The various techniques by which one may effectively treat the scientific data normally obtained in actual analytical procedures are enumerated below :

**STATISTICAL TREATMENT OF FINITE SAMPLES**

The various techniques by which one may effectively treat
the scientific data normally obtained in actual analytical procedures are
enumerated below :

It is the average of a series of results. The mean of a
finite number of measurements, *x*_{l},
*x*_{2}, *x*_{3}, *x*_{4},
........*, x _{n}*, is commonly
represented as

It is evident that the mean of *n* results is * **√**n *times more reliable than any
one of the individual results.* *Therefore,
there exists a diminishing return from accumulating more and more replicate
meaurements. In other words, the mean of 9 results is 3 times as reliable as 1
result in measuring central tendency (*i.e.*,
the value about which the individual results tend to cluster) ; the mean of 16
results is 4 times as reliable etc.

The median of an even number of results is nothing but
the average of the ‘two middle values’ pro-vided the results are listed in
order ; whereas for an odd number of results the median is the ‘middle value’
itself. However, the ‘*mean*’ and the ‘*median*’ are exactly identical in the
case of a truly symmetrical distribu-tion. In short, median is an useful
measure specifically when dealing with very small samples.

It is the average of the differences between the
individual results and the mean. It is regarded as a measure of variability. In
the case of a small number of observations the average deviation is found to be
not quite significant statistically. The average or mean distribution may be
calculated by adopting the following steps, namely :

(*i*) To find the
differences between individual results and the mean, without considering the
+ve or –ve sign,

(*ii*) To add
these individual deviations, and

(*iii*) To divide
by the number of results (*i.e., n*).

Hence, an ‘average deviation’ may be expressed as :

It is the distance from the mean to the point of
inflexion of the normal distribution curve. In compari-son to the average
deviation the ‘**standard deviation**’
is usually considered to be much more useful and meaningful statistically. For
a finite number of values it is normally symbolised as ‘S’, and may be
expressed as follows :

In a situation, where ‘*n*’ is fairly large, say to the extent of 50 or more, it hardly
matters whether the denominator in the above expression is either *n* – 1 or *n*; however, the former (*i.e.,
n* – 1) is strictly correct.

The coefficient of variation (ν)
is simply the standard deviation(s) expressed as a percentage of the mean ( *x’* ) as stated below :

The variance is the square of the standard deviation(s) *i.e.*, *s*^{2}. However, the former is fundamentally more important
in statistics than the latter, whereas the latter is employed more frequently
in the treatment of chemical data.

** Example : **The normality of a solution of
sodium hydroxide as determined by an ‘analyst’ by FOUR

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Pharmaceutical Drug Analysis: Errors In Pharmaceutical Analysis and Statistical Validation : Statistical Treatment of Finite Samples |

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