Mean Median Mode Conditions Examples

Instead of computing a mean, median, or mode from a complete data set, many contest problems give the statistic and ask for a missing value. This runs each definition backward: the mean gives a direct equation for a missing sum, but the median depends on sorted position, which sometimes must be figured out rather than assumed — and every candidate answer should be checked against every condition the problem states.

What you will learn

• Work backward from a stated mean, median, or mode to find missing data
• Combine multiple statistical conditions in one problem
• Check a candidate data set against every stated condition

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Working backward from the mean

Since the mean is the sum divided by the count, a known mean and count immediately give the total sum: $\text{sum}=\text{mean}\times\text{count}$. If all but one value in the data set are known, subtracting their total from this required sum finds the missing value directly. Five numbers with a mean of 20 must have a sum of $20\times5=100$; if four of them are known, the fifth is simply 100 minus their total.

This same idea extends to problems with more than one missing value, as long as enough other information (from a median or mode condition, for instance) is available to pin down the remaining unknowns.

Working backward from the median: position matters

Unlike the mean, the median depends on where a value lands once the data is sorted — so an unknown value's position is not always obvious in advance. If a data set is $4,9,x,15$ with $x>9$ and a stated median of 11, the position of $x$ relative to 15 must be figured out, not assumed. Trying the natural guess that $x$ falls between 9 and 15 gives sorted order $4,9,x,15$, so the median is the average of the two middle values: $\frac{9+x}{2}=11$, giving $x=13$ — which indeed satisfies $9<13<15$, confirming the assumed order was correct.

If solving under an assumed order produces a value that does not actually fit that order (for example, solving under "$x$ is between 9 and 15" but getting an $x$ greater than 15), that case must be discarded and a different ordering tried instead. This is why every solution to a median-backward problem needs a quick check that the found value actually belongs where it was assumed to be.

Combining multiple conditions and checking every one

A single problem often states more than one statistical condition at once — a mean and a mode, or a mean and a median — and each condition contributes its own piece of information toward the unknowns. Solving typically means using whichever condition gives the most direct equation first, then checking that the resulting values are also consistent with every other stated condition, not just the one used to solve for them.

For a data set $3,5,5,8,12,x$ with mean 7, the mean gives $x=6(7)-(3+5+5+8+12)=42-33=9$ directly. Since the problem's data already has a repeated value, 5, it is worth checking that $x=9$ does not accidentally tie or change that mode — sorting the full set $3,5,5,8,9,12$ confirms 5 is still the unique mode, so the answer is consistent with every condition, not just the mean.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Solving a median-backward problem under an assumed order without checking that the resulting value actually satisfies that order.
• Using the mode or median formula in place of the mean formula (or the reverse) when working backward for a missing value.
• Solving for a missing value from one stated condition and reporting it without checking that it is also consistent with every other condition given in the problem.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A known mean and count give the total sum directly; subtract the known values to solve for a single missing one.
• A missing value's contribution to the median depends on its sorted position, which must be assumed, solved for, and then verified.
• When a problem states more than one condition, solve using the most direct one first, then check the result against every other condition.