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Do you calculate uncertainty? Are you tired of looking up statistical functions in books and online?

1. Statistics Cheat Sheet Hypothesis Testing
2. Statistics Cheat Sheet For Dummies
3. Statistics Cheat Sheet College
4. Statistics Cheat Sheet Pdf

Statistics of a Sample or Parameters of a Population. For all of these, remember to use σ or s based on whether you have the whole population or just a sample. Stats of a Plain List of Numbers. Enter numbers in L1, then 1-VarStats L1. Check n before you look at anything else. Details: from a List of Numbers in Chapter 3.

1. Statistics to get a feeling for the data. For example: Where is the center of the data located? How spread out is the data? How correlated are the data from two variables? The most common descriptive statistics are in the following table, along with their formulas and a short description of what each one measures. Statistically Figuring Sample Size.
2. Algebra Cheat Sheet 15 Writing Inequalities Look for ‘clue’ words: 1. For the clue words, ‘the product of’ place the constant before the variable. Do not use a sign. The clue words ‘more than’ and ‘less than’ indicate inverted order. If there are no clue words, write the equation in the order that the words appear.
3. Then, download my handy statistics cheat sheet! My statistics cheat sheet has every statistical function and formula you will need to calculate uncertainty and evaluate your results. All packed into 2 pages! Check out all of the great functions that I packed into this statistics cheat sheet. Average Variance Standard Deviation Determining.

Then, download my handy statistics cheat sheet!

My statistics cheat sheet has every statistical function and formula you will need to calculate uncertainty and evaluate your results. All packed into 2 pages!

Check out all of the great functions that I packed into this statistics cheat sheet.

• 
  Average
  Variance
  Standard Deviation
  Determining Sample Size
  Degrees of Freedom
  Sum of Squares
  Root Sum of Squares
  Pooled Variance
  Effective Degrees of Freedom
  Linear Interpolation
  Linear Regression
  Sensitivity Coefficients
  
• 
  Covariance
  Correlation
  Correlation Coefficient
  Coefficient of Determination
  Central Limit Theorem
  Standard Deviation of the Mean
  Confidence Intervals
  Z-Score
  T-Score
  Student’s T Distribution
  Probability Distributions
  

Plus, I have included a section for probability distributions and divisors to help you quickly reduce uncertainty.

Get the FREE Statistics Cheat Sheet for Uncertainty Analysis now! Just fill out the form to the right and I will immediately email it to you.

Parameter estimation

Definitions

Random sample A random sample is a collection of $n$ random variables $X_1, ..., X_n$ that are independent and identically distributed with $X$.

Estimator An estimator is a function of the data that is used to infer the value of an unknown parameter in a statistical model.

Bias The bias of an estimator $hat{theta}$ is defined as being the difference between the expected value of the distribution of $hat{theta}$ and the true value, i.e.:

Remark: an estimator is said to be unbiased when we have $E[hat{theta}]=theta$.

Estimating the mean

Sample mean The sample mean of a random sample is used to estimate the true mean $mu$ of a distribution, is often noted $overline{X}$ and is defined as follows:

Remark: the sample mean is unbiased, i.e $E[overline{X}]=mu$.

Characteristic function for sample mean The characteristic function for a sample mean is noted $psi_{overline{X}}$ and is such that:

Central Limit Theorem Let us have a random sample $X_1, ..., X_n$ following a given distribution with mean $mu$ and variance $sigma^2$, then we have:

Estimating the variance

Sample variance The sample variance of a random sample is used to estimate the true variance $sigma^2$ of a distribution, is often noted $s^2$ or $hat{sigma}^2$ and is defined as follows:

Remark: the sample variance is unbiased, i.e $E[s^2]=sigma^2$.

Chi-Squared relation with sample variance Let $s^2$ be the sample variance of a random sample. We have:

Confidence intervals

Definitions

Confidence level A confidence interval with confidence level $1-alpha$ is such that $1-alpha$ of the time, the true value is contained in the confidence interval.

Confidence interval A confidence interval $CI_{1-alpha}$ with confidence level $1-alpha$ of a true parameter $theta$ is such that:

With the notation of the example above, a possible $1-alpha$ confidence interval for $theta$ is given by $CI_{1-alpha} = [x_1, x_2]$.

Confidence interval for the mean

When determining a confidence interval for the mean $mu$, different test statistics have to be computed depending on which case we are in. The table below sums it up.

Note: a step by step guide to estimate the mean, in the case when the variance in known, is detailed here.

Confidence interval for the variance

The single-line table below sums up the test statistic to compute when determining the confidence interval for the variance.

Note: a step by step guide to estimate the variance is detailed here.

Hypothesis testing

General definitions

Type I error In a hypothesis test, the type I error, often noted $alpha$ and also called 'false alarm' or significance level, is the probability of rejecting the null hypothesis while the null hypothesis is true. If we note $T$ the test statistic and $R$ the rejection region, then we have:

Type II error In a hypothesis test, the type II error, often noted $beta$ and also called 'missed alarm', is the probability of not rejecting the null hypothesis while the null hypothesis is not true. If we note $T$ the test statistic and $R$ the rejection region, then we have:

p-value In a hypothesis test, the $p$-value is the probability under the null hypothesis of having a test statistic $T$ at least as extreme as the one that we observed $T_0$. We have:

Remark: the example below illustrates the case of a right-sided $p$-value.

Non-parametric test A non-parametric test is a test where we do not have any underlying assumption regarding the distribution of the sample.

Testing for the difference in two means

Statistics Cheat Sheet Hypothesis Testing

The table below sums up the test statistic to compute when performing a hypothesis test where the null hypothesis is:

Testing for the mean of a paired sample

We suppose here that $X_i$ and $Y_i$ are pairwise dependent. By noting $D_i=X_i-Y_i$, the one-line table below sums up the test statistic to compute when performing a hypothesis test where the null hypothesis is:

Testing for the median

Median of a distribution We define the median $m$ of a distribution as follows:

Sign test The sign test is a non-parametric test used to determine whether the median of a sample is equal to the hypothesized median.

By noting $Vunderset{H_0}{sim}mathcal{B}(n,p=frac{1}{2})$ the number of samples falling to the right of the hypothesized median, we have:

$―$ If $npgeqslant5$, we use the following test statistic:

$―$ If $np < 5$, we use the following fact:

$chi^2$ test

Goodness of fit test Let us have $k$ bins where in each of them, we observe $Y_i$ number of samples. Our null hypothesis is that $Y_i$ follows a binomial distribution with probability of success being $p_i$ for each bin.

We want to test whether modelling the problem as described above is reasonable given the data that we have. In order to do this, we perform a hypothesis test:

$chi^2$ statistic for goodness of fit In order to perform the goodness of fit test, we need to compute a test statistic that we can compare to a reference distribution. By noting $k$ the number of bins, $n$ the total number of samples, if we have $np_igeqslant5$, the test statistic $T$ defined below will enable us to perform the hypothesis test:

Trends test

Number of transpositions In a given sequence, we define the number of transpositions, noted $T$, as the number of times that a larger number precedes a smaller one.

Example: the sequence ${1,5,4,3}$ has $T=3$ transpositions because $5>4, 5>3$ and $4>3$

Test for arbitrary trends Given a sequence, the test for arbitrary trends is a non-parametric test, whose aim is to determine whether the data suggest the presence of an increasing trend:

If we note $x$ the number of transpositions in the sequence, the $p$-value is computed as:

Remark: the test for a decreasing trend of a given sequence is equivalent to a test for an increasing trend of the inversed sequence.

Regression analysis

In the following section, we will note $(x_1, Y_1), ..., (x_n, Y_n)$ a collection of $n$ data points.

Simple linear model Let $X$ be a deterministic variable and $Y$ a dependent random variable. In the context of a simple linear model, we assume that $Y$ is linked to $X$ via the regression coefficients $alpha, beta$ and a random variable $esimmathcal{N}(0,sigma)$, where $e$ is referred as the error. We have:

Regression estimation When estimating the regression coefficients $alpha, beta$ by $A, B$, we obtain predicted values $hat{Y}_i$ as follows:

Sum of squared errors By keeping the same notations, we define the sum of squared errors, also known as SSE, as follows:

Statistics Cheat Sheet For Dummies

Method of least-squares The least-squares method is used to find estimates $A,B$ of the regression coefficients $alpha,beta$ by minimizing the SSE. In other words, we have:

Notations Given $n$ data points $(x_i, Y_i)$, we define $S_{XY},S_{XX}$ and $S_{YY}$ as follows:

Least-squares estimates When estimating the coefficients $alpha, beta$ with the least-squares method, we obtain the estimates $A, B$ defined as follows:

Sum of squared errors revisited The sum of squared errors defined above can also be written in terms of $S_{YY}$, $S_{XY}$ and $B$ as follows:

Key results

When $sigma$ is unknown, this parameter is estimated by the unbiased estimator $s^2$ defined as follows:

The estimator $s^2$ has the following property:

The table below sums up the properties surronding the least-squares estimates $A, B$ when $sigma$ is known or not:

Statistics Cheat Sheet College

Correlation analysis

Correlation coefficient The correlation coefficient of two random variables $X$ and $Y$ is noted $rho$ and is defined as follows:

Sample correlation coefficient The correlation coefficient is in practice estimated by the sample correlation coefficient, often noted $r$ or $hat{rho}$, which is defined as:

Testing for correlation In order to perform a hypothesis test with $H_0$ being that there is no correlation between $X$ and $Y$, we use the following statistic:

Statistics Cheat Sheet Pdf

Fisher transformation The Fisher transformation is often used to build confidence intervals for correlation. It is noted $V$ and defined as follows:

By noting $V_1=V-frac{z_{frac{alpha}{2}}}{sqrt{n-3}}$ and $V_2=V+frac{z_{frac{alpha}{2}}}{sqrt{n-3}}$, the table below sums up the key results surrounding the correlation coefficient estimate: