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Statistical considerations on how many observations to make
Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample. The sample
Sample_size_determination
Selection of data points in statistics
number sampling Sample size determination Sampling (case studies) Sampling bias Sampling distribution Sampling error Sortition Survey sampling The textbook
Sampling_(statistics)
Statistical methods for comparing samples
interval as an alternative to the hypothesis testing method. Sample size determination is the act of choosing the number of observations to include in
Two-proportion_Z-test
Statistical error
Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to
Sampling_error
Statistical hypothesis test
x ¯ {\displaystyle {\bar {x}}} is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test
Student's_t-test
Statistical measure of how far values spread from their average
a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N
Variance
Statistical measure of the magnitude of a phenomenon
an effect size is a quantitative measure of the magnitude of a phenomenon. It can refer to the value of a statistic calculated from a sample of data, the
Effect_size
Statistical model validation technique
Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how
Cross-validation_(statistics)
Empirical law on the variance of species in a habitat
the test statistic, s2 is the variance of the sample, m is the mean of the sample and n is the sample size. Jn is asymptotically normally distributed with
Taylor's_law
Middle quantile of a data set or probability distribution
n {\displaystyle n} is the sample size: Sample median ∼ N ( μ = m , σ 2 = 1 4 n f ( m ) 2 ) {\displaystyle {\text{Sample median}}\sim {\mathcal {N}}{\left(\mu
Median
Measure of the shape of a function
expected value of the raw sample moment is equal to the kth raw moment of the population, if that moment exists, for any sample size n. It is thus an unbiased
Moment_(mathematics)
Sampling from a population which can be partitioned into subpopulations
subgroups' sample sizes proportional to the amount of data available from the subgroups, rather than scaling sample sizes to subgroup sizes (or to their
Stratified_sampling
Statistical method
a computer, sampling from it to form a new sample (called a 'resample' or bootstrap sample) that is also of size N. The bootstrap sample is taken from
Bootstrapping_(statistics)
Measure of the asymmetry of random variables
of the sample skewness is thus approximately 6 / n {\displaystyle 6/n} for sufficiently large samples. More precisely, in a random sample of size n from
Skewness
Statistical relationship
the sample means of X {\displaystyle X} and Y {\displaystyle Y} , and s x {\displaystyle s_{x}} and s y {\displaystyle s_{y}} are the corrected sample standard
Correlation
Sampling methodology in statistics
number of interviews and costs given the desired accuracy. For a fixed sample size, the expected random error is smaller when most of the variation in the
Cluster_sampling
Statistical measure of variability
robust or outlier-resistant measure of the variability of a univariate sample of quantitative data. For a univariate data set X1, X2, ..., Xn, the MAD
Median_absolute_deviation
Graphical representation of the distribution of numerical data
(intervals) are adjacent and are typically (but not required to be) of equal size. Histograms give a rough sense of the density of the underlying distribution
Histogram
Statistical hypothesis test
population based on a sample of data, or to compare the locations of two populations using two matched samples. The one-sample version serves a purpose
Wilcoxon_signed-rank_test
Statistical property
the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases
Standard_error
Measure of variation in statistics
square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's
Standard_deviation
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
{\displaystyle c_{\rm {v}}\,} itself. For many practical purposes (such as sample size determination and calculation of confidence intervals) it is s l n {\displaystyle
Coefficient_of_variation
Set of statistical processes for estimating the relationships among variables
the mean square error (MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data
Regression_analysis
Measure of distance between two proportions
Clinical significance Cohen's d Cohen's g Odds ratio Effect size Sample size determination Cohen, Jacob (1988). Statistical Power Analysis for the Behavioral
Cohen's_h
Statistical measure of association
Cramér's V is equal to the absolute value of Phi coefficient. Let a sample of size n of the simultaneously distributed variables A {\displaystyle A} and
Cramér's_V
Statistical hypothesis test
hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether
Chi-squared_test
Inverse of the average of the inverses of a set of numbers
population size and E is the expectation operator. Assuming that the variance is not infinite and that the central limit theorem applies to the sample then
Harmonic_mean
Range to estimate an unknown parameter
population mean and the sample size, respectively. Suppose X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} is an independent sample from a normally distributed
Confidence_interval
Study of health and disease within a population
random error in an epidemiological study. The first is to increase the sample size of the study. In other words, add more subjects to your study. The second
Epidemiology
Method of statistical inference
is false. Such considerations can be used for the purpose of sample size determination prior to the collection of data. An example of Neyman–Pearson
Statistical_hypothesis_test
Estimator for quality of a statistical model
populations are the same. We are given a random sample from each of the two populations. Let m be the size of the sample from the first population. Let m1 be the
Akaike_information_criterion
Complete set of items that share at least one property in common
sample mean may differ from the population mean, especially for small samples. The law of large numbers states that the larger the size of the sample
Statistical_population
Position that there is no relationship between two phenomena
completely. For such a hypothesis the sampling distribution of any statistic is a function of the sample size alone. Composite hypothesis Any hypothesis
Null_hypothesis
Test of normality in frequentist statistics
of this method for higher sample sizes lies in the calculation of the coefficients vector a, more specifically in the size of V which requires the storage
Shapiro–Wilk_test
Experiment methodology
online social-media platforms, obtaining a large sample size is trivial. In other cases, large sample sizes are obtained by increasing the experiment enrollment
A/B_testing
Concept in inferential statistics
problem Sample size Texas sharpshooter fallacy (gives examples of tests where the significance level was set too high) Sirkin, R. Mark (2005). "Two-sample t
Statistical_significance
Human research survey of public opinion
Opinion Research World Association for Public Opinion Research Sample size determination Survey methodology Straw poll Swing (politics) Types of democracy
Opinion_poll
Function of the observed sample results
alone" and that "a p-value, or statistical significance, does not measure the size of an effect or the importance of a result", and "does not provide a good
P-value
Statistic measuring inter-rater agreement for categorical items
"The Kappa Statistic in Reliability Studies: Use, Interpretation, and Sample Size Requirements". Physical Therapy. 85 (3): 257–268. doi:10.1093/ptj/85
Cohen's_kappa
Nonparametric measure of rank correlation
the Pearson correlation coefficient between the rank variables. For a sample of size n , {\displaystyle \ n\ ,} the n {\displaystyle \ n\ } pairs
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Metric for fit of statistical models
the upper limit for bin i, Yl = the lower limit for bin i, and N = the sample size The resulting value can be compared with a chi-square distribution to
Goodness_of_fit
Number taken as representative of a list of numbers
{\displaystyle x_{i}} and w i {\displaystyle w_{i}} are the mean and size of sample i {\displaystyle i} respectively. In other applications, they represent
Average
Measure of linear correlation
The square of the sample correlation coefficient is typically denoted r2 and is a special case of the coefficient of determination. In this case, it estimates
Pearson correlation coefficient
Pearson_correlation_coefficient
Number of values in the final calculation of a statistic that are free to vary
symbolize degrees of freedom but modern usage typically reserves n for sample size. When reporting the results of statistical tests, the degrees of freedom
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Data visualization
interquartile range (IQR) of the sample and is inversely proportional to the square root of the size of the sample. However, there is an uncertainty
Box_plot
Collection of statistical models
Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions
Analysis_of_variance
Statistical test comparing two probability distributions
test whether a sample came from a given reference probability distribution (one-sample K–S test), or to test whether or not two samples came from the same
Kolmogorov–Smirnov_test
Probability distribution of the possible sample outcomes
given sample size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, the sampling procedure
Sampling_distribution
Family of statistical methods based on sampling of available data
approximating the sampling distribution of an estimator. The two key differences to the bootstrap are: the resample size is smaller than the sample size and resampling
Resampling_(statistics)
Type of statistical measure over subsets of a dataset
the initial filling of the FIFO / circular buffer the sampling window is equal to the data-set size thus k = n {\displaystyle k=n} and the average calculation
Moving_average
Study of convergence properties of statistical estimators
statistical tests. Within this framework, it is often assumed that the sample size n may grow indefinitely; the properties of estimators and tests are then
Asymptotic theory (statistics)
Asymptotic_theory_(statistics)
Design of tasks
strategy utilized to carry out research Robust parameter design Sample size determination – Statistical considerations on how many observations to make
Design_of_experiments
Comparison of two distributions
and purpose. One choice, given a sample of size n, is k / n for k = 1, …, n, as these are the quantiles that the sampling distribution realizes. The last
Q–Q_plot
Study of survey methods
concentrating on human-research surveys, survey methodology studies the sampling of individual units from a population and associated techniques of survey
Survey_methodology
Bayes estimator Bayes factor Bayesian inference bias 1. Any feature of a sample that is not representative of the larger population. 2. The difference
Glossary of probability and statistics
Glossary_of_probability_and_statistics
Unit of information
Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power Survey methodology Sampling Cluster
Data
Approximation method in statistics
Laplace, after proving the central limit theorem, used it to give a large sample justification for the method of least squares and the normal distribution
Least_squares
Statistics concept
the errors, S n {\displaystyle S_{n}} represents the sample standard deviation for a sample of size n, and unknown σ, and the denominator term S n / n {\displaystyle
Errors_and_residuals
Type of statistics
human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatment or exposure group)
Descriptive_statistics
Generates a forecast of future values of a time series
It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential
Exponential_smoothing
Method of logical reasoning
because the sample is non-random and the sample size is very small. Statistical generalizations are also called statistical projections and sample projections
Inductive_reasoning
Statistic quantifying the association between two events
{\displaystyle V_{N}} . One could take a random sample of fifty villagers, but quite possibly such a random sample would not include anybody with the disease
Odds_ratio
Processes that maintain quality at a constant level
Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power Survey methodology Sampling Cluster
Quality_control
Term in statistical hypothesis testing
statistic and significance level), the sample size (more data tends to provide more power), and the effect size (effects or correlations that are large
Power_(statistics)
Generalization of the one-dimensional normal distribution to higher dimensions
converges very slowly to the limiting normal distribution. For medium size samples ( 50 ≤ n < 400 ) {\displaystyle (50\leq n<400)} , the parameters of the
Multivariate normal distribution
Multivariate_normal_distribution
Fourth standardized moment in statistics
g_{1}=m_{3}/m_{2}^{3/2}} is the corresponding sample skewness. The variance of the sample kurtosis of a sample of size n from the normal distribution is var
Kurtosis
Measure of the joint variability
probability distribution, and (2) the sample covariance, which, in addition to serving as a descriptor of the sample, also serves as an estimated value of
Covariance
Application of statistical techniques to biological systems
inferences about the population. So, the sample might catch the most variability across a population. The sample size is determined by several things, since
Biostatistics
Non-parametric method for testing whether samples originate from the same distribution
whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends
Kruskal–Wallis_test
Probabilistic problem-solving algorithm
1996 MEZEI, M (December 31, 1986). "Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias". Journal of Computational Physics
Monte_Carlo_method
Statistic which divides a data set into 100 parts and analyzes it as a percentage
will be expressed in kilograms or pounds. In the limit of an infinite sample size, the percentile approximates the percentile function, the inverse of
Percentile
Diagnostic plot of binary classifier ability
example of random guessing is a decision by flipping coins. As the size of the sample increases, a random classifier's ROC point tends towards the diagonal
Receiver operating characteristic
Receiver_operating_characteristic
Kth smallest value in a statistical sample
the kth order statistic of a statistical sample is equal to its kth-smallest value. Given a sample of size n {\displaystyle n} , the kth order statistic
Order_statistic
Statistical property
regression, it retains the R-squared value which is then multiplied by the sample size, and then becomes the test statistic for a chi-squared distribution (and
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Numerical measure of a statistical relationship between variables
may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution
Correlation_coefficient
Concept in machine learning
fit the training data). Or, more precisely, it is the maximum number of samples on which the model/training procedure achieves approximately on average
Double_descent
Form of scientific experiment
statistically significant effect on the treated in a given test. But as the sample size increases, the same RCT may be able to demonstrate a significant effect
Randomized_controlled_trial
Statistical measure to determine how suited data is for factor analysis
to determine how suited data is for factor analysis. The test measures sampling adequacy for each variable in the model and the complete model. The statistic
Kaiser–Meyer–Olkin_test
Statistical sampling technique
hypercube sampling (LHS) is a statistical method for generating a near-random sample of parameter values from a multidimensional distribution. The sampling method
Latin_hypercube_sampling
Statistical phenomenon
to mediocrity) is the phenomenon where if one sample of a random variable is extreme, the next sampling of the same random variable is likely to be closer
Regression_toward_the_mean
Statistical test
variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's t-test may be more appropriate
Z-test
Value that appears most often in a set of data
the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval
Mode_(statistics)
Measure of covariance of components of a random vector
in sample j of the random function X ( t ) {\displaystyle X(t)} . The expected values needed in the covariance formula are estimated using the sample mean
Covariance_matrix
Statistical method for resampling
Given a sample of size n {\displaystyle n} , a jackknife estimator can be built by aggregating the parameter estimates from each subsample of size ( n −
Jackknife_resampling
Study of collection and analysis of data
associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Statistical measurement processes
Statistics
Method used in statistics, pattern recognition, and other fields
regression, discriminant analysis can be used with small sample sizes. It has been shown that when sample sizes are equal, and homogeneity of variance/covariance
Linear_discriminant_analysis
Bias in causal inference
Similarly, "over-stratification" of input data within a study may reduce the sample size in a given stratum to the point where generalizations drawn by observing
Confounding
Fundamental theorem in probability theory and statistics
X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} denote a statistical sample of size n {\displaystyle n} from a population with expected value (average)
Central_limit_theorem
Process of using data analysis for predicting population data from sample data
a limit theorem as being approximately true for large sample sizes, we commit an error the size of which is unknown. [. . .] Realistic information about
Statistical_inference
Method of data analysis
{\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows
Principal_component_analysis
Statistical modeling method
their parameters do not have good interpretations. Furthermore, when the sample size is not large, none of their parameters can be accurately estimated by
Linear_regression
Probability distribution
Calculating the confidence interval Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom
Student's_t-distribution
Experiment in which information about the test is masked to reduce bias
of patients and clinicians reduces effect size. Researchers concluded that unblinding inflates the effect size in antidepressant trials. Some researchers
Blinded_experiment
Statistical hypothesis test
of K {\displaystyle K} groups and N {\displaystyle N} is the overall sample size. This F-statistic follows the F-distribution with degrees of freedom
F-test
Non-parametric statistic used to estimate the survival function
enough sample size, approaches the true survival function for that population. The value of the survival function between successive distinct sampled observations
Kaplan–Meier_estimator
Statistical principle
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic
Sufficient_statistic
Linear regression model with a single explanatory variable
of determination ("R squared") is equal to r x y 2 {\displaystyle r_{xy}^{2}} when the model is linear with a single independent variable. See sample correlation
Simple_linear_regression
Statistical transformation
by 1 N − 3 , {\displaystyle {1 \over {\sqrt {N-3}}},} where N is the sample size, and ρ is the true correlation coefficient. This transformation, and
Fisher_transformation
Type of average of a collection of numbers
arithmetic mean of a sample is always between the largest and smallest values in that sample. The arithmetic mean of any amount of equal-sized number groups
Arithmetic_mean
Variable representing a random phenomenon
mathematical function in which the domain is the set of possible outcomes in a sample space (e.g. the set { H , T } {\displaystyle \{H,T\}} (which are the possible
Random_variable
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