Weighted Mean, Geometric Mean and Mean Seasonal Variation

Giving values different weights

An ordinary arithmetic mean gives each observation an equal contribution. A weighted mean allows values to contribute different amounts. A weight might be the number of people represented by a value, the percentage of a population in a category, or the importance assigned to a test paper.

Multiply each value by its weight, add the products, then divide by the total weight:

$$\text{weighted mean}=\frac{\sum wx}{\sum w}$$

Here $x$ is a value, $w$ is its weight, and $\sum$ means ‘add all the terms’.

For example, Myfanwy scores 64%, 60% and 75% on three papers with weights 25, 35 and 40 respectively.

Her final mark is $6700\div100=67\%$. The third paper contributes most because it has the largest weight.

Weights do not have to add to 100. When combining the means of different groups, the weights are the group sizes. Multiplying a group’s mean by its size reconstructs its total.

For three classes with 28, 25 and 27 students and mean test scores of 64%, 68% and 65%, the combined mean is

$$\frac{28\times64+25\times68+27\times65}{28+25+27}=65.5875\%\approx65.6\%.$$

This gives every student an equal contribution, rather than giving every class an equal contribution.

Frequencies and grouped intervals

A frequency is a weight: if a value occurs $f$ times, it contributes $fx$ to the total. Thus a mean from a discrete frequency table is a weighted mean with $w=f$.

For data grouped into intervals, the individual values are unknown. Represent each interval by its class midpoint, found by adding its endpoints and dividing by two. The resulting weighted mean is an estimate.

For example, these are the recorded weights of 20 puppies:

The estimated mean weight is $87\div20=4.35\text{ kg}$. This treats every puppy in an interval as having the midpoint weight, so it cannot recover the exact mean of the original measurements.

Multiplying values: the geometric mean

The geometric mean of $n$ positive values is found by multiplying them together and taking the $n$th root:

$$G=\sqrt[n]{x_1x_2\cdots x_n}.$$

Two values require a square root, three require a cube root, and so on. For 2, 9 and 12,

$$G=\sqrt[3]{2\times9\times12}=\sqrt[3]{216}=6.$$

The connection is multiplicative: replacing all three values by 6 preserves their product because $6^3=216$. On a calculator, an $n$th root can also be entered as a power of $1/n$.

A frequency table records repeated factors. If a positive value $x$ occurs $f$ times, its contribution to the product is $x^f$. With total frequency $N=\sum f$,

$$G=\left(x_1^{f_1}x_2^{f_2}\cdots x_k^{f_k}\right)^{1/N}.$$

For interval-grouped data, substitute the class midpoints for the unknown values. Using the puppy table gives

$$G\approx\left(3.25^2\times3.75^4\times4.25^6\times4.75^5\times5.25^2\times5.75\right)^{1/20} \approx4.30\text{ kg}.$$

This is an estimated geometric mean. The root is the 20th root because there are 20 puppies, even though there are only six intervals.

Geometric means of percentage changes

Successive percentage changes act by multiplication. Convert each change to a multiplier before calculating its geometric mean: a 2% increase has multiplier 1.02, while a 2% decrease has multiplier 0.98.

For annual profit increases of 1%, 7%, 3% and 12%, the geometric mean multiplier is

$$G=\left(1.01\times1.07\times1.03\times1.12\right)^{1/4} =1.056671\ldots$$

Convert this multiplier back to a percentage change using $(G-1)\times100$. The average annual increase is therefore approximately 5.67%. Repeating that unrounded average increase for four years produces the same overall change as the four different increases.

Separating seasonal effects from the trend

A time series records values over time. Its trend line represents the general direction of change, while a seasonal pattern repeats at corresponding times in each cycle—for example, higher visitor numbers each summer.

Seasonal variation measures how far an actual value lies above or below the trend at the same time:

$$\text{seasonal variation}=\text{actual value}-\text{trend value}.$$

On a graph, locate the required time on the horizontal axis. Read the actual point and the trend line vertically above that same position, using the vertical-axis scale. The difference is positive if the point is above the trend and negative if it is below.

The mean seasonal variation, also called the average seasonal effect, averages these differences for one particular season across different years. It does not average all seasons together.

For example, visitor numbers and trend values for Quarter 1 are:

The mean seasonal variation for Quarter 1 is

$$\frac{-1.34+(-1.81)+(-1.68)}{3}=-1.61\text{ thousand visitors}.$$

This means Quarter 1 visitor numbers are, on average, 1610 below the trend. Quarter 2 variations of 0.84, 0.78 and 0.90 thousand visitors have mean $0.84$ thousand visitors: an average of 840 above the trend.

If variations for a particular season are supplied in a frequency table, average them using their frequencies as weights: $\sum fv\div\sum f$, where $v$ is a variation. If those variations are grouped into intervals, use interval midpoints to estimate the mean. In either case, keep different seasons separate and retain the original units.