Averages and basic statistics

Think in totals

The mean of a list is its sum divided by the number of values. Turned round, it gives the single most useful fact in average problems:

$$\text{sum} = \text{mean} \times \text{count}.$$

Whenever a value is added, removed or changed, convert means into sums, do the arithmetic on the sums, and convert back at the end.

If five numbers have a mean of 14, their sum is 70. If one number is removed and the other four have a mean of 12, their sum is 48, so the removed number was $70 - 48 = 22$. No single number needs to be known.

Mental means with deviations

Without a calculator, pick a convenient reference value close to the data and average the deviations from it. For 98, 103, 101, 96 and 107, take 100 as the reference. The deviations are $-2, +3, +1, -4, +7$, which add to $+5$, an average of $+1$. So the mean is $100 + 1 = 101$. Check: the sum is 505 and $\frac{505}{5} = 101$. The same idea shows that deviations from the mean itself always add to zero.

Weighted averages

Combining groups of different sizes needs a weighted average: each group's mean counts in proportion to its size.

$$\text{combined mean} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}.$$

A group of 20 with mean 65 and a group of 30 with mean 75 have total $1300 + 2250 = 3550$ over 50 people, a combined mean of 71, not the simple mean of 70. The combined mean always lies between the two group means, closer to the larger group: here 71 is nearer to 75, because there are more people in that group.

Weights may come as shares rather than counts. If 40% of a firm's staff earn on average 30 thousand and 60% earn 40 thousand, the overall mean is $0.4 \times 30 + 0.6 \times 40 = 12 + 24 = 36$ thousand.

Median, mode and range

• Median: the middle value once the data are sorted. With an odd number of values it is the middle one; with an even number, the mean of the two middle ones. For 7, 3, 9, 4, 12, 6, sorting gives 3, 4, 6, 7, 9, 12, and the median is $\frac{6 + 7}{2} = 6.5$.
• Mode: the most frequent value. A list may have several modes or none.
• Range: largest minus smallest, a simple measure of spread.

The mean uses every value, so one extreme value can pull it far; the median depends only on the order, so it resists outliers. In the list 2, 3, 3, 4, 48, the mean is 12 but the median is 3. When a question asks which measure "best describes a typical value" in skewed data, this is why the median is often the answer.

Shifting and scaling the data

Changing every value in the same way changes the measures predictably:

Change to every value	Mean, median, mode	Range
add $c$	add $c$	unchanged
multiply by $k > 0$	multiply by $k$	multiply by $k$
multiply by $k$, then add $c$	$\times k$, then $+ c$	$\times k$ only

Adding a constant slides the whole data set along without stretching it, so the gaps between values, and hence the range, stay the same. Multiplying stretches every gap by the same factor.

Quick checks

A mean lies between the smallest and largest values. A combined mean lies between the group means. If your answer breaks either rule, recheck the totals.