Effects of data changes and transformations on averages

The arithmetic mean is the total divided by the number of values. If there are $n$ values with mean $\bar{x}$, their total is $n\bar{x}$. This lets you update the mean without knowing every original value.

Adding one value $v$ gives

$$\text{new mean}=\frac{n\bar{x}+v}{n+1}.$$

The added value pulls the mean towards itself. A value above the original mean increases it; a value below the mean decreases it. Adding a value equal to the mean leaves it unchanged.

Removing one value $v$, provided at least one value remains, gives

$$\text{new mean}=\frac{n\bar{x}-v}{n-1}.$$

Removal has the opposite effect: removing a value above the original mean decreases the mean, while removing a value below it increases the mean. Removing a value equal to the mean leaves it unchanged.

For several additions or withdrawals, use the same total-and-count method: add or subtract the total of the affected values, then divide by the new number of values. For instance, withdrawing several values that are all above the original mean lowers the mean of those remaining.

Why the median needs a fresh look at the middle

The median is determined by position, not by the total. A membership change alters the number of values and can change which observations occupy the middle positions.

Consider these problem-solving times, in seconds, written in order:

$9,\ 10,\ 11,\ 12,\ 12,\ 13,\ 14,\ 14,\ 15,\ 16,\ 17,\ 19,\ 20,\ 21,\ 23$.

There are 15 times, so the median is the eighth value: 14 seconds.

For example, suppose a further time of 24 seconds is added. There are now 16 values, so the median is halfway between the eighth and ninth values, 14 and 15 seconds: 14.5 seconds. Adding a large value has shifted the middle position, rather than making the median depend directly on how large that new value is.

Instead, suppose the original time of 23 seconds is withdrawn. The remaining 14 values have middle positions seven and eight, both containing 14 seconds. The median stays at 14 seconds. Thus a membership change does not necessarily change the median.

How frequency changes affect the mode

The mode is the most frequent value. In the original list of times, both 12 and 14 seconds occur twice, while every other value occurs once. There are therefore two modes: 12 and 14 seconds.

If one of the 14-second observations is withdrawn, 12 seconds becomes the only mode. If another 23-second observation is added instead, 23 seconds joins 12 and 14 seconds as a mode.

To assess a change, update the affected frequencies and compare the highest frequencies again. Adding another occurrence of an existing sole mode keeps that value as the sole mode. Other changes may leave the mode unchanged, replace it, or create or remove a tie.

Translating and scaling every value

A translation adds the same amount to every value. Adding $b$ to every observation adds $b$ to the mean, median and each mode. Subtraction works in the same way.

A scaling multiplies every value by the same factor. Multiplying every observation by a positive factor $a$ multiplies the mean, median and each mode by $a$. Division is scaling by the reciprocal. A 10% increase uses the multiplier 1.1; a 10% decrease uses 0.9.

These rules follow from how the averages work. Adding $b$ to each of $n$ values increases the total by $nb$, so the mean increases by $b$. A positive scaling or translation preserves the order, so the middle values undergo the same operation. It also preserves which observations are equal, so the modal values undergo that operation too.

For a combined transformation $y=ax+b$, with $a>0$:

$$\text{new average}=a\times\text{original average}+b.$$

Apply the operations in their stated order: multiplying and then adding is not generally the same as adding and then multiplying.

Using transformations to simplify calculations

Transformations can make awkward numbers easier to handle. Consider the values $1001.2,\ 1000.8,\ 1002.4,\ 1003.8,\ 1002.4$. Subtracting 1000 from each value and then multiplying by 10 gives $12,\ 8,\ 24,\ 38,\ 24$.

The transformed data have mean 21.2, median 24 and mode 24. To recover the original averages, undo the operations in reverse order: divide by 10, then add 1000. The original mean is 1002.12, and the original median and mode are both 1002.4.

The simplified data are a calculation aid: the final averages must describe the original values.