mean to find the value of $z$z.
Do:
| $4$4 | $=$= | $\frac{1+1+2+4+z+10}{6}$1+1+2+4+z+106 |
| $24$24 | $=$= | $18+z$18+z |
| $24-18$24−18 | $=$= | $z$z |
| $z$z | $=$= | $6$6 |
Worked Examples
QUESTION 1
Six numbers $6$6, $2$2, $7$7, $18$18, $17$17 and an unknown number $x$x have a median of $8.5$8.5.
Determine the value of $x$x
Write all the known numbers in increasing order.
(on one line with each number separated by a comma).
Determine the missing value $x$x
QUESTION 2
Five numbers have a range of $16$16, a mode of $2$2, a median of $7$7 and a mean of $8$8. The minimum number in the set is $2$2.
From the information given, we can determine straight away the following 3 values
minimum median maximum $\editable{}$ ?? $\editable{}$ ?? $\editable{}$ We also know that the mode is $2$2 and the mean is $8$8. Using one of these pieces of information, fill in the 2nd value in the table.
minimum median maximum $2$2 $\editable{}$ $7$7 ?? $18$18 The final score will require a calculation. Find the final score $x$x.
Hence, complete the table.
minimum median maximum $2$2 $2$2 $7$7 $\editable{}$ $18$18
QUESTION 3
Four numbers have a range of $5$5, a median of $9$9 and a mode of $11$11. Write the four numbers of this data set.
(on one line separated by a comma)