#### The Arithmetic Mean

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of all values}}$

The arithmetic mean calculated for sample data is denoted by $$\bar x$$ (read as “x bar”) and the arithmetic mean for population data is denoted by $$\mu$$ (Greek letter mu). Thus, the mean can be expressed by the following equations:

$\bar x = \frac{1}{n}\sum_{i=1}^n x_i$ or $\mu = \frac{1}{N}\sum_{i=1}^N x_i$

where $$\sum_{i=1}^n x_i$$ is the sum of all values ($$x_1, x_2,...,x_n$$), $$N$$ is the population size, and $$n$$ is the sample size.

Let us consider the height variable in the students data set and calculate its arithmetic mean. Be aware that the students data set is a sample and is not the population of all students. Thus, we calculate $$\bar x$$, the sample mean.

students <- read.csv("https://userpage.fu-berlin.de/soga/data/raw-data/students.csv")

stud_heights <- students$height # extract heights vector stud_heigths_n <- length(stud_heights) # get the length of the heights vector, i.e. the number of observations stud_heights_sum <- sum(stud_heights) # sum up the heights vector stud_heights_xbar <- stud_heights_sum / stud_heigths_n sprintf("The arithmetic mean height in the students data set is: %s cm", round(stud_heights_xbar, 1)) ## [1] "The arithmetic mean height in the students data set is: 171.4 cm" Of course, R also already has an in-built function called mean(), which makes things a lot easier for us. mean(stud_heights) ## [1] 171.3808 #### The Geometric Mean When studying phenomena such as inflation or population changes, which involve periodic increases or decreases (known as rates of change), the geometric mean is more appropriate to find the average change over the entire period under investigation. To calculate the geometric mean of a sequence of $$n$$ values $$x_1, x_2,..., x_n$$, we multiply them and then find the $$n^{th}$$ root of this product: $\bar x_{geo} = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}$ which can be rewritten as $\bar x_{geo} = \sqrt[n]{x_1 \cdot x_2 \cdots x_n} =\bigg(\prod_{i=1}^n x_i \bigg)^{1/n} = \sqrt[n]{\prod_{i=1}^n x_i}$ Let us make it clear by calculating an example: We consider the annual growth rates of a swarm of honey bees over a 5-year period. These rates of change are: 14 %, 26 %, 16 %, -38 %, -6 %. Further, we know that at the beginning of the monitoring period there were 5,000 bees in the swarm. We are looking for the mean rate of population change. First, we set up our variable as a vector in R: bees <- c(14, 26, 16, -38, -6) # rate of change in % Now, we apply, against better knowledge, the arithmetic mean: sprintf("The mean rate of population change is: %s percent", mean(bees)) ## [1] "The mean rate of population change is: 2.4 percent" The arithmetic mean indicates that the swarm is growing over the period of five years! Well, we are skeptical, thus we calculate the annual growth of the swarm of bees explicitly. First, we transform the given percentages into relative growth rates (bees_growth_rel). Then we simply calculate the state of the bee population after 5 years by sequentially multiplying the rates of change with the number of bees, which we know was 5,000 at the beginning of the survey. bees_growth_rel <- 1 + bees / 100 # annual rates of growth/decline bees_growth_rel ## [1] 1.14 1.26 1.16 0.62 0.94 round(5000 * bees_growth_rel[1] * bees_growth_rel[2] * bees_growth_rel[3] * bees_growth_rel[4] * bees_growth_rel[5]) ## [1] 4855 Wow, what a surprise! Obviously, there is something wrong. We expected the swarm to grow on average over time. However, we calculated a decline in the absolute number of bees. Let us try the geometric mean! Please note that $$\sqrt[n]{x} = x^{\frac{1}{n}}$$. To calculate the geometric mean explicitly we write: bees_len <- length(bees) # number of observations bees_growth_geom <- (bees_growth_rel[1] * bees_growth_rel[2] * bees_growth_rel[3] * bees_growth_rel[4] * bees_growth_rel[5])^(1 / bees_len) # rounded result round(bees_growth_geom, 3) ## [1] 0.994 Great! The geometric mean indicates that there is a decline in the number of species over time at an average rate of 0.994, which corresponds to -0.006 %. We check that by taking 5,000 bees (the initial number of bees in the swarm) times 0.994 for each year; thus, resulting in 4,971 bees after the first year, 4,942 after the second year, 4,913 after the third year, 4,884 after the fourth year and 4,855 after the fifth year of observation. A perfect match! In contrast to the arithmetic mean, the geometric mean does not over-state the year-to-year growth! Unfortunately, there is no in-built function for the geometric mean in the R-base package. Thus, we install the psych package by calling install.packages("psych") and attaching it to the work space by calling library("psych"). Then, we can access the geometric.mean() function. library("psych") round(geometric.mean(bees_growth_rel), 3) ## [1] 0.994 #### The Harmonic Mean The harmonic mean is best used when we want to average inverse metrics, such as speed (km/h) or population density (pop/km2). Consider the following example: The distance from your house to the next lake is 40 km. You drove to the lake at a speed of 20 km per hour and returned home at a speed of 80 km per hour. What was your average speed for the whole trip? Let us first calculate the arithmetic mean. one_way <- 20 back_way <- 80 mean(c(one_way, back_way)) ## [1] 50 The arithmetic mean of the two speeds you drove at is 50 km per hour. However, this is not the correct average speed. It ignores the fact that you drove at 20 km per hour for a much longer time than you drove at 80 km per hour. To find the correct average speed, we must instead calculate the harmonic mean. The harmonic mean $$\bar x_h$$ for the positive real numbers $$x_1 , x_2 , ... , x_n$$ is defined by $\bar x_h = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}, \qquad x_i > 0 \text{ for all } i.$ Unfortunately, there is no in-built function for the harmonic mean in the R-base package. So we either write our own function my.harmonic.mean <- function(vals) { length(vals) / sum(1 / vals) } my.harmonic.mean(c(one_way, back_way)) ## [1] 32 or use the harmonic.mean() function provided by the psych package # function from the psych package harmonic.mean(c(one_way, back_way)) ## [1] 32 Perfect, both implementations give the same result. However, is this result correct? We can verify it by reasoning. In the example above the distance from the lake to your home is 40 km. So the trip from A to B at a speed of 20 km/h will take 2 hours. The trip from B to A at a speed of 80 km/h will take 0.5 hours. The total time taken for the round trip of 2x40 km will be 2.5 hours. The average speed will then be $$\frac{80}{2.5} = 32$$ km/h. #### The Weighted Mean There are applications, where certain values in a data set may be considered more important than others. In general, for a sequence of $$n$$ data values $$x_1, x_2,..., x_n$$ and their corresponding weights $$w_1, w_2,..., w_n$$ the weighted (arithmetic) mean is given by $\bar x_w = \frac{\sum_{i=1}^n w_ix_i }{\sum_{i=1}^n w_i}$ where $$\sum_{i=1}^n w_ix_i$$ is obtained by multiplying each data value by its weight and then adding the products. For example, to determine the grades of students in a course an instructor may assign a weight to the final exam that is three times as much as that of the other exams. Let us find the weighted mean for a student who scores 45 and 68 on the first two exams and 74 on the final. First, we calculate the weighted mean explicitly: sum(45 * 1, 68 * 1, 74 * 3) / sum(1, 1, 3) ## [1] 67 Now, we repeat the calculation by applying the weighted.mean function of R: stud_scores <- c(45, 68, 74) stud_scores_weights <- c(1, 1, 3) weighted.mean(stud_scores, stud_scores_weights) ## [1] 67 Just for the ease of comparison we calculate the arithmetic mean, too: round(mean(stud_scores)) ## [1] 62 Please note that the weighting of input values is a principle that is applicable to other measures of the mean as well. For example, we may weight the input variable for calculating the geometric mean. #### The Weighted Geometric Mean $\bar x_{geo_w} = \bigg(\prod_{i=1}^n x_i^{w_i} \bigg)^{1/ \sum_{i=1}^n w_i}$ where $$x_1,x_2,...,x_n$$ correspond to the data values and $$w_1,w_2,...,w_n$$ to the weights, respectively. Based on the equation above it is straight forward to write an implementation for the weighted geometric mean in R: my.weighted.geometric.mean <- function(vals, weights) { prod(vals)^(1 / sum(weights)) } To make sure that our implementation is correct we recompute the example of the swarm of bees from the section above. Recall, that in the example we observed a swarm of bees over 5 years and noted the rates of change in the bee population. The annual rates of change were 1.14, 1.26, 1.16, 0.62, 0.94, which correspond to $$x_1,x_2,...,x_n$$. To reproduce the result from above we set all weights $$w_1,w_2,...,w_n$$ to be 1. round(my.weighted.geometric.mean(bees_growth_rel, rep(1, 5)), 3) ## [1] 0.994 Correct; we get the same result! #### The Weighted Harmonic Mean The weighted harmonic mean $$\bar x_{h_w}$$ for the positive real numbers $$x_1 , x_2 , ... , x_n$$ is defined as $\bar x_{h_w} = \frac{w_1+w_2+\cdots +w_n}{\frac{w_1}{x_1} + \frac{w2}{x_2} + \cdots + \frac{w_n}{x_n}} = \frac{\sum_{i=1}^n w_i}{\sum_{i=1}^n \frac{w_i}{x_i}}, \qquad x_i > 0 \text{ for all } i.$ Let us implement the weighted harmonic mean function in R. The coding is straightforward, however, we include an if-statement to expand the functionality of our function. This if-statement in the code will normalize the weights if the weights are not given in proportions. When the if-statement is run, it prints a user-feedback, otherwise there will be no feedback. my.weighted.harmonic.mean <- function(vals, weights) { if (sum(weights) != 1) { weights <- weights / sum(weights) print("Weights do not sum up to 1. Weights are normalized...") } sum(weights) / sum(weights / vals) } Let us try our function my.weighted.harmonic.mean on a fairly complex data set. The data set cities consists of all state capitals of Germany, their population size and area. The goal is to calculate the mean population density for the state capitals of Germany. You may download the cities.csv file here. The data is retrieved from this website. First, we load the data set, assign it a proper name and take a look at it. cities <- read.csv("https://userpage.fu-berlin.de/soga/data/raw-data/cities.csv") cities ## name state area_km2 pop_size ## 1 Berlin Land Berlin 891.85 3520100 ## 2 Bremen Freie Hansestadt Bremen 325.42 546450 ## 3 Dresden Freistaat Sachsen 328.31 525100 ## 4 D\xfcsseldorf Land Nordrhein-Westfalen 217.41 593682 ## 5 Erfurt Freistaat Th\xfcringen 269.17 203480 ## 6 Hamburg Freie und Hansestadt Hamburg 755.26 1751780 ## 7 Hannover Land Niedersachsen 204.14 514130 ## 8 Kiel Land Schleswig-Holstein 118.60 239860 ## 9 Magdeburg Land Sachsen-Anhalt 200.97 229924 ## 10 Mainz Land Rheinland-Pfalz 97.76 202750 ## 11 M\xfcnchen Freistaat Bayern 310.71 1388300 ## 12 Potsdam Land Brandenburg 187.27 159450 ## 13 Saarbr\xfccken Saarland 167.07 176990 ## 14 Schwerin Land Mecklenburg-Vorpommern 130.46 91260 ## 15 Stuttgart Land Baden-W\xfcrttemberg 207.36 597939 ## 16 Wiesbaden Land Hessen 203.90 272630 Second, we create a new column and calculate the population density (inhabitant per square kilometer). cities$pop_density <- cities$pop_size / cities$area_km2
cities
##              name                        state area_km2 pop_size pop_density
## 1          Berlin                  Land Berlin   891.85  3520100   3946.9642
## 2          Bremen      Freie Hansestadt Bremen   325.42   546450   1679.2146
## 3         Dresden            Freistaat Sachsen   328.31   525100   1599.4030
## 4   D\xfcsseldorf     Land Nordrhein-Westfalen   217.41   593682   2730.7024
## 5          Erfurt       Freistaat Th\xfcringen   269.17   203480    755.9535
## 6         Hamburg Freie und Hansestadt Hamburg   755.26  1751780   2319.4397
## 7        Hannover           Land Niedersachsen   204.14   514130   2518.5167
## 8            Kiel      Land Schleswig-Holstein   118.60   239860   2022.4283
## 9       Magdeburg          Land Sachsen-Anhalt   200.97   229924   1144.0713
## 10          Mainz         Land Rheinland-Pfalz    97.76   202750   2073.9566
## 11     M\xfcnchen             Freistaat Bayern   310.71  1388300   4468.1536
## 12        Potsdam             Land Brandenburg   187.27   159450    851.4444
## 13 Saarbr\xfccken                     Saarland   167.07   176990   1059.3763
## 14       Schwerin  Land Mecklenburg-Vorpommern   130.46    91260    699.5248
## 15      Stuttgart    Land Baden-W\xfcrttemberg   207.36   597939   2883.5793
## 16      Wiesbaden                  Land Hessen   203.90   272630   1337.0770

Third, we calculate the weight for each city according its population size.

cities$pop_weight <- cities$pop_size / sum(cities$pop_size) cities ## name state area_km2 pop_size pop_density ## 1 Berlin Land Berlin 891.85 3520100 3946.9642 ## 2 Bremen Freie Hansestadt Bremen 325.42 546450 1679.2146 ## 3 Dresden Freistaat Sachsen 328.31 525100 1599.4030 ## 4 D\xfcsseldorf Land Nordrhein-Westfalen 217.41 593682 2730.7024 ## 5 Erfurt Freistaat Th\xfcringen 269.17 203480 755.9535 ## 6 Hamburg Freie und Hansestadt Hamburg 755.26 1751780 2319.4397 ## 7 Hannover Land Niedersachsen 204.14 514130 2518.5167 ## 8 Kiel Land Schleswig-Holstein 118.60 239860 2022.4283 ## 9 Magdeburg Land Sachsen-Anhalt 200.97 229924 1144.0713 ## 10 Mainz Land Rheinland-Pfalz 97.76 202750 2073.9566 ## 11 M\xfcnchen Freistaat Bayern 310.71 1388300 4468.1536 ## 12 Potsdam Land Brandenburg 187.27 159450 851.4444 ## 13 Saarbr\xfccken Saarland 167.07 176990 1059.3763 ## 14 Schwerin Land Mecklenburg-Vorpommern 130.46 91260 699.5248 ## 15 Stuttgart Land Baden-W\xfcrttemberg 207.36 597939 2883.5793 ## 16 Wiesbaden Land Hessen 203.90 272630 1337.0770 ## pop_weight ## 1 0.31960740 ## 2 0.04961492 ## 3 0.04767644 ## 4 0.05390334 ## 5 0.01847496 ## 6 0.15905283 ## 7 0.04668042 ## 8 0.02177808 ## 9 0.02087594 ## 10 0.01840868 ## 11 0.12605067 ## 12 0.01447726 ## 13 0.01606980 ## 14 0.00828595 ## 15 0.05428986 ## 16 0.02475343 sprintf("The sum of the weight vector is %s", sum(cities$pop_weight))
## [1] "The sum of the weight vector is 1"

Now, we finally apply our implementation of the weighted harmonic mean (my.weighted.harmonic.mean) and compare it to the arithmetic mean.

We start by giving the function the cities$pop_weight vector as an input parameter. my.weighted.harmonic.mean(cities$pop_density, cities$pop_weight) ## [1] 2386.186 Then we test the functionality of our function by providing the cities$pop_size vector as an input parameter.

my.weighted.harmonic.mean(cities$pop_density, cities$pop_size)
## [1] "Weights do not sum up to 1. Weights are normalized..."
## [1] 2386.186

Awesome, the function works as expected. The results are identical.

We can conclude that the mean population density in the state capitals of Germany is about 2,386 inhabitants/km2.

For comparison, we may calculate the arithmetic mean of the population density.

mean(cities\$pop_density)
## [1] 2005.613

Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.