Mean and Standard Deviation

Anytime a process is mentioned, there are two separate attributes that needs to be understood. Process ‘population’ and process ‘sample’. Population of a process includes all individual units of that process while sample of a process means a random sample of individual units of the process.

In most cases, the process is measured based on a sample since it is practically impossible, time consuming and unnecessary to measure ALL individual units of a process. So the process attribute is determined based on the measurement of a sample set of units taken from the population.

It is assumed that the sample of the population reflects the characteristics of the population. Sample selection should be done carefully in order to meet this assumption.

Mean

Mean of a process is nothing but the average value of individual units. Mean of the population is called the population mean (μ) and mean of the sample is called the sample mean (Xbar)

Standard Deviation

Standard deviation is the spread of the individual values about the mean. It can also be stated as the average difference of individual values from the mean. A low standard deviation indicates that the individual data points lie close to the mean while a high standard deviation indicates that the data points are spread out further from the mean.

The symbol ‘σ’ is used to denote the standard deviation for the population while the symbol ‘s’ is used to denote the standard deviation of the sample.

See attached figure for the mathematical calculation of σ and s.

Figure1: Calculation of σ and s

There is a slight difference in the mathematical formula use to calculate σ and s.

If the population size is N, N is used in the denominator to calculate the σ. But for a sample size of n, n-1 is used in the denominator to calculate s.

Explanation for this is here. Consider a population. The spread of the population is normal. Now consider a sample. Due to the normal property of the population, the probability of the samples to be closer to the mean is high. The samples have a tendency to lie closer to the centre.

Therefore, the variation of the sample is less than the variation of the population and is therefore considered ‘biased’. Therefore we use the denominator n-1 to compensate for this and make the variation bigger (and closer to that of the population) and hence remove this bias. This is statistically proven.

As the sample size increases, the sample variance will get closer to the population and n vs. n-1 will become less significant.

Variance

Variance is the measure of the amount of variation for a set of values. Variance is the square of std. deviation. We use standard deviation instead of variance due to the addictive power of the standard deviation. Variance is squared (a higher degree than the unit level) and is not in the same unit level as the individual units.