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Mean, median mode and range (find missing data)


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.


$x$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.


$x$x $=$= $16-15$1615
$x$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.


$3$3 $=$= $\frac{2+y}{2}$2+y2
$6$6 $=$= $2+y$2+y
$6-2$62 $=$= $y$y
$y$y $=$= $4$4

Think: Lastly, we need to use the mean to find the value of $z$z.


$4$4 $=$= $\frac{1+1+2+4+z+10}{6}$1+1+2+4+z+106
$24$24 $=$= $18+z$18+z
$24-18$2418 $=$= $z$z
$z$z $=$= $6$6

Worked Examples


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

  1. Write all the known numbers in increasing order.

    (on one line with each number separated by a comma).

  2. Determine the missing value $x$x


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.

  1. From the information given, we can determine straight away the following 3 values

    minimum median maximum
    $\editable{}$ ?? $\editable{}$ ?? $\editable{}$
  2. 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
  3. The final score will require a calculation. Find the final score $x$x.

  4. Hence, complete the table.

    minimum   median   maximum
    $2$2 $2$2 $7$7 $\editable{}$ $18$18


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)



Plan and conduct surveys and experiments using the statistical enquiry cycle:– determining appropriate variables and measures;– considering sources of variation;– gathering and cleaning data;– using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;– comparing sample distributions visually, using measures of centre, spread, and proportion;– presenting a report of findings

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