Chapter 11: Random Errors in Chemical Analysis

Question 1

Usually, most contributors to __ __ cannot be positively identified. Even if we can identify __ __ __, it is often impossible to measure them because most are so small that they cannot be detected individually. The accumulated effect of the individual uncertainties, however, causes replicate results to __ __ around the __ of the set.

Answer 1

• random error
• random error sources
• fluctuate randomly
• mean

Question 2

• a type of continuous probability distribution for a real-valued random variable.
• a plot that shows the frequency distribution for measurements containing random uncertainties.

Answer 2

Gaussian curve or a normal error curve

Question 3

• ___ in a set of replicate measurements is the difference between the highest and lowest.
  __ of data results directly from an accumulation of all random uncertainties in the experiment.

Answer 3

• spread
• spread

Question 4

a curve that shows a symmetrical distribution of data around the mean of an infinite set of data

Answer 4

Gaussian curve or a normal error curve

Question 5

Sources of random uncertainties in the calibration of a pipet include (4)

Answer 5

1) visual judgement
2) drainage time
3) temperature fluctuations
4) vibrations and drafts

Question 6

A __ is the collection of all measurements of interest to the experimenter, while a __ is a subset of measurements selected from the population.

Answer 6

• population
• sample

Question 7

Gaussian curves can be described
by an equation that contains two
parameters:

Answer 7

• population mean µ
• population standard deviation σ.

Question 8

The term __ refers to quantities such as µ and σ that define a population or distribution. Data values such as x are variables.

Answer 8

parameter

Question 9

The term __ refers to an estimate of a parameter that is made from a sample of data.

Answer 9

statistic

Question 10

If there is no systematic error in the population, the __ __ _is also the true value for the measured quantity.

Answer 10

• population mean µ

Question 11

More often than not, particularly when N is small, x̄ differs from µ because a small sample of data may not exactly represent its __.

Answer 11

• population

Question 12

In most cases we do not know µ and must infer its value from x̄. The probable difference between x̄ and µ decreases rapidly as the number of measurements making up the sample increases; usually, by the time N reaches __ to __, this difference is negligible. Note that the __ __ _is a statistic that estimates the population parameter µ.

Answer 12

• 20 to 30
• sample mean x̄

Question 13

The __ __ __ _, which is a measure of the precision of the population, is given by summing the squares of the deviations from the mean (xi - µ), dividing by the number of measurements N, and taking the square root of the result

Answer 13

population standard deviation σ

Question 14

population standard deviation σ formula

Answer 14

σ = √∑(xi-µ)^2 / N

Question 15

A normal error curve has several general properties:

a) The __ occurs at the central point of maximum frequency,

b) there is a symmetrical distribution
of __ and __ deviations about the maximum

c) there is an exponential decrease in
__ as the __ __ __ increase. Thus, small uncertainties are observed much more often than very large ones.

Answer 15

a.
- mean
b.
- positive
- negative
c.
- frequency
- magnitude of deviations

Question 16

sample standard deviation, s formula

Answer 16

s = √∑(xi-x̄)^2 / N-1

Question 17

The Sample Standard Deviation: A Measure of Precision:

As N becomes larger, __ and __ become better estimators of µ and σ.

Answer 17

• x̄
• s

Question 18

The standard deviation of each mean is known as the __ __ _ _ __ and is given the symbol __.

Answer 18

• standard error of the mean
• s sub m

Question 19

Reliability of s as a Measure of Precision:

When N is greater than about __, s is usually a good estimator of __, and these quantities can be assumed to be identical for most purposes.

Answer 19

• 20
• σ

Question 20

as the sample size increases, the __ __ __ _ tends to provide a more accurate estimate of the __ __ __ _.

Answer 20

• sample standard deviation, s
• population standard deviation, σ

Question 21

Reliability of s as a Measure of Precision:

If we have several subsets of data, a better estimate of the population standard deviation can be obtained by __ (__) the data instead of using only one data set.

Again, we must assume the same sources of random error in all the measurements.

This assumption is usually valid if the samples have similar compositions and have been analyzed in exactly the same way.

We must also assume that the samples are randomly drawn from the same population and thus have a common value of σ.

Answer 21

pooling (combining)