How Deviant and Mean Are Your Data?

Intro and Basics Part 2
Just the Facts

DOFPro Team

How Deviant and Mean Are Your Data?

Intro and Basics Part 2
The Full Story

DOFPro Team

Scale Video TBA

Ruler Video TBS

Voltmeter Video TBA

Thermistor Video TBA

Reporting Uncertainty

\(m = 1.03 \pm 0.03\ \mathrm{kg}\text{ (95% confidence)}\)

\(u = 2.36 \pm 0.04\ \mathrm{\frac{m}{s}}\text{ (95% confidence)}\)

\(mu = 2.43 \pm 0.08\ \mathrm{\frac{(kg\cdot m)}{s} }\text{ (95% confidence)}\)

\(\frac{1}{2}mu^2 = 2.87 \pm 0.13\ \mathrm{J}\text{ (95% confidence)}\)

A future video will teach you how to propagate the uncertainty. \(\textbf{Remember!}\) A responsible engineer will report uncertainty in the calculated quantity due to the uncertainties present in measured quantities.

Reporting Functional Uncertainty

CC BY-SA 3.0, link

\(F_\mathrm{s} = k(x_\mathrm{n}-x_\mathrm{f})\)

\(F_\mathrm{_s} = 30.00 \pm 2.50\ \mathrm{(95\%\ conf.)\ N/m}(x\mathrm{_n}-x\mathrm{_f})\)

True Mean and Standard Deviation

True value of a measured quantity \(:=\)1 True Mean or Population Mean.

\(\mu \equiv\) True or Population Mean,

i.e., \(\mu =\ \)average value from infinite-precision, full population measurement (almost never known).

Distribution of data about the true mean \(:=\) True Standard Deviation or Population Standard Deviation.

\(\sigma \equiv\) True or Population Standard Deviation

i.e., \(\sigma =\ \)standard deviation given true mean and infinite-precision data points (also almost never known).

Sample Mean and Residuals

Set of Measurements

\(\{x_1,\ x_2,\ \ldots,\ x_N\}\)

e.g., \(x_1 = 43.15\)

A set of \(N\) measurements

implies a set of

\(N\) independent equations

\(\bar{x}\)

\(=\frac{\sum_{i=1}^Nx_i}{N}\)

\(\approx\mu\)

\(\bar{x} \equiv\) sample mean

How do we estimate \(\mu - \bar{x}\)?

Can we calculate the error

\(\varepsilon_i =\mu-x_i\)?

Set of Errors

\(\{\varepsilon_1,\ \varepsilon_2,\ \ldots,\ \varepsilon_N \}\)

We don’t know \(\mu\), so…

Calculate the residuals:

\(e\mathrm{_i} = \bar{x} - x_i\)

Set of Residuals

\(\{e_1,\ e_2,\ \ldots,\ e_N \}\)

estimate of the errors

based on \(\bar{x}\), not \(\mu\)

Sample Variance and Standard Deviation

True variance

\(\sigma^2=\frac{\displaystyle{\sum_{i=1}^N \varepsilon^2_i}}{N}\)

Sample variance

\(S^2=\frac{\displaystyle{\sum_{i=1}^N e^2_i}}{N-1}\)

\(\bar{x},\ e,\) and \(S^2\) are calculated from the \(x_i\).

\(\mu,\ \varepsilon,\) and \(\sigma^2\) are not.

We lose one degree of freedom, or an independent equation, by using the sample mean in our calculation.

The true standard deviation, \(\sigma=\sqrt{\sigma^2}\).

The sample standard deviation, \(S=\sqrt{S^2}\).

Are We There Yet?

Can we estimate \(\mu - \bar{x}\) yet?

No, but we can estimate the spread of our data from \(S\).

The Takeaways

  1. Responsible engineers always estimate and report the uncertainty in any measurement or quantity calculated from measurements.
  2. The most common estimate for the central value of a set of measurements is the sample mean.
  3. The most commonly reported value for the spread of a set of measurements is the sample standard deviation.
  4. Most people don’t know or care about the difference between the sample mean and standard deviation and the population or true mean and standard deviation.

The discussion continues in How Deviant and Mean Are Your Data? Intro and Basics Part 2






Thanks for watching!
The Just The Facts companion video is in the link in the upper left. The next video in the series is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description to find previous and following videos in this series.

The DOFPro Team






Thanks for watching!
The Full Story companion video is in the link in the upper left. The first of the How To videos, Basic Statistics by Hand is in the upper right. To learn more about Chemical and Thermal Processes, visit the website linked in the description to find additional previous and following videos in this series.

The DOFPro Team