Work backwards from the mean or median

Missing values

In 5 of 20 recent exams
Paper 1 or 2
2 to 5 marks

How it is set in the exam

Paper 2 (calculator) mostly, 2–5 marks, a short question on its own or one part of a longer one. Typical (5 of 20 exam versions, 1 of them on Paper 1 (no calculator)). Two of the five are inside a regression question.

Spot it in the exam

Exam phrases to spot (4)

“the mean is …; find $p$ and $q$ ”
“two more values are added and the mean becomes …”
“the median is …; find $k$ ”
“find the missing value”

In disguise (3)

a value was recorded wrongly and is corrected, and the new mean is given: the total changes by the size of the correction
"how many more values of … must be added": the unknown is the number of values, and it appears in the total and in $n$
two values are added beyond both ends of the data (one below the old least value, one above the old greatest, as on a box plot), and the new range is given instead of their difference: the range is the larger added value minus the smaller, so it is still a sum and a difference

Variants (2)

given: the total frequency and the mean; the mean and a relation between the unknowns; the old mean, the new mean and the difference between the added values; the median
context: inside a regression question, where the mean of $y$ comes from the regression line

Looks like it, but isn't (3)

the unknowns are probabilities in a table, with a total of $1$ and $\mathrm{E}(X)$ given: Probability distribution tables
one value is added on each side of the median, neither of them between the two middle values, and nothing is unknown: the median doesn't move (Averages)
two regression lines are given and both means are asked for, with nothing missing: The mean point (solve the two lines simultaneously)

Worked example

Two unknown frequencies from the total and the mean

Paper 1 (no calculator)

Question

$25$ office workers were asked how many cups of coffee they drank yesterday.

Cups	Frequency
$0$	$4$
$1$	$p$
$2$	$6$
$3$	$q$
$4$	$3$

Cups	$0$	$1$	$2$	$3$	$4$
Frequency	$4$	$p$	$6$	$q$	$3$

The mean number of cups is $2$ . Find the value of $p$ and the value of $q$ . [5]

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Which method?

Given	Method
a mean	total $= n \times$ mean, one equation for each fact, then solve them together
a median	the median's position, $\frac{n + 1}{2}$ : count through the running totals of the frequencies until the middle value(s) are the median. A median halfway between two table values means equal counts on the two sides; a median that is itself a table value only needs the middle position to fall among its copies, which gives an inequality (Your turn 4)
a mean inside a regression question	$\bar{y}$ from the line at $\bar{x}$ first (The mean point), then total $= n \times \bar{y}$

Steps

Totals, not means:
total $= n \times$ mean, before and after any change. Never add two means. Averaging two means is right only when the groups are the same size; totals always work.
One equation per fact:
the frequencies add up to $n$ ; $\sum fx = n \times$ mean; any stated relation between the unknowns. A mean equation can keep an unknown $n$ in the denominator.
Solve
the equations together (by hand, or simultaneous linear equations on ).
Median:
the median's position fixes how many values lie below and above it. A median halfway between two table values means the counts on the two sides are equal. For the median to be a table value $v$ that is in the data: fewer than half of the values must lie below $v$ , and fewer than half above it. For a least (or greatest) possible frequency, find the smallest (or largest) value that makes both true, then work out the median with one less (or one more) to check that it is no longer $v$ .
Check:
frequencies must be whole numbers, $0$ or more; recompute the mean or the median from the completed data.

Common mistakes

Working with means instead of totals
the mean of the $20$ parcels is not the average of $4.2$ and the new masses.
$\sum f$ instead of $\sum fx$
"the mean is $2$ " is about the $50$ cups, not the $25$ workers.
Treating a median like a mean
setting $\frac{\sum fx}{n} = 2.5$ in Example 2.3, or using $2.5$ as a position.

More mistakes (3)

In a regression question, putting the missing point's $x$ into the regression line
this earns nothing, because the line need not pass through any data point; only the mean point is certain to be on it. Use the mean point: $\bar{y}$ from $\bar{x}$ , then the total.
No working
correct values alone earn $2$ of $5$ marks.
Not checking
a frequency that comes out negative or fractional means an equation is wrong.

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