Two weeks of organic sessions, a Black Friday in the middle, and the report says: \u00ab385 sessions a day, on average\u00bb. Except that no normal day came anywhere near it: the typical day had 233.
It is not a calculation error (the mean is right), it is the wrong measure for those data. Choosing the number that sums up a distribution is the first move of any analysis, and also the first way to be fooled.<\/p>\n\n\n\n
Measures of position<\/strong> (or measures of central tendency<\/strong>) are there for exactly this: telling us, with a single value, where the data concentrate. Let’s look at the various means, the median, the mode, quartiles and percentiles, when to use which, and how to compute them in R in a moment.<\/p>\n\n\n\n\n\n\n\n In short:<\/strong> measures of position<\/strong> sum up with a single number where a distribution concentrates. The main ones are the mean<\/strong> (the sum of the values divided by how many there are), the median<\/strong> (the middle value of the sorted data) and the mode<\/strong> (the most frequent value); quartiles, deciles and percentiles<\/strong> split the sorted data into 4, 10 and 100 parts. The practical rule: with skewed data or outliers, the median describes the \u00abtypical case\u00bb better than the mean.<\/p>\n\n\n\t\t\t\t A mean is a measure of the central tendency of a set of values. There are several, and each answers a slightly different question: let’s go through them one by one.<\/p>\n\n\n\n The arithmetic mean, as we all learned in school, is the sum of the values in a data set divided by the number of values. Note: For this discussion, we will consistently refer to population parameters.<\/p>\n\n\n\n Here is the formula to calculate the population mean:<\/p>\n\n\n\n\\( \n\\mu = \\frac {\\Sigma X}{N} \\\\ \\\\ \n\\)\n \nIn R, the function is mean()<\/strong>.\nA simple example:\n<\/p>\n\n\n\n Let’s imagine we have n<\/em> data points grouped into k<\/em> classes. We’ll call xc<\/sub> the central value of each class and f<\/em>i<\/sub> the observed frequency of each class.<\/p>\n\n\n\n The formula to calculate the mean in this case is:<\/p>\n\n\n\n\\(\n\\bar x = \\frac{\\sum^{k}_{i=1} x_c f_i}{N} \\ \\\n\\)\n\n\n\n Let’s look at an example:<\/p>\n\n\n\n Let’s use the formula and calculate the mean:<\/p>\n\n\n\n\\(\n\\bar x = \\frac{(65) + (1011) + (1614) + (247)}{37} \\ \\ = \\frac{532}{37} = 14.38\n\\)\n\n\n\n The weighted mean<\/strong>, or weighted average<\/strong>, is an arithmetic mean where each value is weighted according to its importance<\/strong> within the group. Let’s look at an example. Imagine a sporting discipline where the jury’s score for each exercise is “weighted” with a certain difficulty coefficient. We would have these scores for the various exercises: Let’s multiply each score by its respective coefficient:<\/p>\n\n\n\n (9.3 * 4) + (9.8 * 2.8) + (8.8 * 3.3) = 93.68<\/p>\n\n\n\n The sum of the coefficients (i.e., the “weights”) is:<\/p>\n\n\n\n 4 + 2.8 + 3.3 = 10.1<\/p>\n\n\n\n And calculate the weighted mean:<\/p>\n\n\n\n 93.68 \/ 10.1 = 9.275<\/p>\n\n\n\n \nN.B. The capital Greek letter pi is the symbol for product<\/strong>. The formula therefore equals this:\n<\/p>\n\\(\n\\\\ \nGeometric \\ Mean = \\sqrt[n]{x_{1} x_{2} \\cdots x_{n}} \\\\ \\\\\n\\)\n\n\n\n R doesn’t have a built-in function for calculating the geometric mean, but calculating this measure of position is very simple. Let’s look at an example:<\/p>\n\n\n\n The harmonic mean of a dataset x1<\/sub>, x2<\/sub>, …, xn<\/sub> is the reciprocal of the arithmetic mean of the reciprocals of the data<\/strong>. In formula:<\/p>\n\n\n\n\\(\nHarmonic \\ mean = \\frac{1}{\\frac{1}{n} \\vc{\\Sigma}{n}{i=1} \\frac{1}{x_{i}}} \\\\ \\\\\n\\)\n\n\n\n Similarly, for the harmonic mean, there is no specific built-in function available. However, since the harmonic mean is the reciprocal of the mean of the reciprocals of the data, the calculation is straightforward:<\/p>\n\n\n\n Using the very simple example mentioned above:<\/p>\n\n\n\n A trimmed mean (e.g., at 5%) is an arithmetic mean of all ordered values after excluding the lowest 5% and the highest 5% from the dataset. In the example chosen, this mean is obtained by calculating the arithmetic mean of the central 90% of the population in the sorted series of observations.<\/p>\n\n\n\n In R, the trimmed mean can be calculated by specifying the trim=proportionToExclude<\/em><\/strong> option in the mean()<\/strong> function, where proportionToExclude<\/em> is the proportion (between 0 and 1) of the smallest and largest values to exclude before calculating the arithmetic mean. (Note: with 11 values a 5% trim removes nothing: 5% of 11 is 0.55, which R rounds down to zero, so the result equals the arithmetic mean. At 10% it drops one value per side, the minimum and the maximum.)<\/p>\n\n\n\n The median of a group of elements is the value of the central element when all the elements in the group are arranged in ascending or descending order of value.<\/p>\n\n\n\n In practice, the median of a dataset is the value that divides the series into two equal parts<\/strong>: as many values above the median as below it.The mean and its variants<\/h2>\n\n\n\n
The Arithmetic Mean<\/h3>\n\n\n\n
In statistics, a parameter of a population<\/strong> is represented by a Greek letter<\/strong>, while a descriptive measure of a sample<\/strong> is denoted with a Roman letter<\/strong>.<\/p>\n\n\n\nFor the mean, we use the symbol \\( \\mu \\) for the mean of a population of values, and the letter \\( \\overline X \\) (read as “x-bar”) for the mean of a sample of values.\n\n\n\nvariable = c(-3,1,2,4,5,2,8);\nmean(variable);\n\n[1] 2.714286<\/code><\/pre>\n\n\n\nThe Mean of Grouped Data<\/h3>\n\n\n\n
Class<\/td> xc<\/sub><\/td> fi<\/sub><\/td> xc<\/sub> * fi<\/sub><\/td><\/tr> 4<x\u2264<\/strong>8<\/td> 6<\/td> 5<\/td> 30<\/td><\/tr> 8<x\u2264<\/strong>12<\/td> 10<\/td> 11<\/td> 110<\/td><\/tr> 12<x\u2264<\/strong>20<\/td> 16<\/td> 14<\/td> 224<\/td><\/tr> 20<x\u2264<\/strong>28<\/td> 24<\/td> 7<\/td> 168<\/td><\/tr> Total<\/td> <\/td> 37<\/strong><\/td> 532<\/strong><\/td><\/tr><\/tbody><\/table>
\n\n\n\n\n\nThe Weighted Mean<\/h3>\n\n\n\n
Each value in the group (X) is multiplied by the appropriate weight factor \\( \\omega \\) and the various products are summed and divided by the sum of the weights:<\/p>\n\n\n\n\\(\n\\mu_{w} = \\frac{\\Sigma(w X)}{\\Sigma w}\n\\)\n\n\n\n
<\/p>\n\n\n\nJury Score<\/td> Coefficient<\/td><\/tr> 9.3<\/td> 4<\/td><\/tr> 9.8<\/td> 2.8<\/td><\/tr> 8.8<\/td> 3.3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n The Geometric Mean<\/h3>\n\n\n\nGiven a set of positive numbers x1<\/sub>,x2<\/sub>,…,xn<\/sub> the geometric mean is the nth root<\/i> of the product of the n<\/i> numbers. In formula:\n
\n\n\n\n\\(\n\\\n\\newcommand{\\vc}[3]{\\overset{#2}{\\underset{#3}{#1}}}\nGeometric \\ Mean = \\sqrt[n]{\\vc{\\Pi}{n}{i=1} \\ x_{i}}\n\\)\n
\ndata = c(2,9,12) # My values\nn = length(data) # the number of values\n \n# prod calculates the product of vector elements\n# Calculate the geometric mean\nprod(data)^(1\/n)\n\n[1] 6<\/code><\/pre>\n\n\n\nThe Harmonic Mean<\/h3>\n\n\n\n
1\/mean(1\/data)<\/code><\/pre>\n\n\n\ndata = c(2,9,12) # My values\nn = length(data) # The number of values\n\n# Calculate the harmonic mean\n1\/mean(1\/data)\n\n[1] 4.32<\/code><\/pre>\n\n\n\nThe Trimmed Mean<\/h3>\n\n\n\n
A simple R example to clarify:<\/p>\n\n\n\nvariable = c(-3,1,2,4,5,2,5,0.8,2.4,6,8);\nmean(variable) ### arithmetic mean\n[1] 3.018182\nmean(variable, trim = 0.05) ### 5% trimmed\n[1] 3.018182\nmean(variable, trim = 0.10) ### 10% trimmed\n[1] 3.133333<\/code><\/pre>\n\n\n\n
\n\n\n\nThe Median<\/h2>\n\n\n\n
To find the median, the rule known as even-odd rule<\/em> is used:
<\/p>\n\n\n\n