Midterm 1

Midterm 1#

Answer key for MT1: here

Exam logistics#

  • FRIDAY, October 24th, 2-3:15pm, MR 1128

  • 100% closed to all resources: notes, books, internet, classmates

Study guide#

This page provides a list of the key concepts that you should focus your studying on for this exam. Definitions for some, but not all, of the key concepts are provided at the bottom of the page in a glossary.

Unit 00: Introduction#

Why we need statistical methods / quantitative data analysis methods

Unit 01: Numeracy and data cleaning#

numeracy

Components of numerate claims:

  • quantitative: they use numbers derived from actual data

  • uncertainty quantification: confidence intervals

  • methodology: they describe how the numbers and their associated uncertainties were determined

metadata: data about your data. Location, times, technical specs of the instruments, valid ranges, calibration issues, understanding of changes in a record

Looking out for and dealing with spurious values

Unit 02: Descriptive statistics and data visualization#

Measures of central tendency: mean, median, mode

Measures of variation: two categories, measures of (1) dispersion or (2) shape

Measures of dispersion: term{range}, term{IQR}, term{standard deviation}, term{variance}, term{coefficient of variation}

Measures of shape: term{skewness}, term{kurtosis}

Plot types and their interpretation: box and whiskers; histograms; scatter plots; timeseries

Unit 03: Probability theory#

The three axioms of probability

  1. Non-negativity: \(P(E)\geq0\) for any event \(E\)

  2. Certainty: \(P(S)=1\) for the sample space \(S\)

  3. Additivity: If \(E_1\) and \(E_2\) are mutually exclusive, then \(P(E_1\cup E_2)=P(E_1)+P(E_2)\).

Outcomes vs. events

Mutually exclusive events

Simple events vs. compound events

Complement: for an event \(E\), its complement \(E^C\) means that \(E\) does not occur

Sets: collections of zero or more unordered, unique elements. They’re used to express the math underlying probability.

Sets: union, intersection, subsets

Independent events

Conditional probability

Joint probability: probability of two events both occurring. Note: this is not the same of the union of two events occurring.

Probability of union

Frequentist vs. Bayesian interpretations of probability

Empirical probabilities

Unit 04: Probability distributions (and populations vs. samples)#

Types of data: continuous vs. discrete

Probability density functions

Probability mass functions

Cumulative distribution functions

Parametric probability distributions and how they differ from empirical distributions

Uniform distribution

The normal distribution, a.k.a. Gaussian distribution. How the mean and standard deviation of a Gaussian influence its shape

Expectation: sum up all possible outcomes, each one weighted by its probability. For parametric distributions, this is the mean.

Central Limit Theorem: the sum of a large number of IID random variables will be normally distributed (i.e. its PDF will be a Gaussian) as the number of random variables being summed tends toward infinity.

Populations vs samples. Samples: the values we get. Population: the distribution from which the sample values we get were drawn from. We’re always trying to infer the properties of the population; but we almost always only have access to the samples.

Glossary (incomplete)#

coefficient of variation#

The term{standard deviation} divided by the term{mean}.

Always dimensionless and therefore unitless, making it meaningful to compare between variables that don’t have the same dimensions.

expectation#

Probability-weighted sum over all possible outcomes of a probability distribution

event#
IQ #

Inter-Quartile Range: 75th percentile minus the 25th percentile.

kurtosis#

A measure of how “fat-tailed” a dataset is.

This combines both the positive and negative term{tail}; in other words, it is independent of term{skewness}

For our purposes, we’ll only be concerned with its relative size:

  • smaller kurtosis: fewer values far away from the mean (i.e. “skinny” tails)

  • larger kurtosis: lots of values far away from the mean (i.e. “fat” tails)

mean#

Average.

median#

The value at which 50% of a dataset is larger (more positive) and 50% is smaller (less positive). AKA 50th percentile.

metadata#

Data about data. Information regarding how a particular piece of data was collected: location, date/time, instruments, other relevant factors.

mode#

The value that occurs most frequently in a dataset.

mutually exclusive#
numeracy#

The ability to intelligently evaluate claims involving numbers

percentile#

The \(N\)th percentile, \(0\leq N\leq100\), is the value at which \(N\)% of the values in the dataset fall below, and \((1-N)\)% fall at or above.

See also quantile: percentiles are simply quantiles represented as percent from 0 to 100 rather than a fraction 0.0 to 1.0

population#
quantile#

The \(N\)th quantile, \(0\leq N\leq1\), is the value at which a fraction \(N\) of the values in the dataset fall below, and a fraction \((1-N)\) fall at or above.

See also percentile: percentiles are simply quantiles represented as percent from 0 to 100 rather than a fraction 0.0 to 1.0

range#

The largest value minus the smallest value

sample#
sample space#

The set of all possible outcomes of an experiment.

Note that the “sample” in sample space has a different meaning than the term sample when used in the context of population vs. sample, e.g. in quantities such as the sample mean and sample variance.

For more: the Wikipedia page.

skewness#

A measure of how asymmetric values are about the mean value.

We will only be concerned with its sign.

  • Zero skew: values are evenly split above and below the mean.

  • Positive skew: more values are below the mean than above it, and there’s a long positive tail

  • Negative skew: more values are above the mean than below it, and there’s a long negative tail

standard deviation#
tail#

The far reaches of a distribution

uncertainty quantification#
variance#
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