Probability and Statistical Inference - 2024

Environmental Data Analysis:
Probability Distribution and
Statistical Inference
Vitor Vieira Vasconcelos
João Marcelo Borovina Josko
Federal University of ABC (UFABC)
São Bernardo do Campo-SP, Brazil
October 2024

What We Will Study in Today's Class
• Populations and Samples
• Normal Curve
• Frequency and Probability Distributions
• Standard scores
• Calculating Probability Under the Normal Curve
• Central Limit Theorem and Standard Error
• Confidence Interval
• Practice in R

As researchers, we are interested in
investigating questions that apply to a
whole population of people or things
 The population can be general (all humans) or small
(all rabbits in field)
 We rarely have access to data from the entire
population, but only from a subset of a sample,
which we use to infer things about the entire
population
Populations & Samples

 The larger the sample, the more likely it is to reflect
the entire population
 Random samples from the same population may
give slightly different results
 On average, results from large samples should be
quite similar
Populations & Samples

 POPULATION: It is the set of all the elements or
results under investigation.
 SAMPLE: Any subset of the population.
 PARAMETER: It is a numerical measure that
describes a population
 STATISTICS: It is a numerical measure that
describes a sample
 ESTIMATOR: It is a sample statistic used to
approximate a population parameter
Concepts
Source: Slides Prof. Marcos Pó, UFABC

Scientific method for drawing conclusions about
population parameters from the collection,
treatment, and analysis of data from a sample
collected from that population
Statistical Inference

Statistical Inference boiling
down to an equation...
Outputi = (Modeli) + errori
That is, the data we observe can be predicted by
the model we choose to fit the data plus an error

Frequency Distributions
HISTOGRAM: Graph with the values observed on the
horizontal axis, with bars showing how many times
each value occurred in the dataset
Useful for evaluating the properties of a set of values
Mode
Most frequently
occurring score in the
dataset
Frequency
Values

Cummulative Distribution Function
https://statisticsbyjim.com/probability/cumulative-distribution-function-cdf/
Median

Cummulative
Frequency
Distribution
https://statisticsbyjim.com/probability/cumulative-distribution-function-cdf/

Cummulative
Frequency
Distributions
Wang, J., Lai, S., Ke, Z., Zhang, Y., Yin, S.,
& Zheng, J. (2014). Exposure assessment,
chemical characterization and source
identification of PM 2.5 for school children
and industrial downwind residents in
Guangzhou, China. Environmental
geochemistry and health, 36, 385-397.
Histograms and cumulative
frequency curves of the daily
average potential exposure
doses of PM 2.5 for the school
children at the school (a) and
the industrial downwind
residents in the community (b)

Normal Curve
Most of the scores are around the center of the distribution.
As we move away from the center (average), the frequency of
the scores decreases.
Frequência
Valores

Examples of normal distribution in
Environmental Sciences
 Daily air, surface or water temperature in a
location
 Noise in environmental sensors or measurements
 Soil pH Levels (or another environmental
characteristic) in a homogeneous region
 Species body size
 Noise level in urban environments
 Daily or monthly averaged wind speeds
 Canopy height in mature forests

Properties of Frequency Distributions
A distribution can deviate from a normal in 2 main ways:
(1) Lack of symmetry
ASYMMETRY
(2) Flattening
KURTOSIS
Leptokurtic Platykurtic
Positively Asymmetrical Negatively Asymmetrical
Frequency
Values
Frequency
Values
Frequency
Values
Frequency
Values
GREATER
STANDARD
DEVIATION
LOWER
STANDARD
DEVIATION

Pearson’s coefficient of skewness
Where
= the mean,
Md = the median and
s = the standard deviation for the sample

Fisher’s moment coeficient of
skewness
Where
X = random variable (central moment – median)
μ = mean
σ = the standard deviation for the sample
E = expectation operator

Central Trend Measures vs Distribution Frequences
MODE (Mo): Most frequent value in a distribution
MEDIAN (Me): Measure that separates the distribution into two equal parts
MEAN (X): Sum of a set of scores divided by the total number of scores in
the set
Negativelly Asymmetrical Positively Asymmetrical

Kurtosis
Leptokurtic Mesokurtic Platykurtic
Tomy, L., Chesneau, C., & Madhav, A. K. (2021). Statistical Techniques for Environmental
Sciences: A Review. Mathematical and Computational Applications, 26(4), 74.

Pearson’s measure of kurtosis
 Forth moment (squared of skewness – 1)
 Calculated based on the extremity of the deviations
(outliers) and not of data near the mean
 Kurtosis is not “peakdness”
 Kurtosis is ““tailedness”
Where
X = random variable (central moment – median)
μ = mean
σ = the standard deviation for the sample
E = expectation operator

Normal Curve
Symmetric. Average, median and fashion coincide!
Neither leptokurtic, neither platykurtic  Mesokurtic
From the central peak, the curve gradually drops at both ends, getting
closer and closer to the basic straight, without ever touching it
Frequency
Values

C’mon guys! Can’t
we behave normally?

Probability Distributions
Frequency distributions can be used to get a rough idea of the
probability of a score occurring (or interval)
Considering that the distribution of the number of fish offspring
per birth of a certain species has a normal distribution, what
would be the probability of having, in a birth, 4 offsprings or
less?
PROBABILITY:
AN IMPORTANT NOTION FOR DECISION MAKING!!

To facilitate our work, statisticians
have devised a mathematical form
that specifies idealized versions of
the distributions:
PROBABILITY DISTRIBUTIONS

The probability distribution associates a probability
with each numerical outcome of an experiment, that
is, it gives the probability of each value (or range of
values) of a random variable.
 It is analogous to a frequency distribution, except that it is
based on theory rather than empirical data (real-world
observations)
 The probabilities represent the chance of each score
occurring, directly analogous to the percentages in a
frequency distribution.

You cannot prove anything from
samples or by using statistical
inference
You always estimate a probability

The normal curve as a probability
distribution
 The normal curve is a theoretical ideal
 However, there are many actual data distributions that
approximate the shape of the normal curve
It is always important to check!!!
Building a histogram is a good start!

Normal distribution
μ = the mean or expectation of the distribution
(and also it’s the median and the mode)
σ2 = variance.
σ (sigma) = standard deviation

distribution

distribution
Offsprings per birth
Mean = 2.6; s = 1.14
1,46 3,74
2,6

Interactive Practice
https://distribution-explorer.github.io/continuous/normal.html

Returning to our question:
less?
1,46 3,74
2,6
4
Offsprings per birth
Mean = 2,6; s = 1,14

Standard Normal Distribution
 They have already calculated the probability of certain scores
occurring in a normal distribution with
Mean = 0 & Standard Deviation = 1
STANDARD NORMAL DISTRIBUTION

BUT... The distribution of my data does not show
mean = zero and standard deviation = 1!
So????
ANY DATASET CAN BE CONVERTED TO A DATASET
THAT HAS ZERO MEAN AND STANDARD DEVIATION 1!
YEAH!!!!
How to do:
(1) To center data at zero, we took each score and
subtracted from it the mean of all the scores.
(2) We divided the resulting score by the standard
deviation to ensure that the results will have SD = 1
Z score

Returning to our question:
less?
Considering that the distribution of the data can be described as a normal
distribution, with mean = 2.6 and standard deviation = 1.14
Z score
Convert the value 4 into a z-score
(4 - 2.6)/1.14 = 1.23

1,23

Some z-scores are cutoff
points that highlight
important points of the
distribution.
1,96
-1,96
z = 1.96
z = -1.96
Separates the top and tail
2.5%
That is, 95% of the scores are
between -1.96 and 1.96

z = -2,58
z = +2,58
99% of the scores
are between -2.58
and 2.58
Some z-scores
are cutoff points
that highlight
important points
of the
distribution.
z = -3.29
z = +3.29
99.9% of the scores
are between -3.29
and 3.29

Is my sample
representative of my
population?
Conventions:
μ = population mean
X = sample mean
We used samples to estimate the
behavior/characteristics of a population.
For example, we use the sample mean
(X), to estimate the population mean
(μ).
If we take many samples from the same
population, each sample will have its
own mean and in several of these
samples the means will be different.
Population

Is my sample
representative of my
population?
We can construct a frequency
distribution with the averages of these
samples!
SAMPLE DISTRIBUTION
Frequency distribution of the means of
all samples of the same population. It is
centered on the same value as the
population mean.
SAMPLE
DISTRIBUTION
Conventions:
μ = population mean
X = sample mean
σ = standard deviation
of the population
s = standard deviation of
the sample

3. The standard deviation of a sample
distribution (σX) is smaller than that of the
population (σ). The sample mean is more
stable than the scores that compose it.
Characteristics of a
sample distribution
SAMPLE
DISTRIBUTION
1. Aproximates a normal curve
(provided the sample size is reasonably
large – N > 30)
2. The mean of a sample distribution (the
mean of the means) is equal to the true
population mean (μ).

DURATION OF
INTER-
MUNICIPAL
PHONE CALLS
Distribution of the population Distribution of a sample (N = 200)
Sample distribution (100 samples) Theoretical sample distribution (for
infinite samples) – Normal curve

Standard Error of the
Mean
SAMPLE
DISTRIBUTION
STANDARD ERROR
It measures variability
between the means
of different samples.
In fact, it should be the standard deviation of the population
divided by the square root of the sample size; however, for large
samples, this approximation is reasonable.
STANDARD ERROR OF THE
MEAN(σX )
Standard deviation of sample means.
Measure of how representative the
sample may be of the population
In reality, we cannot select hundreds of
samples to construct a sample
distribution.
Technique for estimating standard error
from the standard deviation of the
sample(s):
Divide s by the square root of the sample
size (N)

Standard Error of the Mean
RECAP:
 We use the sample mean as an estimate of the value of the
population mean.
 Different samples give different values than the population
mean.
 The Standard Error can be used to get an idea of the
difference between the sample mean and the population
mean.
 The Standard Error can be estimated  higher when the
standard deviation of the population is greater (in the
absence of the standard deviation of the population, we used
the standard deviation of the sample); smaller when the
sample number is bigger.

https://istats.shinyapps.io/sampdist_cont/

Try with different number of samples, sample size, and
different datasets (real and simulated examples).

Interactive practice
https://stats.libretexts.org/Learning_Objects/02%3A_Interactive_Statistics/15%3A_Discover_the_Central_Limit_Theorem_Activity

 Visualize the sample means with the different sample
sizes.
 What happens when you increase sample size?

 Click on “New Dist” button and try the same
procedure with other distribution functions

Standard Error of the Mean
In addition to providing us with an idea of the
difference between the sample mean (X) and the
population mean (μ)...
 With the help of the Standard Error of the Mean we
can estimate the probability that our population
mean is within a range of mean values.
Concept of CONFIDENCE INTERVAL

Confidence Intervals
An approach to determining the accuracy of
the Sample Mean:
Calculate the thresholds between which we believe
the value of the true mean will be
CONFIDENCE INTERVAL
Range of values (thresholds) between which we
think the population value (parameter) will be
(in this case, the value of the true average)

Point Estimate Confidence Interval
https://moderndive.com/8-confidence-intervals.html
Ismay, C.; Kim, A. Y. 2019. Statistical Inference via Data Science: A ModernDive into
R and the Tydiverse. Chapman & Hall / XRC The R Series
Which one has higher chance of getting the fish?

A 95% confidence interval (CI)
How do I interpret it???
If we select 100 samples, calculate the mean, and
after we determine the confidence interval for
that mean, 95% of the confidence intervals will
contain the actual value of the population mean

OK! Now let's
see how the CI
is calculated...
SAMPLE
DISTRIBUTION
OF MEANS
THE MEAN OF OUR
SAMPLE IS
SOMEWHERE IN THE
DISTRIBUTION

Remember why the value 1.96
is an important z-value???
Also remember how we can
convert scores into z-scores:
Z scores
And 2,58?
And 3,29?
Because 95% of z-scores are
between -1.96 and 1.96!!

If we know that our limits will be -1.96 and 1.96, in z-
scores, what are the corresponding scores in values
of our data?
[It is the inverse of how to calculate the Z score]
To find this, let's put z back into the equation
Z score

Z - scores
-
-
Lower threshold of the confidence interval
Upper threshold of the confidence interval
SE)
SE)

SE)
Z - scores
We use Standard Error rather
than Standard Deviation
because we are interested in the
variability of sample means
rather than the variability of
within-sample observations
SE)

Example – 95% CI
Let's say we have collected data on the
emissions of CO2 (million tons per year) per
country in the world. We have a sample of
100 countries (N=100), with mean = 3800 and
standard deviation(s) = 1500.
Standard Error (SE) calculation:
SE

Exemple – 95% CI
Let's say we have collected data on the emissions of CO2 (million
tons per year) per country in the world. We have a random sample
of 100 countries (N=100), with mean = 3800 and
Lower threshold of the confidence interval =
3800 – (1.96*150) = 3506
Upper threshold of the Confidence Interval =
3800 + (1.96*150) = 4094
SE)
SE)

Example – 95% CI
Let's say we have collected data on the emissions of CO2 (million
tons per year) per country in the world. We have a random sample
of 100 countries (N=100), with mean = 3800 and
Lower threshold of the confidence interval = 3800 – (1.96*150) = 3506
Upper threshold of the Confidence Interval = 3800 + (1.96*150) = 4094
Considering that 95% of the confidence intervals contain the
population mean, we can say that this interval between 3506 and
4094 has a 95% chance of containing the real average emission of
CO2 per country.

https://stats.libretexts.org/Learning_Objects/02%3A_Interactive_Statistics/22%3A_Intera
ctively_Observe_the_Effect_of_Changing_the_Confidence_Level_and_the_Sample_Size
• Try different sample sizes and confidence levels.
• What happens when you increase or decrease the sample size?
• What happens when you increase or decrease the confidence level?

More Accurate Confidence Intervals
For small samples, where s is a less reliable estimate of σ, we
should build our confidence interval a little differently.
Instead of using 1.96 (z-score), we used a slightly higher value to
reflect our reduction in confidence. This value is based on the t
distribution.

Degrees of freedom
“The number of values in a final calculation of a statistic that are free to
vary”
“The number of independent pieces of information that go into the
estimate of a parameter”.
“The number of independent scores that go into the estimate minus the
number of parameters used as intermediate steps (restrictions) in the
estimation of the parameter itself”
“The number of unique pieces of information that are used as input into
the analysis, called knowns, minus the number of parameters that are
uniquely estimated, called unknowns.”
“The number of dimensions of the domain of a random vector, or
essentially the number of "free" components (how many components
need to be known before the vector is fully determined)”
“Summarizes the flexibility of a model”
“The denominator of a variance estimate”

Degrees of freedom
VARIANCE
"mean of the square of deviations"
Estimation of population variance using n random
samples xi where i = 1, 2, ..., n.

https://probstats.org/studentt.html

https://distribution-explorer.github.io/continuous/student_t.html

Comparison of Confidence Intervals
We measured heavy metal pollution concentration in soils
of sites in a group of industrial areas and in another group
of agricultural areas. We can construct a 95% confidence
interval for the mean for each of the groups, and then
construct a graph with these intervals against a common
axis to see if there is an intersection (i.e. if there are any
values in common). If the ranges do not overlap, then we
have (at least) 95% confidence that the true means are not
equal.

Comparison of Confidence Intervals

Margins of Error in Ellections
https://www.surveypractice.org/article/11736-sample-size-and-uncertainty-when-
predicting-with-polls-the-shortcomings-of-confidence-intervals

Margins of Error in Ellections
I don’t
know /
refuse
to
answer
Candidate
S
Candidate
M
https://www.surveypractice.org/article/11736-sample-size-and-uncertainty-when-
predicting-with-polls-the-shortcomings-of-confidence-intervals

Margins of Error
https://www.ft.com/content/710b8677-2ed2-4a68-9600-4ecb4b22d0d6

Error Margins
https://ig.ft.com/us-elections/2024/polls/

Symmetrical vs Asymmetrical
Laufer, I. (2013). Statistical analysis of CPT tip resistances. Periodica Polytechnica
Civil Engineering, 57(1), 45-61.

Distribution by Bootstrapping
https://towardsdatascience.com/bootstrapping-statistics-what-it-is-and-why-its-used-e2fa29577307

https://www.khstats.com/blog/tmle/tutorial

Poisson distribution
Probability of a given
number of events occurring
in a fixed interval of time
or space, under the
assumption that these
events happen independently
of one another.

Examples in environmental sciences
 Rare event counts (e.g., disasters)
 Rainfall events in a time period
 Number of parasites on host species
 Occurrence of endangered species
sightings
 Number of species in a quadrant

where
k is the number of occurrences ( k = 0 , 1 , 2 , …)
e is Euler's number ( e = 2.71828 …)
k! = k(k–1) ··· (3)(2)(1) is the factorial.
The positive real number λ (lambda) is equal to the
expected value of X and also to its variance.
λ = E (X) = Var (X)

https://probstats.org/poisson.html

https://distribution-explorer.github.io/discrete/poisson.html

Binomial Distribution
Discrete probability distribution
of the number of successes in a
sequence of n independent
experiments, each asking a yes–
no question

What is the probability that a toast with jelly will fall with
topping side down?

Binomial probability tree
Up
Down
Up
Down
Up
Down
50% vs 50%
0.5
0.5
0.5
0.5
0.5
0.5
0.5 * 0.5 = 0.25
2x UP
Up +
Down
2x Down 0.5 * 0.5 = 0.25
2 * (0.5 * 0.5) = 0.5

Binomial probability tree
Up
Down
Up
Down
Up
Down
10% vs 90%
0.1
0.9
0.1
0.9
0.1
0.9
0.1 * 0.1 = 0.01
2x UP
Up +
Down
2x Down 0.9 * 0.9 = 0.81
2 * (0.1 * 0.9) = 0.18

Examples of Binomia Distributions
in Environmental Sciences
(yes/no questions)
• Disaster occurrence / absence
• Species distribution modeling
(presence / absence)
• Survival of seedlings / animal / humans
• Contaminant presence / absence
• Probability of a certain environmental condition
(e.g., wet days in a month) being met

for k = 0, 1, 2, ..., n, where
The probability of getting exactly k successes in n
independent Bernoulli trials (with the same rate p) is given
by the probability mass function:

https://commons.wikimedia.org/wiki/File:Binomial_distribution_pmf.svg

https://probstats.org/binomial.html

https://stats.libretexts.org/Learning_Objects/02%3A_Interactive_Statistics/17%3A_
Observe_the_Relationship_Between_the_Binomial_and_Normal_Distributions

Negative binomial
Discrete probability distribution that
models the number of failures in a
sequence of independent and identically
distributed Bernoulli trials before a
specified (non-random) number of
successes (denoted r) occurs

Example of negative binomial
distribution
How many lottery tickets I
would probably need to buy
before I win for two times?

Negative binomial
where
r is the number of successes,
k is the number of failures, and
p is the probability of success on each trial.

Negative binomial
where
r is the number of successes,
k is the number of failures,
and
p is the probability of
success on each trial.

https://distribution-explorer.github.io/discrete/negative_binomial.html

Examples of Negative Binomial distribution
 Number of failed seedlings until you reach a certain
number of mature plants in a restoration site
 Number of failed attacks before a predator acquires
the amount of prey food
 Number of events when the mean is different from
the variance
(Poisson distribution cannot apply)
 Species abundance per unit of area
 Infections disease spread
 Microbial counts in water samples

Beta Binomial Distribution
Used to model binomial distributions
(probability of success of yes/no events)
when the probability of success (p) is unknown.
https://commons.wikimedia.org/wiki/File:Beta-binomial_distribution_pmf.png

Beta Binomial Distribution
Used to model number of events when the mean is
different from the variance
(Poisson distribution is not valid in this case)
https://commons.wikimedia.org/wiki/File:Beta-binomial_distribution_pmf.png

Examples of Beta Binomial Distribution in
 Species abundance in habitat patches with
different environmental conditions (food
availability, light, temperature, etc.)
 Germination rates (success of seeds) in regions
with heterogeneous soil types and moisture.
 Survival rates of animal populations
 Number of positive samples testing positive for
bacterial contamination (e.g., E. coli), Where the
source of contamination changes from site to
site

Multinomial Distribution
 Generalization of the binomial distribution, used
to model outcomes that fall into multiple
categories
https://jramkiss.github.io/2020/05/15/beta-and-dirichlet-distributions/

Examples of multinomial distribution in
 Classification of land use types in remote sensing or
ecological studies
 Proportions of different species in ecological
communities
 Pollutant source attribution (industrial, agricultural,
vehicle traffic, natural background)
 Soil classification
 Waste composition analysis (organic, plastic, paper,
metal, and glass).
 Species functional groups (herbivore, carnivores,
decomposers, pollinators, etc.)

Normalized difference vegetation index
vs
Land cover
Azizah, S. N. N., June, T., Salmayenti, R., Ma'rufah, U., & Koesmaryono, Y. (2022).
Land use change impact on normalized difference vegetation index, surface albedo, and
heat fluxes in Jambi province: Implications to rainfall. Agromet, 36(1), 51-59.

Continuous Uniform Distribution
https://commons.wikimedia.org/wiki/File:Uniform_Distribution_PDF_SVG.svg

Log-Normal Distribution
 The logarithm of the variable is normally distributed
 Many small common events, but sometimes a very large
one happens
https://commons.wikimedia.org/wiki/File:Log-normal-pdfs.png

https://probstats.org/lognormal.html

Examples of Log-Normal Distribution in
 Concentrations of pollutants, such as particulate matter
(PM2.5 or PM10) or heavy metals (lead, mercury) in air,
water or soil.
 Distribution of species abundance, where a few species
are highly abundant, but many species have low counts.
 Rainfall intensity of storm events
(mm of rain / storm time)
 River discharge
 Wildfire size
 Economic Losses from Environmental Disasters

Exponential distribution
Time between
independent
events occurring
at a constant
average rate
https://commons.wikimedia.org/wiki/File:Exponential_distribution_pdf_-_public_domain.svg

https://probstats.org/exponential.html

Examples of Exponential distribution in
 Time intervals between natural events, like
earthquakes, extreme weather events (e.g.,
storms, droughts, wildfires), or lightning strikes.
 The decay rate of pollutants or radioactive
materials in ecosystems.
 The recovery time of ecosystems following a
disturbance (e.g., deforestation, flooding, or a
wildfire)

Gamma distribution
Often used for
non-negative
variables
https://commons.wikimedia.org/wiki/File:Gamma_distribution_pdf.svg

https://probstats.org/gamma.html

Examples of Gamma distribution
 Precipitation data
 Hydrological processes
 Drought duration
 Time to ecological recovery
 Time to disasters events

Gumbell distribution
Used for modeling extreme values, such as the
maximum or minimum of sample groups
https://commons.wikimedia.org/wiki/File:Gumbel-Density.svg

https://rdzudzar-distributionanalyser-main-45cc69.streamlit.app/
Try also with gumbel_l

Examples of Gumbell distribution in
 Modeling extreme weather events, such as
maximum daily rainfall or highest river
discharge during floods.
 Hailstorm frequency
 Landslide events caused by extreme rainfall
 Analysis of extreme temperatures
(heatwaves or cold spells)
 Peaks of severe air pollution

Weibull
Distribution
https://upload.wikimedia.org/wikipedia/commons/5/58/Weibull_PDF.svg

https://probstats.org/weibull.html

Examples of Weibull distribution in
environmental sciences
 Wind speed modelling
 Disaster recurrence time, such as
wildfire and landslides
 Species survival rates under a changing
environmental condition (pollution,
temperature, etc.)

Beta Distribution
https://commons.wikimedia.org/wiki/File:Beta_distribution_pdf.svg
X is a
continuous
probability from
0 (zero) to 1

https://probstats.org/beta.html

Examples of Beta distribution in
 Proportion of vegetation types in an
Ecosystems
 Soil moisture content
 Proportion of land use cover in regions
(e.g., municipalities)
 Proportion of water samples meeting
regulatory standards
 Probability of adverse affects from
pollutants

Cauchy Distribution
https://commons.wikimedia.org/wiki/File:Cauchy_pdf.svg
Ratio of two independent normally distributed random
variables with mean zero

Cauchy Distribution
where :
x0 is the location
parameter, specifying the
location of the peak of
the distribution
γ (gamma) is the scale
parameter which
specifies the half-width at
half-maximum (HWHM)

Cauchy Distribution
• Mode and median = x0
• Symmetrical distribution (skewness = 0)
• Heavy tails (many extreme values for both sides)

https://distribution-explorer.github.io/continuous/cauchy.html

Examples of Cauchy distribution in
• Extreme weather events including
scarcity vs excess
 rainstorms vs droughts
 heatwaves vs cold spells
• Remote sensing reflectance data

Uses of distribution functions in Statistics
• Calculate
 Probabilities
 Uncertainty and variability
 Recurrence times
• Simulate data
• Planning for sampling design and error margins
• Generalized models
 Extend models based on normal distributions to
other distributions
 Allow more precise estimates and less effect of
outliers

Probability and Statistical Inference - 2024

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