Most of the time the population mean differs from the mean of each
individual sample taken from the population.

Consider the following example where absorbance of a solution
containing a known concentration of substance A was determined by U.V./Visible
spectrometer. The absorbance of the solution was measured three times during
each experiment and the average value, standard deviation s and 2s was
calculated (see Table I.1).

**Table I.1: Absorbance values measured for a solution A using a U.V./Visible spectrometer. Three consecutive measurements were recorded during each experiment.**

The mean values
from Table I.1 were plotted in the graph shown in Fig. 1 below and 2s (2 *
standard deviation s) is shown for each mean. As can you see, the mean value
calculated for the absorbance of A in each experiment (sample of the
population) differs from the mean of the population (red line). Please also
note that the interval - solid line above and below the mean in each experiment
(Fig. 1) - of each mean (the range of values the mean can
take with a certain probability) does not always contain the population mean
(experiments 5, 9, 10, 13).

From the above
discussion the following question arises:

*How can we assess the accuracy of the population mean? Within wich boundaries the true value of the population mean is contained?*
Such boundaries are
called confidence intervals or confidence levels. Confidence
intervals in a sense give us the range of values that the population mean can
take with a certain degree of confidence – usually 90%, 95% or 99%.

Most of the time
we look at 95% confidence intervals but all of them have similar
interpretation:

they are limits
constructed such that a certain percentage of the time 95% in this case the
value of the population mean will fall within these limits.

*How can we calculate confidence intervals?*
In order to calculate the confidence interval, we need to
know the limits within which 95% of means will fall. If we will assume a normal
distribution with a mean = 0 and s = 1
we can use the z-scores with values between -1.96 and +1.96 (remember that 95% of z-scores fall
between these two values). Remember also that we can convert values to z-scores
using the formula:

z = (x - x̅ )/ s
(1)

If we know that the upper limit will be z = +1.96 then from (1) we
get:

(x - x̅ )/ s = 1.96 and x
= x̅ + 1.96 * s (this is the upper boundary – limit)

and

(x - x̅ )/ s = -1.96 and x
= x̅ - 1.96 * s (this is the lower boundary – limit)

Therefore, the confidence interval can easily be calculated once the
standard deviation s of the mean and the mean are known. The general form of
the confidence interval is given below:

_{critical}* s (2)

where x is the upper or lower value the mean of the population can take with a certain degree of confidence, x̅ is the mean value of the population of measurements, z

_{critical}is the z critical value from statistical tables (see Table I.2) at a certain confidence level (usually 95%) and s is the standard deviation of the measurements.Confidence Level
(%) |
z-critical value |

99 |
2.58 |

95 |
1.96 |

90 |
1.645 |

50 |
0.675 |

**Table I.2: Critical values of z at different confidence levels**

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