Mean and Standard Deviation

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Mean and standard deviation

The median is known as a measure of location; that is, it tells us where the data are. As stated in , we do not need to know all the exact values to calculate the median; if we made the smallest value even smaller or the largest value even larger, it would not change the value of the median. Thus the median does not use all the information in the data and so it can be shown to be less efficient than the mean or average, which does use all values of the data. To calculate the mean we add up the observed values and divide by the number of them. The total of the values obtained in Table 1.1 was 22.5 , which was divided by their number, 15, to give a mean of 1.5. This familiar process is conveniently expressed by the following symbols:

(pronounced "x bar") signifies the mean; x is each of the values of urinary lead; n is the number of these values; and σ , the Greek capital sigma (our "S") denotes "sum of". A major disadvantage of the mean is that it is sensitive to outlying points. For example, replacing 2.2 by 22 in Table 1.1 increases the mean to 2.82 , whereas the median will be unchanged.
As well as measures of location we need measures of how variable the data are. We met two of these measures, the range and interquartile range, in Chapter 1.
The range is an important measurement, for figures at the top and bottom of it denote the findings furthest removed from the generality. However, they do not give much indication of the spread of observations about the mean. This is where the standard deviation (SD) comes in.
The theoretical basis of the standard deviation is complex and need not trouble the ordinary user. We will discuss sampling and populations in Chapter 3. A practical point to note here is that, when the population from which the data arise have a distribution that is approximately "Normal" (or Gaussian), then the…...

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