Intro and Basics Reference Page
How Deviant and Mean Are Your Data?

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Intro to Basic Statistics

Engineers must collect, analyze, and interpret large amounts of data. To evaluate a data set, a number of basic statistical calculations are commonly used, including the determination of the mean, the standard deviation, and the confidence interval around the mean.

This set of videos introduces the fundamentals of these calculations and demonstrates how to perform them by hand, on a spreadsheet, with MATLAB, and with R.

Part 1 Just the Facts

Part 1 of Just the Facts introduces the concepts of the population (or true) mean and standard deviation, as well as the sample mean and sample standard deviation, and explains how to calculate them.

Visuals

Part 1 The Full Story

Part 1 of The Full Story examines common conceptions and misconceptions about the mean and standard deviation, explains how to calculate both population and sample statistics, and prepares the viewer for their later use in uncertainty analysis.

Visuals

How Deviant and Mean Are Your Data? Part 2

Part 2 explains how to calculate the standard error, confidence interval, and Student’s \(t\), and how those quantities are used in practice.

Visuals

Basic Statistics by Hand

Basic Statistics by Hand demonstrates how to calculate the sample mean, sample standard deviation, standard error, and confidence interval by hand using only a calculator and a Student’s \(t\)-table.

Visuals

Basic Statistics on a Spreadsheet

Basic Statistics on a Spreadsheet demonstrates how to calculate the sample mean, sample standard deviation, standard error, and confidence interval first by using the spreadsheet as a calculator and then by using built-in statistical functions.

Visuals

Basic Statistics with MATLAB

ImportantNot a DOFPro video

This is NOT a DOFPro video. It may contain copyrighted material and references to proprietary software.

Basic Statistics in MATLAB demonstrates how to calculate the sample mean, sample standard deviation, standard error, and confidence interval using both the command line and an .m file.

Basic Statistics in R

Basic Statistics in R demonstrates how to calculate the sample mean, sample standard deviation, standard error, and confidence interval using both the command line and a script.

Examples and Definitions

Definitions

True or Population Mean, \(\mu\)
The arithmetic mean of an entire population. In practice, it is usually unknown and must be estimated from sample data.
Sample Mean, \(\bar{x}\)
The arithmetic mean of a sample of \(N\) measurements:

\[ \bar{x}=\frac{1}{N}\sum_{i=1}^{N} x_i \]

True Variance, \(\sigma^2\)
A measure of the spread of an entire population about the population mean:

\[ \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2 \]

True or Population Standard Deviation, \(\sigma\)
The square root of the population variance:

\[ \sigma = \sqrt{\sigma^2} \]

Sample Variance, \(S^2\)
A measure of the spread of a sample about the sample mean:

\[ S^2=\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2 \]

Sample Standard Deviation, \(S\)
The square root of the sample variance:

\[ S = \sqrt{S^2} \]

Error, \(\varepsilon_i\)
The difference between an observed value and the true or population mean:

\[ \varepsilon_i = x_i - \mu \]

Residual, \(e_i\)
The difference between an observed value and the sample mean:

\[ e_i = x_i - \bar{x} \]

Estimated Standard Error, \(S_{\bar{x}}\)
An estimate of the standard deviation of the sample mean:

\[ S_{\bar{x}}=\frac{S}{\sqrt{N}} \]

Student’s \(t\)
A probability distribution used when estimating confidence intervals from small samples, especially when the population standard deviation is unknown. As the sample size increases, the \(t\)-distribution approaches the normal distribution.
Confidence Interval, \(\bar{x} \pm \lambda\)
An interval centered on the sample mean that is expected, with a specified probability, to contain the true mean.

For the form used in these videos,

\[ \lambda = t S_{\bar{x}} \]

so that the confidence interval is

\[ \bar{x} \pm t S_{\bar{x}} \]