Analytical Chemistry module 2

For purely random errors, what do the results of many measurements of an experimental quantity follow?

a Gaussian distribution, a bell shaped curve

for infinite measurements, what do we define?

true population mean as μ (mu) with a spread of standard deviation σ (sigma)

for finite number of measurements, what do we define?

mathematical mean x̄, with a spread of standard deviation s

for a large number of replicate measurements,

• calculated mean x̄ approaches true mean μ

• calculated spread s approaches true spread σ

what is the formula for mean/average?

x̄ = (Σxi) / n

what do different symbols mean in the mean/average formula?

• x̄ is the sample mean/average

• Σ is the addition of things ( of all measured values xi)

• xi or x are the individual values in the data set

• n is the total count of values in the sample

what is the standard deviation formula?

s = √ [ Σ(xi - x̄)² / (n - 1) ]

what do different symbols mean in the standard deviation formula?

• s is the standard deviation

• Σ is the sum of ( addition of everything together)

• xi is the individual value in the data set

• x̄ is the sample mean/average

• n is the number of values in the sample

the smaller s ( standard deviation)

equals to a more precise measurement

• this doesnt relate to accuracy

what is variance?

its the square of standard deviation

s²

what is another name for percent relative standard deviation ( %RSD)?

coefficient of variation

what is the percent relative standard deviation formula?

(s / x̄) * 100

how do you calculate the range of data set?

highest value minus lowest value

what can you use %RSD for?

when you need to compare relative spread of two very different means

• both means can have different s, but the same %RSD

what is the formula for Gaussian distribution?

• σ is the standard deviation

• μ is the mean

• x is the input value

what is z = +2 ?

it is a measure value at x̄ + 2s

what is z = -2?

is a measured value at x̄ - 2s

what is the probability of measuring a value in a certain range equal to?

to area under normal error curve in that range

what does 𝝁±𝟏𝝈 cover in standard deviation and probability?

it covers 68.3% of expected values

what does 𝝁±2𝝈 cover in standard deviation and probability?

it covers 94.5% of expected values

what does 𝝁±3𝝈 cover in standard deviation and probability?

covers 99.7% of expected values

what does standard deviation measure?

measures amount of dispersion for a subject set of data from the mean

what does standard uncertainty measure?

measures how far the sample mean of the data is likely to be from the true population mean

what is the formula for standard uncertainty?

u_x becomes smaller with what?

with increased number of replicate measurements (n)

what is more replicate measurements equal to?

to a higher certainty ( smaller u_x )

what does confidence interval represent?

a spread around the sample mean

what can you say with a certain confidence? ( like 90%, 95%, or 99% confidence)

that true population mean is found within the confidence interval

what is the confidence interval formula?

value “t” is called student’s t

where can student’s t be found?

can be found as a tabled value for certain degrees of freedom and the desired confidence level

student’s t values get lower with what?

with more replicate measurements

student’s t values get higher with what?

with increasing confidence level

degrees of freedom

related to number of replicate measurements (n) used to calculate mean

what is the formula for degrees of freedom?

deg. of freedom = n - 1

when will u_x and tu_x decrease?

with more replicates

s approaches true standard deviation when?

when there are more replicates

x̄ changes and approaches true mean when?

when there are more replicates

standard uncertainty (u_x ) or confidence intervals often used to quantify what?

random error associated to a measured sample mean

how do you state sample mean and its error ( uncertainty) ?

x̄ +- u_x ( n = ……)

or x̄ +- tu_x (n =……)

also state the type of error used and number of replicate measurements used

both u_x and tu_x are both what?

they are both absolute errors (e)

what is the formula for % relative error? ( %e)

%e = ( e / x̄ ) x 100

how does the absolute error of a mean determines number f sig figs?

error is rounded to 1 sig fig

• if tu_x is rounded to two sig figs, then x̄ is rounded to two sig figs as well

what is the rounding exception?

error with first digit of “1” can round to two sig figs

• if tu_x is rounded to two sig figs, then x̄ is rounded to two sig figs as well

what is the Null Hypothesis? H₀

is a statement that two data sets are drawn from populations with properties such as standard deviation and mean

if H₀ is true …

variations between different data sets arise from random variations in measurements

what is alternative hypothesis? H_A

where data sets are different

• different mean and standard deviation

when do we reject H_o

reject for less than 5% probability that two data sets are drawn from same population

• same mean and standard deviation

if H_o is rejected, then …

we accept H_A

what is an outliner?

a point that is far from other points of the same measured quantity

what is Grubbs test used for?

is a statistical test used to decide if an outliner should be rejected from data set

what is the formula for Grubbs test?

standard deviation and mean are calculated including x_out

when do you keep x_out?

if G_calc < G_crit

when do you reject x_out ?

if G_calc > G_crit

what do we do if x_out is rejected?

then mean and standard deviation need to be recalculated without x_out

what is the F-test used for?

its used to decide if standard deviation of two data sets is significantly different from each other

how do you calculate critical F-value?

• larger standard deviation should be divided by smaller one

if F_calc > F_table then what happens next?

we reject H_o

• this means that both sets have different standard deviation

what is the student’s t-test?

it is used to decide if the means of two or more data sets are significantly different from each other

• compare a measured result with a known value

• compare replicate measures

• paired t-test for comparing individual differences

what is F-test used for?

used to compare the variances of two or more populations or samples to see if they are significantly different

what is the t-value formula?

what is the formula for s_pooled ?

what is the formula to calculate t value with s_pooled ?

• usually used when standard deviations are different

what are the two tailed t-test?

are unusual t-tests

• H_O is rejected when the result of measurement has less than 5% probability of being observed

a measured value is significantly different from known value in two tailed t test

it happens when the value is beyond the 2.5% line on either end

one tailed t-test

• H₀ is rejected when the result of measurement less than 5% probability of being observed on one particular end of the probability curve