{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "*The codes were developed on Windows 10, and were not tested on other machines. Anaconda 5.2.0 is chosen as a Python interpreter.*\n", "\n", "This notebook introduces the concept and methodology of implementing Bootstrap Simulation.\n", "\n", "# Non-Parametric Confidence Intervals with Bootstrap\n", "\n", "> **Bootstrap** is a non-parametric statistical technique to resample from known samples to estimate uncertainty in summary statistics. When there are small, limited number of samples, it gives a more accurate forecast model than directly obtaining a forecast model from the limited sample pool (assuming that the sample set of data is reasonable representation of the population). It is non-parametric because it does not require any prior knowledge of the distribution (shape, mean, standard devation, etc..).\n", "\n", "\n", "** Advantages of Bootstrap **\n", "\n", "One great thing about Bootstrapping is that it is **distribution-free**. You do not need to know distribution shape, mean, standard devation, skewness, kurtosis, etc... All you need is just a set of sample data that is representative of a population. The fact that Bootstrapping does not depend on a type of distribution leads to another great advantage - **It can calculate uncertainty in any confidence interval of any kind of distribution**. \n", "\n", "For example, the analytical solution to calculate a confidence interval in any statistics of a distribution is as follows:\n", "\n", "
It is extremly important that the N size for each Bootstrap realization matches the N size of the original sample pool. \n",
"
We use Bootstrap to numerically estimate the confidence interval (CI). It's an alternative tool to analytically solve for CI. Observing how CI is analytically calculated may help one to understand why the value of N is important. Let's take the CI of a mean for example. Recall that the CI of a mean represents how far a sample mean can deviate from the true population mean.
In case of a Gaussian, or Gaussian-like distribution (ex: student-t), the equation to analytically solve for confidence interval of a mean is as follows:
\n", "By 95% chance, the following statistics will fall within the range of:
\n", "\n", " | Male 1 | \n", "Male 2 | \n", "Male 3 | \n", "Male 4 | \n", "Male 5 | \n", "Male 6 | \n", "Male 7 | \n", "Male 8 | \n", "Male 9 | \n", "Male 10 | \n", "... | \n", "Male 91 | \n", "Male 92 | \n", "Male 93 | \n", "Male 94 | \n", "Male 95 | \n", "Male 96 | \n", "Male 97 | \n", "Male 98 | \n", "Male 99 | \n", "Male 100 | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Height (in) | \n", "70.8 | \n", "72.8 | \n", "72.5 | \n", "67.3 | \n", "72.7 | \n", "73.6 | \n", "65.0 | \n", "67.1 | \n", "70.8 | \n", "70.6 | \n", "... | \n", "71.7 | \n", "66.4 | \n", "72.9 | \n", "74.5 | \n", "73.5 | \n", "70.5 | \n", "73.1 | \n", "63.6 | \n", "68.7 | \n", "73.0 | \n", "
1 rows × 100 columns
\n", "\n", " | Height (in) | \n", "
---|---|
mean | \n", "69.881971 | \n", "
std | \n", "3.169548 | \n", "
min | \n", "63.143732 | \n", "
max | \n", "77.762886 | \n", "
median | \n", "69.894434 | \n", "
skew | \n", "-0.059779 | \n", "
kurtosis | \n", "-0.700743 | \n", "
IQR | \n", "5.154145 | \n", "
\n", " | Real 1 | \n", "Real 2 | \n", "Real 3 | \n", "Real 4 | \n", "Real 5 | \n", "Real 6 | \n", "Real 7 | \n", "Real 8 | \n", "Real 9 | \n", "Real 10 | \n", "... | \n", "Real 91 | \n", "Real 92 | \n", "Real 93 | \n", "Real 94 | \n", "Real 95 | \n", "Real 96 | \n", "Real 97 | \n", "Real 98 | \n", "Real 99 | \n", "Real 100 | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | \n", "66.0 | \n", "65.5 | \n", "68.6 | \n", "68.5 | \n", "74.1 | \n", "69.1 | \n", "69.0 | \n", "72.9 | \n", "67.2 | \n", "68.4 | \n", "... | \n", "65.0 | \n", "73.5 | \n", "68.3 | \n", "69.9 | \n", "69.1 | \n", "68.7 | \n", "73.0 | \n", "70.5 | \n", "69.9 | \n", "67.3 | \n", "
1 | \n", "69.5 | \n", "74.1 | \n", "66.4 | \n", "66.3 | \n", "65.3 | \n", "64.7 | \n", "71.6 | \n", "70.8 | \n", "73.6 | \n", "69.0 | \n", "... | \n", "64.1 | \n", "68.4 | \n", "66.4 | \n", "67.2 | \n", "68.6 | \n", "74.1 | \n", "72.1 | \n", "67.9 | \n", "71.6 | \n", "74.1 | \n", "
2 | \n", "72.7 | \n", "77.8 | \n", "73.1 | \n", "66.1 | \n", "64.9 | \n", "70.8 | \n", "73.0 | \n", "66.8 | \n", "69.5 | \n", "65.5 | \n", "... | \n", "67.2 | \n", "66.3 | \n", "67.9 | \n", "69.9 | \n", "64.5 | \n", "66.1 | \n", "70.8 | \n", "70.8 | \n", "70.5 | \n", "70.5 | \n", "
3 | \n", "72.9 | \n", "70.6 | \n", "65.0 | \n", "71.6 | \n", "73.6 | \n", "77.8 | \n", "73.1 | \n", "72.7 | \n", "72.2 | \n", "73.0 | \n", "... | \n", "66.8 | \n", "67.6 | \n", "73.3 | \n", "66.0 | \n", "70.8 | \n", "71.7 | \n", "71.6 | \n", "69.2 | \n", "68.7 | \n", "69.9 | \n", "
4 | \n", "69.2 | \n", "64.5 | \n", "67.3 | \n", "74.4 | \n", "67.1 | \n", "67.1 | \n", "66.4 | \n", "72.5 | \n", "72.2 | \n", "68.6 | \n", "... | \n", "71.6 | \n", "73.0 | \n", "72.9 | \n", "73.5 | \n", "74.5 | \n", "72.5 | \n", "67.3 | \n", "69.5 | \n", "72.3 | \n", "68.7 | \n", "
5 | \n", "73.0 | \n", "73.5 | \n", "74.4 | \n", "68.4 | \n", "69.5 | \n", "66.8 | \n", "73.5 | \n", "72.2 | \n", "63.1 | \n", "66.1 | \n", "... | \n", "70.8 | \n", "72.5 | \n", "72.7 | \n", "68.5 | \n", "67.9 | \n", "70.5 | \n", "70.8 | \n", "66.6 | \n", "73.6 | \n", "67.1 | \n", "
6 | \n", "73.0 | \n", "66.6 | \n", "67.9 | \n", "70.5 | \n", "64.5 | \n", "73.1 | \n", "72.8 | \n", "71.4 | \n", "73.5 | \n", "66.8 | \n", "... | \n", "73.0 | \n", "73.1 | \n", "74.5 | \n", "70.6 | \n", "73.5 | \n", "64.1 | \n", "70.9 | \n", "70.8 | \n", "68.7 | \n", "70.6 | \n", "
7 | \n", "68.6 | \n", "64.9 | \n", "73.3 | \n", "69.2 | \n", "70.6 | \n", "70.2 | \n", "70.3 | \n", "70.3 | \n", "72.2 | \n", "72.2 | \n", "... | \n", "63.6 | \n", "67.3 | \n", "74.1 | \n", "73.6 | \n", "68.5 | \n", "69.5 | \n", "72.2 | \n", "72.4 | \n", "73.6 | \n", "65.0 | \n", "
8 | \n", "68.3 | \n", "68.7 | \n", "69.5 | \n", "69.5 | \n", "72.7 | \n", "72.5 | \n", "72.5 | \n", "72.8 | \n", "70.8 | \n", "66.4 | \n", "... | \n", "66.3 | \n", "71.7 | \n", "68.3 | \n", "66.4 | \n", "67.9 | \n", "66.4 | \n", "70.6 | \n", "70.2 | \n", "73.6 | \n", "74.1 | \n", "
9 | \n", "73.5 | \n", "66.2 | \n", "73.0 | \n", "69.1 | \n", "64.9 | \n", "73.6 | \n", "74.5 | \n", "72.9 | \n", "73.0 | \n", "64.5 | \n", "... | \n", "67.9 | \n", "67.1 | \n", "72.4 | \n", "71.3 | \n", "70.8 | \n", "70.8 | \n", "67.1 | \n", "68.8 | \n", "65.0 | \n", "67.9 | \n", "
10 rows × 100 columns
\n", "\n", " | Real 1 | \n", "Real 2 | \n", "Real 3 | \n", "Real 4 | \n", "Real 5 | \n", "Real 6 | \n", "Real 7 | \n", "Real 8 | \n", "Real 9 | \n", "Real 10 | \n", "... | \n", "Real 91 | \n", "Real 92 | \n", "Real 93 | \n", "Real 94 | \n", "Real 95 | \n", "Real 96 | \n", "Real 97 | \n", "Real 98 | \n", "Real 99 | \n", "Real 100 | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
mean | \n", "69.1 | \n", "69.6 | \n", "69.7 | \n", "69.9 | \n", "70.1 | \n", "70.2 | \n", "70.3 | \n", "70.1 | \n", "70.1 | \n", "69.7 | \n", "... | \n", "69.8 | \n", "69.9 | \n", "69.7 | \n", "69.6 | \n", "70.0 | \n", "70.3 | \n", "69.7 | \n", "69.7 | \n", "69.8 | \n", "69.6 | \n", "
std | \n", "3.1 | \n", "3.2 | \n", "3.3 | \n", "3.2 | \n", "2.9 | \n", "3.2 | \n", "3.1 | \n", "3.2 | \n", "3.1 | \n", "3.1 | \n", "... | \n", "3.1 | \n", "2.9 | \n", "3.2 | \n", "3.5 | \n", "3.3 | \n", "3.3 | \n", "3.5 | \n", "3.1 | \n", "3.1 | \n", "3.1 | \n", "
min | \n", "63.1 | \n", "63.1 | \n", "63.6 | \n", "63.1 | \n", "64.5 | \n", "64.1 | \n", "63.6 | \n", "63.1 | \n", "63.1 | \n", "63.1 | \n", "... | \n", "63.1 | \n", "63.1 | \n", "63.6 | \n", "63.1 | \n", "64.1 | \n", "63.1 | \n", "63.1 | \n", "64.1 | \n", "63.1 | \n", "63.1 | \n", "
max | \n", "76.1 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "... | \n", "77.8 | \n", "76.1 | \n", "77.8 | \n", "76.1 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "77.8 | \n", "74.4 | \n", "77.8 | \n", "
median | \n", "68.8 | \n", "69.4 | \n", "69.1 | \n", "69.9 | \n", "70.6 | \n", "70.2 | \n", "70.3 | \n", "70.5 | \n", "70.5 | \n", "70.4 | \n", "... | \n", "69.4 | \n", "69.7 | \n", "69.2 | \n", "69.2 | \n", "69.7 | \n", "70.6 | \n", "70.4 | \n", "69.9 | \n", "70.2 | \n", "69.9 | \n", "
skew | \n", "0.3 | \n", "0.1 | \n", "0.2 | \n", "0.1 | \n", "-0.2 | \n", "0.1 | \n", "-0.0 | \n", "-0.2 | \n", "-0.2 | \n", "-0.2 | \n", "... | \n", "0.2 | \n", "-0.2 | \n", "0.1 | \n", "-0.1 | \n", "0.3 | \n", "-0.1 | \n", "0.0 | \n", "-0.0 | \n", "-0.3 | \n", "-0.1 | \n", "
kurtosis | \n", "-0.8 | \n", "-0.4 | \n", "-0.8 | \n", "-0.4 | \n", "-0.6 | \n", "-0.6 | \n", "-0.4 | \n", "-0.5 | \n", "-0.6 | \n", "-0.7 | \n", "... | \n", "-0.5 | \n", "-0.7 | \n", "-1.0 | \n", "-1.0 | \n", "-0.6 | \n", "-0.4 | \n", "-1.0 | \n", "-0.8 | \n", "-1.1 | \n", "-0.9 | \n", "
IQR | \n", "5.2 | \n", "5.1 | \n", "5.4 | \n", "5.2 | \n", "4.2 | \n", "4.7 | \n", "4.3 | \n", "4.4 | \n", "4.9 | \n", "5.0 | \n", "... | \n", "5.0 | \n", "4.2 | \n", "5.6 | \n", "5.9 | \n", "5.3 | \n", "4.9 | \n", "6.0 | \n", "5.0 | \n", "5.3 | \n", "5.5 | \n", "
8 rows × 100 columns
\n", "Each line in the plot represents one Bootstrap realization. There are 100 realizations, each having 100 random samples.
\n", "\n", " | P2.5 | \n", "P50 | \n", "P97.5 | \n", "
---|---|---|---|
mean | \n", "69.2 | \n", "69.8 | \n", "70.3 | \n", "
std | \n", "2.8 | \n", "3.1 | \n", "3.5 | \n", "
min | \n", "63.1 | \n", "63.1 | \n", "64.5 | \n", "
max | \n", "75.4 | \n", "77.8 | \n", "77.8 | \n", "
median | \n", "69.0 | \n", "69.7 | \n", "70.6 | \n", "
skew | \n", "-0.4 | \n", "-0.0 | \n", "0.3 | \n", "
kurtosis | \n", "-1.1 | \n", "-0.7 | \n", "-0.2 | \n", "
IQR | \n", "4.1 | \n", "5.1 | \n", "6.0 | \n", "
\n", " | Permeability (mD) | \n", "
---|---|
mean | \n", "161.008972 | \n", "
std | \n", "80.900128 | \n", "
min | \n", "43.534147 | \n", "
max | \n", "573.461883 | \n", "
median | \n", "144.329837 | \n", "
skew | \n", "1.625086 | \n", "
kurtosis | \n", "5.498080 | \n", "
IQR | \n", "102.580432 | \n", "
Outlier removal on rock permeability cannot be done directly, as this is a lognormal distribution. Recall that the typical outlier removal method assumes the distribution to be Gaussian. If you want to detect outliers for non-Gaussian distributions, you have to first transform the distribution into Gaussian.
\n", "\n", " | Real 1 | \n", "Real 2 | \n", "Real 3 | \n", "Real 4 | \n", "Real 5 | \n", "Real 6 | \n", "Real 7 | \n", "Real 8 | \n", "Real 9 | \n", "Real 10 | \n", "... | \n", "Real 91 | \n", "Real 92 | \n", "Real 93 | \n", "Real 94 | \n", "Real 95 | \n", "Real 96 | \n", "Real 97 | \n", "Real 98 | \n", "Real 99 | \n", "Real 100 | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | \n", "179.7 | \n", "168.4 | \n", "132.7 | \n", "123.3 | \n", "258.4 | \n", "116.9 | \n", "144.4 | \n", "170.5 | \n", "144.7 | \n", "141.8 | \n", "... | \n", "56.3 | \n", "128.9 | \n", "224.4 | \n", "205.1 | \n", "77.8 | \n", "210.8 | \n", "83.3 | \n", "84.9 | \n", "138.8 | \n", "184.8 | \n", "
1 | \n", "132.7 | \n", "58.8 | \n", "166.9 | \n", "206.6 | \n", "216.7 | \n", "258.4 | \n", "53.4 | \n", "286.8 | \n", "117.6 | \n", "168.4 | \n", "... | \n", "145.6 | \n", "264.2 | \n", "58.8 | \n", "84.7 | \n", "213.7 | \n", "84.9 | \n", "214.9 | \n", "108.6 | \n", "126.3 | \n", "213.7 | \n", "
2 | \n", "83.9 | \n", "286.8 | \n", "79.4 | \n", "103.5 | \n", "80.4 | \n", "138.8 | \n", "170.5 | \n", "97.7 | \n", "138.8 | \n", "265.5 | \n", "... | \n", "227.4 | \n", "89.6 | \n", "257.4 | \n", "133.2 | \n", "162.9 | \n", "58.8 | \n", "227.4 | \n", "116.6 | \n", "141.8 | \n", "210.8 | \n", "
3 | \n", "131.6 | \n", "187.9 | \n", "311.6 | \n", "125.9 | \n", "80.4 | \n", "85.0 | \n", "131.6 | \n", "190.9 | \n", "132.7 | \n", "214.9 | \n", "... | \n", "125.9 | \n", "108.6 | \n", "92.7 | \n", "129.1 | \n", "134.7 | \n", "131.6 | \n", "135.8 | \n", "224.4 | \n", "104.0 | \n", "234.5 | \n", "
4 | \n", "43.5 | \n", "142.7 | \n", "77.8 | \n", "142.7 | \n", "123.3 | \n", "59.1 | \n", "151.0 | \n", "58.8 | \n", "61.0 | \n", "193.1 | \n", "... | \n", "216.7 | \n", "132.7 | \n", "116.9 | \n", "160.5 | \n", "161.0 | \n", "573.5 | \n", "146.0 | \n", "156.1 | \n", "183.1 | \n", "143.7 | \n", "
5 | \n", "56.3 | \n", "134.7 | \n", "86.8 | \n", "279.8 | \n", "143.7 | \n", "279.8 | \n", "263.5 | \n", "92.1 | \n", "133.2 | \n", "53.4 | \n", "... | \n", "58.8 | \n", "206.6 | \n", "84.7 | \n", "117.6 | \n", "85.0 | \n", "61.9 | \n", "112.3 | \n", "83.9 | \n", "128.9 | \n", "162.3 | \n", "
6 | \n", "102.7 | \n", "84.7 | \n", "162.3 | \n", "77.8 | \n", "265.5 | \n", "179.7 | \n", "77.8 | \n", "141.8 | \n", "106.4 | \n", "61.0 | \n", "... | \n", "43.5 | \n", "61.9 | \n", "234.5 | \n", "132.1 | \n", "258.4 | \n", "131.6 | \n", "146.0 | \n", "285.3 | \n", "80.4 | \n", "116.6 | \n", "
7 | \n", "214.9 | \n", "224.4 | \n", "311.6 | \n", "321.1 | \n", "86.8 | \n", "79.4 | \n", "116.9 | \n", "187.9 | \n", "210.8 | \n", "144.4 | \n", "... | \n", "234.5 | \n", "234.5 | \n", "77.5 | \n", "144.7 | \n", "179.7 | \n", "123.3 | \n", "184.8 | \n", "214.1 | \n", "265.5 | \n", "170.5 | \n", "
8 | \n", "171.4 | \n", "54.3 | \n", "244.1 | \n", "151.5 | \n", "126.3 | \n", "136.9 | \n", "136.9 | \n", "258.4 | \n", "58.8 | \n", "83.3 | \n", "... | \n", "213.7 | \n", "205.1 | \n", "286.8 | \n", "112.3 | \n", "216.7 | \n", "145.6 | \n", "59.1 | \n", "117.6 | \n", "106.4 | \n", "80.4 | \n", "
9 | \n", "171.4 | \n", "84.7 | \n", "227.4 | \n", "240.6 | \n", "135.8 | \n", "128.9 | \n", "162.3 | \n", "244.1 | \n", "116.6 | \n", "161.0 | \n", "... | \n", "125.9 | \n", "77.5 | \n", "240.6 | \n", "92.7 | \n", "123.3 | \n", "258.6 | \n", "227.4 | \n", "102.7 | \n", "141.8 | \n", "380.1 | \n", "
10 rows × 100 columns
\n", "\n", " | Real 1 | \n", "Real 2 | \n", "Real 3 | \n", "Real 4 | \n", "Real 5 | \n", "Real 6 | \n", "Real 7 | \n", "Real 8 | \n", "Real 9 | \n", "Real 10 | \n", "... | \n", "Real 91 | \n", "Real 92 | \n", "Real 93 | \n", "Real 94 | \n", "Real 95 | \n", "Real 96 | \n", "Real 97 | \n", "Real 98 | \n", "Real 99 | \n", "Real 100 | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
mean | \n", "154.3 | \n", "154.3 | \n", "165.7 | \n", "168.1 | \n", "161.8 | \n", "159.8 | \n", "165.2 | \n", "159.9 | \n", "152.7 | \n", "170.9 | \n", "... | \n", "173.8 | \n", "158.4 | \n", "166.7 | \n", "156.7 | \n", "166.1 | \n", "149.5 | \n", "153.1 | \n", "163.7 | \n", "152.6 | \n", "156.9 | \n", "
std | \n", "66.8 | \n", "67.4 | \n", "84.1 | \n", "91.6 | \n", "78.5 | \n", "73.7 | \n", "85.7 | \n", "83.1 | \n", "70.5 | \n", "69.9 | \n", "... | \n", "100.3 | \n", "66.0 | \n", "84.5 | \n", "75.1 | \n", "65.2 | \n", "82.4 | \n", "78.1 | \n", "81.5 | \n", "71.9 | \n", "71.6 | \n", "
min | \n", "43.5 | \n", "43.5 | \n", "43.5 | \n", "43.5 | \n", "53.4 | \n", "53.4 | \n", "53.4 | \n", "43.5 | \n", "43.5 | \n", "53.4 | \n", "... | \n", "43.5 | \n", "54.3 | \n", "53.4 | \n", "43.5 | \n", "43.5 | \n", "54.3 | \n", "53.4 | \n", "43.5 | \n", "43.5 | \n", "53.4 | \n", "
max | \n", "380.1 | \n", "380.1 | \n", "573.5 | \n", "573.5 | \n", "573.5 | \n", "573.5 | \n", "573.5 | \n", "573.5 | \n", "380.1 | \n", "380.1 | \n", "... | \n", "573.5 | \n", "321.1 | \n", "573.5 | \n", "573.5 | \n", "380.1 | \n", "573.5 | \n", "573.5 | \n", "380.1 | \n", "573.5 | \n", "380.1 | \n", "
median | \n", "144.3 | \n", "143.7 | \n", "151.0 | \n", "144.4 | \n", "144.3 | \n", "144.4 | \n", "144.4 | \n", "142.7 | \n", "138.8 | \n", "162.9 | \n", "... | \n", "144.7 | \n", "144.7 | \n", "151.0 | \n", "141.8 | \n", "156.1 | \n", "132.0 | \n", "136.9 | \n", "144.4 | \n", "141.8 | \n", "143.7 | \n", "
skew | \n", "0.7 | \n", "1.0 | \n", "1.3 | \n", "1.8 | \n", "1.7 | \n", "1.9 | \n", "1.6 | \n", "1.5 | \n", "0.9 | \n", "0.5 | \n", "... | \n", "1.9 | \n", "0.4 | \n", "1.4 | \n", "2.1 | \n", "0.8 | \n", "2.0 | \n", "1.8 | \n", "0.7 | \n", "2.1 | \n", "0.9 | \n", "
kurtosis | \n", "0.5 | \n", "1.2 | \n", "3.9 | \n", "5.6 | \n", "6.3 | \n", "8.3 | \n", "4.4 | \n", "4.5 | \n", "0.9 | \n", "0.2 | \n", "... | \n", "5.4 | \n", "-0.6 | \n", "4.1 | \n", "8.4 | \n", "0.4 | \n", "6.6 | \n", "6.8 | \n", "-0.2 | \n", "10.1 | \n", "0.4 | \n", "
IQR | \n", "80.8 | \n", "81.6 | \n", "111.4 | \n", "113.2 | \n", "90.1 | \n", "80.9 | \n", "90.1 | \n", "121.1 | \n", "86.7 | \n", "85.2 | \n", "... | \n", "100.1 | \n", "92.8 | \n", "111.4 | \n", "91.4 | \n", "80.8 | \n", "98.0 | \n", "121.1 | \n", "121.7 | \n", "86.9 | \n", "88.2 | \n", "
8 rows × 100 columns
\n", "\n", " | P5.0 | \n", "P50 | \n", "P95.0 | \n", "
---|---|---|---|
mean | \n", "149.9 | \n", "161.4 | \n", "171.2 | \n", "
std | \n", "64.3 | \n", "79.9 | \n", "97.6 | \n", "
min | \n", "43.5 | \n", "43.5 | \n", "56.3 | \n", "
max | \n", "311.6 | \n", "573.5 | \n", "573.5 | \n", "
variance | \n", "4129.3 | \n", "6387.1 | \n", "9528.9 | \n", "
skew | \n", "0.4 | \n", "1.5 | \n", "2.3 | \n", "
kurtosis | \n", "-0.7 | \n", "4.9 | \n", "8.4 | \n", "
IQR | \n", "74.3 | \n", "98.1 | \n", "121.8 | \n", "