Statistics for Quant: averages, medians and spread

Think in sums

The mean of a list is its sum divided by the number of values, so

$$\text{sum} = \text{mean} \times \text{number of values}$$

Almost every averages question becomes easy once you switch from means to sums. If five numbers average 18, they add to 90. If a sixth number lifts the average to 20, the six add to 120, so the new number is $120 - 90 = 30$. A second way to see the same thing: the new number must equal the new mean (20) plus enough to lift each of the five old numbers by 2, that is $20 + 5 \times 2 = 30$.

Weighted averages

When groups of different sizes are combined, the overall mean is a weighted average:

$$\text{mean} = \frac{n_1 m_1 + n_2 m_2}{n_1 + n_2}$$

It lies between the two group means, closer to the larger group. The balance method from mixtures applies: the distances from the overall mean to the group means are in the inverse ratio of the group sizes. Averaging the two means directly is right only when the groups are the same size.

Median, mode and range

The median is the middle value once the list is sorted. With an even number of values it is the mean of the two middle values. The median ignores how far the extreme values are from the middle, so one huge value moves the mean but not the median. The mode is the most frequent value (a list can have several). The range is the largest value minus the smallest.

Evenly spaced sets

In a list with a constant gap between consecutive values (consecutive integers, consecutive multiples of 5, and so on), the mean equals the median, and both equal the average of the first and last values:

$$\text{mean} = \text{median} = \frac{\text{first} + \text{last}}{2}$$

The number of values is $\frac{\text{last} - \text{first}}{\text{gap}} + 1$; forgetting the $+1$ is the usual off-by-one slip. The sum is the number of values times the mean, so the integers from 11 to 40 add to $30 \times 25.5 = 765$.

Standard deviation without computing it

Standard deviation measures how far the values typically lie from the mean. You will rarely need to compute it; you need to know how it reacts:

Change to every value	Effect on the standard deviation
add or subtract the same constant	none: the values move together
multiply by $k$	multiplied by $\lvert k \rvert$
add a new value equal to the mean	decreases (unless it is already 0)
add a new value far from the mean	increases

To compare two sets, compare the distances of their values from their own means. $\{1, 5, 9\}$ and $\{101, 105, 109\}$ have the same standard deviation; $\{1, 9, 17\}$ has twice as much as either.

Extremes under constraints

"What is the greatest possible value of the largest number?" asks you to make every other number as small as the conditions allow, and then let the sum do the rest. Read every condition: whether the numbers must be different, positive, integers, and where the median sits.