Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

Statistics (Gray)

Last updated

Jan 29, 2021
Save as PDF
- Standardization
- Statistics (Mullaugh)

Contributor
Analytical Sciences Digital Library

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

In-class Exercises, Class 2

Name: _________________________

1) How many significant figures in each of the following numbers?

82.059
0.0003
200300
300.0

2) Write the numbers above in scientific notation maintaining the same number of significant figures.

3) To how many significant figures should each answer be rounded?

40.5 / 1020.2 = 0.039698
0.002 kg + 98.3 kg = 98.3002 kg
1001 cm – 20.86 cm = 980.14 cm

4) Calculate the answer to the correct number of significant figures.

10^2.259 =?

What is the pH if the [H⁺] = 7.245 x 10^-6 M

In-class Exercises, Class 3

Name: _________________________

1) Find the absolute and percent relative uncertainty and express each answer with the correct number of significant figures.

9.23 (±0.03) + 4.21 (±0.02) – 3.26 (±0.06) = ?

91.3 (±1.0) * 40.3 (±0.2) / 21.1 (±0.2) = ?

[6.2 (±0.2) – 4.2 (±0.1)] / 9.43 (±0.05) =?

2) Why is the data reported below incorrect? Report the data correctly.

5.4359 M ± 0.00671 M

In-class Exercises, Class 4

Name: _________________________

1) For the following data set, calculate the 99% confidence interval.

Sample	Value
1	3.0541
2	2.9845
3	3.0512
4	2.99584

2) A standard reference material is certified to contain 94.6 ppm of an organic contaminant in soil. Your analysis gives values of 98.6, 98.4, 97.2, 94.6 and 96.2 ( $\bar{x}$ = 97.0, s = 1.65₅). Do your results differ from the expected result at the 95% confidence level? If you made one more measurement at 94.5, would your conclusion change (new $\bar{x}$ = 96.5₈₃, s = 1.79₈)?

In-class Exercises, Class 5 and 6

Name: _________________________

1) Given the following data, are the precisions of the two datasets significantly different at the 95% confidence level? What is the t_calc for these two means? Are the two means significantly different at the 95% confidence level?

$\bar{X}_1$ = 80.34 s=0.0548 N=4

$\bar{X}_2$ = 80.46 s=0.2793 N=5

2) Using both the Grubbs and the Q-test, can the last data point be excluded at the 95% CI?

Data
0.1503
0.1505
0.1496
0.1493
0.1496
0.1497
0.1507
0.1617

$\bar{x}$ = 0.1514

s = 0.004181

In-class Exercises, Class 7

Name: _________________________

1) Match the formula below to the statistical method.

Methods:

Mean
Standard deviation
Variance
Confidence intervals
Relative standard deviation
t-test, case 1
t-test, case 2; variances are equal
t-test, case 2; variances are not equal
compare 2 variances
reject an outlier

$\text{Degrees of freedom}=\dfrac{(s_1^2/n_1 + s_2^2/n_2)^2}{\dfrac{(s_1^2/n_1)^2}{n_1-1}+\dfrac{(s_2^2/n_2)^2}{n_2-1}} \nonumber$
$s_{pooled}=\sqrt{\dfrac{s_1^2(n_1-1)+s_2^2(n_2-1)}{n_1+n_2-2}} \nonumber$	$s = \sqrt{\dfrac{\sum_{i=1}^{N}(X_i-\bar{X})^2}{N-1}} \nonumber$
$=100\times\dfrac{s}{\bar{x}} \nonumber$	$=s^2 \nonumber$
$\mu=\bar{x}\pm \dfrac{ts}{\sqrt{N}} \nonumber$	$\|\mu-\bar{x}\|>\dfrac{ts}{\sqrt{N}} \nonumber$
$t_{cal}=\dfrac{\bar{x}_1-\bar{x}_2}{s_{pooled}}\sqrt{\dfrac{n_1n_2}{n_1+n_2}} \nonumber$	$t_\text{calculated}=\dfrac{\|\bar{x}_1-\bar{x}_2\|}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} \nonumber$
$\bar{X}=\dfrac{\sum_{i=1}^{N}X_i}{N} \nonumber$	$Q=\dfrac{gap}{range} \nonumber$
$F_{cal}=\dfrac{s_1^2}{s_2^2} \nonumber$	$G_{calc} = \dfrac{\|questionable\: value - \bar{x}\|}{s} \nonumber$

Contributors and Attributions

Sarah Gray, Stockton University (sarah.gray@stockton.edu)
Sourced from the Analytical Sciences Digital Library

In-class Exercises, Class 2

In-class Exercises, Class 3

In-class Exercises, Class 4

In-class Exercises, Class 5 and 6

In-class Exercises, Class 7

Contributors and Attributions

Support Center

How can we help?