Instant Notes: Analytical Chemistry

found within ± 1 sof the mean, approximately 95% will be found within ± 2 sand

approximately 99.7% within ± 3 s. More practically convenient levels, as shown

in Figure 3, are those corresponding to 90%, 95% and 99% of the population,

which are defined by ±1.64s, ±1.96sand ±2.58srespectively. Many statistical

tests are based on these probability levels.

The value of the population standard deviation, s, is given by the formula

s=



(1)

where xirepresents any individual value in the population and Nis the total

number of values, strictly infinite. The summation symbol, S, is used to show

that the numerator of the equation is the sum for i = 1 to i =Nof the squaresof

the deviationsof the individual xvalues from the population mean, m. For very

large sets of data (e.g., when N >50), it may be justifiable to use this formula as

the difference between sand swill then be negligible. However, most analytical

data consists of sets of values of less than ten and often as small as three.

Therefore, a modified formula is used to calculate an estimated standard

deviation, s, to replace s, and using an experimental mean, x

_

, to replace the

population mean, m:

s=



(2)



i=N

i= 1

(xi−x

_

)^2





N− 1



i=N

i= 1

(xi−m)^2





N

B2 – Assessment of accuracy and precision 29

–4s –3s –2s –1s 01 s 2 s 3 s 4 s

Relative

frequency (

y/

N

)

–1.29s +1.29s

80%

–4s–3s–2s–1s 01 s 2 s 3 s 4 s

Relative

frequency (

y/

N

)

–1.64s +1.64s

90%

–4s –3s –2s –1s 01 s 2 s 3 s 4 s

Relative

frequency (

y/

N

)

–1.96s +1.96s

95%

–4s–3s–2s–1s 01 s 2 s 3 s 4 s

Relative

frequency (

y/

N

)

–2.58s +2.58s

99%

Fig. 3. Proportions of a population within defined limits of the mean.