Find the missing value when the median of six numbers is known

Key takeaway: Median questions look simple, but the ordering matters a lot. With an even number of data points, the middle two values control everything, so the missing time has to fit that structure.

We have 6 race times with median $4.9$ hours.

For an even number of data points, the median is the average of the 3rd and 4th numbers after sorting.

The known times are:

$4.7,\ 8.3,\ 3.5,\ 4.6,\ 5.5,\ x$

Sort the five known values:

$3.5,\ 4.6,\ 4.7,\ 5.5,\ 8.3,\ x$

Now we need the two middle values to average to $4.9$. There are two useful cases:

• If $x$ is between $4.7$ and $5.5$, then the middle two values are $4.7$ and $x$.
• If $x$ is smaller than $4.7$, the middle pair changes, but it will not produce the required median.

So use

$\frac{4.7+x}{2}=4.9$

$4.7+x=9.8$

$x=5.1$

Thus the 6th time must be

$\boxed{5.1\text{ hours}}.$

Check the order:

$3.5,\ 4.6,\ 4.7,\ 5.1,\ 5.5,\ 8.3$

The middle two are $4.7$ and $5.1$, and their average is $4.9$.

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Pitfalls the pros know 👇 A frequent mistake is trying to average all six numbers. The median is not the mean. Also, you have to sort the data before using the middle values.

What if the problem changes? If the median had been $4.8$ instead, then the same reasoning would give $(4.7+x)/2=4.8$, so $x=4.9$. The exact missing value shifts with the target median, but the middle-two rule stays the same.

Tags: median, ordered data, middle values