Pearson's coefficient of skewness<\/strong>:<\/p>\n\n\\(\nSkewness = \\frac{3(\\mu - med)}{\\sigma} \\\\ \\\\\n\\)\n\nA perfectly symmetric distribution has a skewness value of 0. A right-skewed (positive) distribution has a positive value, while a left-skewed distribution has a negative value.<\/p>\n\n
Skewness values typically fall between -3 and 3, and the fact that the standard deviation appears in the denominator makes the value independent of the unit of measurement<\/strong>.<\/p>\n\nHow do we calculate the skewness index in R? The simplest way is to use a library that provides the functions we need \"ready to go\":<\/p>\n\n
library(moments)\ndata <- c(24, 17, 21, 23, 15, 30, 24, 21, 24, 19,\n 25, 28, 22, 20, 14, 19, 26, 29, 23, 25,\n 24, 18, 27, 21)\nskewness(data)\n# [1] -0.1918578<\/code><\/pre>\n\nThe moments<\/code> library serves us well. Let us see, however, how to calculate the index without relying on a library. It is very simple. The first step is to remember that R uses n-1<\/em> in the denominator of the variance<\/strong>. We are reasoning about a population, however, with n<\/em> in the denominator. So, let us define a function that gives us the value we need:<\/p>\n\npop_var <- function(x) { var(x) * (1 - 1\/length(x)) }<\/code><\/pre>\n\nAt this point we can calculate the skewness index:<\/p>\n\n
data <- c(24, 17, 21, 23, 15, 30, 24, 21, 24, 19,\n 25, 28, 22, 20, 14, 19, 26, 29, 23, 25,\n 24, 18, 27, 21)\npop_var <- function(x) { var(x) * (1 - 1\/length(x)) }\nz <- (data - mean(data)) \/ sqrt(pop_var(data))\nskew <- mean(z^3)\nskew\n# [1] -0.1918578<\/code><\/pre>\n\nKurtosis<\/h2>\n\n Kurtosis<\/strong> compares the shape of a distribution with that of the normal: how peaked it is at the centre and, above all, how \u00abheavy\u00bb its tails are<\/strong>, that is how often values far from the mean turn up.<\/p>\n\nWe have three cases:<\/p>\n\n
\na tall<\/strong> curve, called leptokurtic<\/strong><\/em>, which is highly concentrated around its mean<\/li>\na normal<\/strong> curve, called mesokurtic<\/strong><\/em><\/li>\na low and flat<\/strong> curve, called platykurtic<\/strong><\/em>, with little concentration around its mean<\/li>\n<\/ol>\n\nKurtosis can be measured by dividing the fourth moment by the standard deviation raised to the fourth power. Sounds difficult? It is easier done than said. Here is the formula:<\/p>\n\n\\(\nKurtosis = \\frac{\\Sigma f(X -\\mu)^{4}}{N \\sigma^{4}} \\\\ \\\\\n\\)\n\n
The kurtosis of a mesokurtic curve has a value of 3. Naturally, a kurtosis coefficient less than 3 indicates a platykurtic curve, while a value greater than 3 indicates a leptokurtic curve.excess kurtosis<\/strong>, that is the value minus 3 (with a small correction for samples): there the normal is 0. The 2.48 we are about to obtain corresponds to an excess of about \u22120.52: a slightly platykurtic curve.<\/p>\n\nAs with the skewness index, the moments<\/code> library provides a convenient ready-made function:<\/p>\n\nlibrary(moments)\ndata <- c(24, 17, 21, 23, 15, 30, 24, 21, 24, 19,\n 25, 28, 22, 20, 14, 19, 26, 29, 23, 25,\n 24, 18, 27, 21)\nkurtosis(data)\n# [1] 2.480035<\/code><\/pre>\n\nBut we are not averse to computing it ourselves:<\/p>\n\n
data <- c(24, 17, 21, 23, 15, 30, 24, 21, 24, 19,\n 25, 28, 22, 20, 14, 19, 26, 29, 23, 25,\n 24, 18, 27, 21)\npop_var <- function(x) { var(x) * (1 - 1\/length(x)) }\nz <- (data - mean(data)) \/ sqrt(pop_var(data))\nkurt <- mean(z^4)\nkurt\n# [1] 2.480035<\/code><\/pre>\n\nWe have seen that measures of central tendency alone are not enough: two datasets can have the same mean yet behave in profoundly different ways. Variability tells us how much<\/em> the data fluctuate, and the shape of the distribution tells us how<\/em> they are arranged. In the next article, we will take the first steps into the world of probability\u2014the essential bridge between descriptive statistics and the inferential tools that let us draw conclusions from data.<\/p>\n\nA Practical SEO Case: the Coefficient of Variation to Compare Two Sites<\/h2>\n\n The standard deviation, on its own, can mislead when we compare quantities of different scale. Take two sites: a big one, around 5,000 sessions a day, and a small one, around 300. Let's measure the daily variability of each in R.<\/p>\n\n
big <- c(4800, 5300, 4600, 5500, 5100, 4400, 5600, 4900, 5200, 4700)\nsmall <- c(260, 340, 220, 380, 300, 240, 360, 280, 320, 300)\n\nsd(big) # 395.7\nsd(small) # 51.6\nsd(big) \/ mean(big) # CV = 0.079 (7.9%)\nsd(small) \/ mean(small) # CV = 0.172 (17.2%)<\/code><\/pre>\n\nIn absolute terms the big site swings by almost 400 sessions a day, the small one by just 52: one would be tempted to call the big site far more “jumpy”. But that is an illusion of scale. The coefficient of variation<\/strong> — the standard deviation divided by the mean, a pure number read as a percentage — tells the opposite story: 7.9%<\/strong> for the big site, 17.2%<\/strong> for the small one. Relative to its own size, the small site is more than twice as unstable. This is exactly what the CV is for: it makes comparable variabilities that, in raw units, would not be.<\/p>\n\n\n\n\n
Same data, two readings: in raw sessions the small site looks flat, but as a percentage of the mean it swings twice as much \u2014 the illusion of scale that the coefficient of variation dispels.<\/figcaption><\/figure>\n\n<\/div><\/div>\n\n\nThe operational rule is clear: when we compare the variability of metrics on different scales<\/strong> — two sites, two channels, two periods with very different volumes — the coefficient of variation is the tool to use, not the bare standard deviation. The latter remains irreplaceable when the scale is the same.<\/p>\n\nTry It Yourself<\/h2>\n\n Let's try a similar case. We have a week of daily clicks for two channels: organic, robust, and a niche referral. Which of the two is steadier, net of the difference in volume?<\/p>\n\n
organic: 1200 1350 1100 1280 1420 1180 1320\nreferral: 80 140 60 170 95 130 110<\/pre>\n\nThe task: for each channel, compute the CV with sd(x) \/ mean(x)<\/code>.<\/p>\n\nTo check the numbers: organic has a CV of 8.7%<\/strong>, the referral one of 33.5%<\/strong>. Although it brings ten times fewer clicks, the referral is nearly four times more volatile in relative terms: a channel on which to base forecasts is far riskier. The raw standard deviation (110 against 38) would have suggested the exact opposite.<\/p>\n\nFrequently asked questions<\/h2>\n\nWhat are the main measures of dispersion?<\/h3>\n\n The range, the variance, the standard deviation, the interquartile range and the coefficient of variation. Variance, standard deviation and coefficient of variation are based on the distance of the data from the mean; the interquartile range on the quartiles, which is why it is robust to outliers.<\/p>\n\n
What is the difference between variance and standard deviation?<\/h3>\n\n The standard deviation is the square root of the variance. They measure the same thing, but the variance is expressed in the square of the data's unit (sessions squared), the standard deviation in the same unit (sessions): that is why it is much easier to read and communicate.<\/p>\n\n
Why does R divide by n \u2212 1 and not by n?<\/h3>\n\n Because var() and sd() estimate the variability of a population from a sample: dividing by n would systematically underestimate the true variance, and n \u2212 1 corrects that bias. If the data are the whole population, you divide by n.<\/p>\n\n
When should you use the coefficient of variation?<\/h3>\n\n When comparing the variability of quantities on different scales: two sites with different volumes, two channels, metrics with different units. The CV is the standard deviation divided by the mean, a pure number read as a percentage. It loses meaning with means close to zero or with data that can be negative.<\/p>\n\n
What does platykurtic mean?<\/h3>\n\n A distribution flatter than the normal, with lighter tails: the kurtosis is below 3 (the excess kurtosis below 0). The opposite is a leptokurtic curve, more peaked and with heavier tails; the normal is mesokurtic.<\/p>\n\n
Which measure of dispersion should you use when there are outliers?<\/h3>\n\n The interquartile range, which measures the width of the middle half of the data and therefore ignores the tails. Variance, standard deviation and range are instead inflated even by a single extreme value, such as a traffic spike or a session left open by a bot.<\/p>\n\n
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