Statistics | Gentle Bedtime Reading For Sleep

Drift off with calm bedtime reading about statistics, designed to ease you into sleep while gently supporting those with insomnia. This calm bedtime reading continues with simple ideas from statistics, offering a peaceful path toward sleep and comfort for restless minds and insomnia as you settle in for the night. In this episode, you’ll explore the basics of statistics—patterns, averages, probability, and how we make sense of data—while learning something new as your body and mind relax.

Benjamin reads in a steady, reassuring cadence that’s intentionally not whispered, just calm, fact-filled education meant to soothe, helping ease stress, quiet anxiety, and make sleeplessness feel less overwhelming. Press play, get comfortable, and allow yourself to drift off naturally.

Happy sleeping!

Transcript

Welcome to the I Can't Sleep Podcast,

Where I help you drift off one fact at a time.

I'm your host,

Benjamin Boster,

And today's episode is about statistics.

This episode is actually a surprise happy birthday to Zach from Hillary.

In fact,

All episodes I've recorded this month are one big birthday present.

I hope you have a wonderful birthday,

Zach.

Now,

Let's learn about statistics.

Statistics is the discipline that concerns the collection,

Organization,

Analysis,

Interpretation,

And presentation of data.

In applying statistics to a scientific,

Industrial,

Or social problem,

It is conventional to begin with a statistical population or a statistical model to be studied.

Populations can be diverse groups of people or objects,

Such as all people living in a country,

Or every atom composing a crystal.

Statistics deals with every aspect of data,

Including the planning of data collection in terms of the design of surveys and experiments.

When census data comprising every member of the target population cannot be collected,

Statisticians collect data by developing specific experiment designs and survey samples.

Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole.

An experimental study involves taking measurements of the system under study,

Manipulating the system,

And then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.

In contrast,

An observational study does not involve experimental manipulation.

Two main statistical methods are used in data analysis.

Descriptive statistics,

Which summarize data from a sample using indexes,

Such as the mean or standard deviation,

And inferential statistics,

Which draw conclusions from data that are subject to random variation,

E.

G.

,

Observational errors,

Sampling variation.

Descriptive statistics are most often concerned with two sets of properties of a distribution,

A sample or population.

Central tendency or location seeks to characterize the distribution's central or typical value,

While dispersion or variability characterizes the extent to which members of the distribution depart from its center and each other.

Inferences made using mathematical statistics employ the framework of probability theory,

Which deals with the analysis of random phenomena.

A standard statistical procedure involves the collection of data leading to a test of the relationship between two statistical datasets,

Or a dataset and synthetic data drawn from an idealized model.

A hypothesis is proposed for the statistical relationship between the two datasets,

An alternative to an idealized null hypothesis of no relationship between two datasets.

Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false,

Given the data that are used in the test.

Working from a null hypothesis,

Two basic forms of error are recognized.

Type I errors.

Null hypothesis is rejected when it is in fact true,

Giving a false positive.

Type II errors.

Null hypotheses fail to be rejected when it is in fact false,

Giving a false negative.

Multiple problems have come to be associated with this framework,

Ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.

Statistical measurement processes are also prone to error in regards to the data that they generate.

Many of these errors are classified as random noise or systematic bias,

But other types of errors,

E.

G.

Blunder,

Such as when an analyst reports incorrect units,

Can also occur.

The presence of missing data or censoring may result in biased estimates,

And specific techniques have been developed to address these problems.

Statistics is the discipline that deals with data,

Facts,

And figures with which meaningful information is inferred.

Data may represent a numerical value in form of quantitative data,

Or a label,

As with qualitative data.

Data may be collected,

Presented,

And summarized in one of two methods,

Called descriptive statistics.

Two elementary summaries of data,

Singularly called a statistic,

Are the mean and dispersion.

Whereas inferential statistics interprets data from a population sample to induce statements and predictions about a population.

Statistics is regarded as a body of science,

Or a branch of mathematics.

It is based on probability,

A branch of mathematics that studies random events.

Statistics is considered the science of uncertainty.

This arises from the ways to cope with measurement and sampling error,

As well as dealing with uncertainties in modeling.

Although probability and statistics were once paired together as a single subject,

They are conceptually distinct from one another.

The former is based on deducing answers to specific situations from a general theory of probability.

Meanwhile,

Statistics induces statements about a population based on a dataset.

Statistics serves to bridge the gap between probability and applied mathematical fields.

Some consider statistics to be a distinct mathematical science,

Rather than a branch of mathematics.

While many scientific investigations make use of data,

Statistics is generally concerned with the use of data in the context of uncertainty and decision-making in the face of uncertainty.

Statistics is indexed at 62,

A subclass of probability theory and stochastic processes in the mathematics subject classification.

Mathematical statistics is covered in the range 276-280 of subclass QA in the Library of Congress classification.

The word statistics ultimately comes from the Latin word status,

Meaning situation or condition in society,

Which in late Latin adopted the meaning state.

Derived from this,

Political scientist Gottfried Achenwall coined the German word statistik,

A summary of how things stand.

In 1770,

The term entered the English language through German and referred to the study of political arrangements.

The term gained its modern meaning in the 1790s in John Sinclair's works.

In modern German,

The term statistic is synonymous with mathematical statistics.

The term statistics,

In singular form,

Is used to describe a function that returns its value of the same name.

When full census data cannot be collected,

Statisticians collect sample data by developing specific experiment designs and survey samples.

Statistics itself also provides tools for prediction and forecasting through statistical models.

To use a sample as a guide to an entire population,

It is important that it truly represents the overall population.

Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole.

A major problem lies in determining the extent that the sample chosen is actually representative.

Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures.

There are also methods of experimental design that can lessen these issues at the outset of a study,

Strengthening its capability to discern truths about the population.

Sampling theory is part of the mathematical discipline of probability theory.

Probability is used in mathematical statistics to study the sampling distributions of sample statistics and,

More generally,

The properties of statistical procedures.

The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.

The difference in point of view between classic probability theory and sampling theory is roughly that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples.

Statistical inference,

However,

Moves in the opposite direction,

Inductively inferring from samples to the parameters of a larger or total population.

A common goal for a statistical research project is to investigate causality and,

In particular,

To draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables.

These are two major types of causal statistical studies,

Experimental studies and observational studies.

In both types of studies,

The effect of differences of an independent variable or variables on the behavior of the dependent variable are observed.

The difference between the two types lies in how the study is actually conducted.

Each can be very effective.

An experimental study involves taking measurements of the system under study,

Manipulating the system,

And then taking additional measurements with different levels,

Using the same procedure to determine if the manipulation has modified the values of the measurements.

In contrast,

An observational study does not involve experimental manipulation.

Instead,

Data are gathered and correlations between predictors and response are investigated.

While the tools of data analysis work best on data from randomized studies,

They are also applied to other kinds of data,

Like natural experiments and observational studies,

For which a statistician would use a modified,

More structured estimation method,

E.

G.

Difference-and-differences estimation.

And instrumental variables,

Among many others,

That produce consistent estimators.

The basic steps of a statistical experiment are,

One,

Planning the research,

Including finding the number of replicates of the study using the following information.

Preliminary estimates regarding the size of treatment effects,

Alternative hypotheses,

And the estimated experimental variability.

Consideration of the selection of experimental subjects and the ethics of research is necessary.

Statisticians recommend that experiments compare at least one new treatment with a standard treatment or control,

To allow an unbiased estimate of the difference in treatment effects.

Two,

Design of experiments,

Using blocking to reduce the influence of confounding variables,

And randomized assignment of treatments to subjects,

To allow unbiased estimates of treatment effects and experimental error.

At this stage,

The experimenters and statisticians write the experimental protocol that will guide the performance of the experiment,

And which specifies the primary analysis of the experimental data.

Three,

Performing the experiment,

Following the experimental protocol,

And analyzing the data following the experimental protocol.

Four,

Further examining the data set in secondary analyses,

To suggest new hypotheses for future study.

Five,

Documenting and presenting the results of the study.

Experiments on human behavior have special concerns.

The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company.

The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers.

The researchers first measured the productivity in the plant,

Then modified the illumination in an area of the plant,

And checked if the changes in illumination affected productivity.

It turned out that productivity indeed improved under the experimental conditions.

However,

The study is heavily criticized today for errors in experimental procedures,

Specifically for the lack of a control group and blindness.

The Hawthorne effect refers to finding that an outcome,

In this case worker productivity,

Changed due to observation itself.

Those in the Hawthorne study became more productive not because the lighting was changed,

But because they were being observed.

An example of an observational study is one that explores the association between smoking and lung cancer.

This type of study typically uses a survey to collect observations about the area of interest,

And then performs statistical analysis.

In this case,

The researchers would collect observations of both smokers and non-smokers,

Perhaps through a cohort study,

And then look for the number of cases of lung cancer in each group.

A case control study is another type of observational study in which people with and without the outcome of interest,

E.

G.

Lung cancer,

Are invited to participate,

And their exposure histories are collected.

Various attempts have been made to produce a taxonomy of levels of measurement.

The psychophysicist Stanley Smith Stevens defined nominal,

Ordinal,

Interval,

And ratio scales.

Nominal measurements do not have meaningful rank order among values,

And permit any one-to-one injective transformation.

Ordinal measurements have imprecise differences between consecutive values,

But have a meaningful order to those values,

And permit any order-preserving transformation.

Interval measurements have meaningful distances between measurements defined,

But the zero value is arbitrary,

As in the case with longitude and temperature measurements in Celsius or Fahrenheit,

And permit any linear transformation.

Ratio measurements have both a meaningful zero value,

And the distance between different measurements defined,

And permit any rescaling transformation.

Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically,

Sometimes they are grouped together as categorical variables,

Whereas ratio and interval measurements are grouped together as quantitative variables,

Which can be either discrete or continuous due to their numerical nature.

Such distinctions can often be loosely correlated with data type in computer science,

In that dichotomous categorical variables may be represented with the Boolean data type,

Polytomous categorical variables with arbitrarily assigned integers in the integral data type,

And continuous variables with the real data type involving floating-point arithmetic.

But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented.

Other categorizations have been proposed.

For example,

Mosteller and Tukey,

1977,

Distinguished grades,

Ranks,

Counted fractions,

Counts,

Amounts,

And balances.

Nelder,

1990,

Described continuous counts,

Continuous ratios,

Count ratios,

And categorical modes of data.

The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions.

The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values,

Which are not invariant under some transformations.

Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer.

A descriptive statistic,

In the count-noun sense,

Is a summary statistic that quantitatively describes or summarizes features of a collection of information,

While descriptive statistics,

In the mass-noun sense,

Is the process of using and analyzing those statistics.

Descriptive statistics is distinguished from inferential statistics or inductive statistics in that descriptive statistics aims to summarize a sample rather than use the data to learn about the population that the sample of data is thought to represent.

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.

Inferential statistical analysis infers properties of a population,

For example by testing hypotheses and deriving estimates.

It is assumed that the observed dataset is sampled from a larger population.

Inferential statistics can be contrasted with descriptive statistics.

Descriptive statistics is solely concerned with properties of the observed data,

And it does not rest on the assumption that the data come from a larger population.

Consider independent,

Identically distributed random variables with a given probability distribution.

Standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables.

The population being examined is described by a probability distribution that may have unknown parameters.

A statistic is a random variable that is a function of the random sample,

But not a function of unknown parameters.

The probability distribution of the statistic,

Though,

May have unknown parameters.

Consider now a function of the unknown parameter.

An estimator is a statistic used to estimate such function.

Commonly used estimators include sample mean,

Unbiased sample variance,

And sample covariance.

A random variable that is a function of the random sample and of the unknown parameter,

But whose probability distribution does not depend on the unknown parameter,

Is called a pivotal quantity or pivot.

Widely used pivots include the z-score,

The chi-square statistic,

And student's t-value.

Between two estimators of a given parameter,

The one with lower mean squared error is said to be more efficient.

Furthermore,

An estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated,

And asymptotically unbiased if its expected value converges at the limit to the true value of such parameter.

Other desirable properties for estimators include UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated.

This is usually an easier property to verify than efficiency,

And consistent estimators which converges in probability to the true value of such parameter.

This still leaves the question of how to obtain estimators in a given situation and carry the computation.

Several methods have been proposed.

The method of moments,

The maximum likelihood method,

The least squares method,

And the more recent method of estimating equations.

Interpretation of statistical information can often involve the development of a null hypothesis,

Which is usually,

But not necessarily,

That no relationship exists among variables,

Or that no change occurred over time.

The alternative hypothesis is the name of the hypothesis that contradicts the null hypothesis.

The best illustration for a novice is the predicament encountered by a criminal trial.

The null hypothesis,

H0,

Asserts that the defendant is innocent,

Whereas the alternative hypothesis,

H1,

Asserts that the defendant is guilty.

The indictment comes because of suspicion of the guilt.

The H0,

The status quo,

Stands in opposition to H1,

And is maintained unless H1 is supported by evidence beyond a reasonable doubt.

However,

Failure to reject H0 in this case does not imply innocence,

But merely that the evidence was insufficient to convict.

So,

The jury does not necessarily accept H0,

But fails to reject H0.

While one cannot prove a null hypothesis,

One can test how close it is to being true with a power test,

Which tests for Type II errors.

Working from a null hypothesis,

Two broad categories of error are recognized.

Type I errors,

Where the null hypothesis is falsely rejected,

Giving a false positive.

Type II errors,

Where the null hypothesis fails to be rejected,

And an actual difference between population is missed,

Giving a false negative.

Standard deviation refers to the extent to which individual observations in a sample differ from a central value,

Such as the sample or population mean,

While standard error refers to an estimate of difference between sample mean and population mean.

A statistical error is the amount by which an observation differs from its expected value.

A residual is the amount an observation differs from the value the estimator of the expected value assumes on a given sample,

Also called prediction.

Mean-squared error is used for obtaining efficient estimators,

A widely used class of estimators.

Root-mean-square error is simply the square root of mean-squared error.

Many statistical methods seek to minimize the residual sum of squares.

These are called methods of least squares,

In contrast to least absolute deviations.

The latter gives equal weight to small and big errors,

While the former gives more weight to large errors.

Residual sum of squares is also differentiable,

Which provides a handy property for doing regression.

Least squares applied to linear regression is called ordinary least squares method,

And least squares applied to non-linear regression is called non-linear least squares.

Also,

In a linear regression model,

The non-deterministic part of the model is called error term,

Disturbance,

Or more simply,

Noise.

Both linear regression and non-linear regression are addressed in polynomial least squares,

Which also describes the variance in a prediction of the dependent variable y-axis as a function of the independent variable x-axis,

And the deviations,

Errors,

Noise,

Disturbances from the estimated fitted curve.

Measurement processes that generate statistical data are also subject to error.

Many of these errors are classified as random,

Noise,

Or systematic bias,

But other types of errors,

E.

G.

,

Blunder,

Such as when an analyst reports incorrect units,

Can also be important.

The presence of missing data or censoring may result in biased estimates,

And specific techniques have been developed to address these problems.

Reviews

I enjoyed statistics classes in college so this was interesting. Somehow I didn’t make it through the entire reading though!
Thank you, Benjamin! 😻

Statistically, I have a higher than normal chance of sleeping when I listen to your recordings.

I was actually interested. In fact I was so interested, I didn’t realize I’d dozed off…