refer to table 28-4. what is the adult male population in meditor?

The median is known as a measure of location; that is, it tells us where the data are. Equally stated in , nosotros do non need to know all the verbal values to calculate the median; if we fabricated the smallest value even smaller or the largest value fifty-fifty larger, it would not alter the value of the median. Thus the median does not use all the information in the data and so information technology can be shown to be less efficient than the mean or average, which does use all values of the data. To summate the mean we add together up the observed values and divide by the number of them. The total of the values obtained in Tabular array ane.ane was 22.v

, which was divided by their number, 15, to give a mean of one.5. This familiar process is
conveniently expressed by the following symbols:

(pronounced "ten bar") signifies the mean; x is each of the values of urinary lead; due north 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 ane.ane increases the hateful to ii.82 , whereas the median will exist 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 one.

The range is an of import measurement, for figures at the top and bottom of it denote the findings furthest removed from the generality. However, they do not requite much indication of the spread of observations nigh the hateful. This is where the standard departure (SD) comes in.

The theoretical ground of the standard deviation is complex and need not trouble the ordinary user. Nosotros will discuss sampling and populations in Chapter three. 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 standard deviation provides a useful basis for interpreting the data in terms of probability.

The Normal distribution is represented by a family of curves divers uniquely by two parameters, which are the mean and the standard difference of the population. The curves are ever symmetrically bell shaped, but the extent to which the bong is compressed or flattened out depends on the standard deviation of the population. However, the mere fact that a bend is bell shaped does not mean that it represents a Normal distribution, because other distributions may accept a similar sort of shape.

Many biological characteristics conform to a Normal distribution closely enough for it to be usually used – for instance, heights of adult men and women, claret pressures in a healthy population, random errors in many types of laboratory measurements and biochemical data. Effigy 2.1 shows a Normal curve calculated from the diastolic claret pressures of 500 men, mean 82 mmHg, standard deviation 10 mmHg. The ranges representing [+-1SD, +12SD, and +-3SD] near the hateful are marked. A more extensive gear up of values is given in Tabular array A of the print edition.

Figure ii.1

The reason why the standard difference is such a useful measure out of the besprinkle of the observations is this: if the observations follow a Normal distribution, a range covered by one standard deviation above the hateful and one standard deviation below information technology

includes virtually 68% of the observations; a range of 2 standard deviations above and two below (

) about 95% of the observations; and of 3 standard deviations above and three below (

) about 99.7% of the observations. Consequently, if nosotros know the mean and standard deviation of a set of observations, we tin can obtain some useful data by unproblematic arithmetic. By putting ane, two, or 3 standard deviations to a higher place and beneath the mean nosotros can estimate the ranges that would exist expected to include about 68%, 95%, and 99.seven% of the observations.

Standard deviation from ungrouped data

The standard deviation is a summary measure of the differences of each observation from the mean. If the differences themselves were added up, the positive would exactly rest the negative and then their sum would exist goose egg. Consequently the squares of the differences are added. The sum of the squares is and so divided past the number of observations minus oneto give the hateful of the squares, and the square root is taken to bring the measurements back to the units we started with. (The division past the number of observations minus oneinstead of the number of observations itself to obtain the mean foursquare is because "degrees of freedom" must be used. In these circumstances they are one less than the total. The theoretical justification for this need not trouble the user in practice.)

To gain an intuitive feel for degrees of freedom, consider choosing a chocolate from a box of n chocolates. Every time we come to cull a
chocolate we accept a choice, until we come up to the last ane (unremarkably 1 with a nut in it!), and so we have no pick. Thus nosotros have north-1 choices, or "degrees of freedom".

The calculation of the variance is illustrated in Table 2.1 with the 15 readings in the preliminary report of urinary lead concentrations (Table i.2). The readings are ready out in cavalcade (1). In column (2) the difference betwixt each reading and the mean is recorded. The sum of the differences is 0. In column (iii) the differences are squared, and the sum of those squares is given at the bottom of the column.

Table 2.1

The sum of the squares of the differences (or deviations) from the hateful, ix.96, is now divided by the total number of observation minus ane, to give the variance.Thus,

In this instance we detect:

Finally, the square root of the variance provides the standard deviation:

from which nosotros get

This procedure illustrates the structure of the standard deviation, in particular that the ii farthermost values 0.1 and 3.2 contribute most to the sum of the differences squared.

Estimator procedure

Most inexpensive calculators have procedures that enable one to calculate the mean and standard deviations straight, using the "SD" mode. For instance, on modern Casio calculators i presses SHIFT and '.' and a little "SD" symbol should appear on the display. On earlier Casios one presses INV and MODE , whereas on a Sharp second F and Stat should be used. The data are stored via the K+ push button. Thus, having set the figurer into the "SD" or "Stat" mode, from Tabular array ii.1 nosotros enter 0.1 One thousand+ , 0.4 M+ , etc. When all the data are entered, we can cheque that the correct number of observations have been included past Shift and due north, and "fifteen" should be displayed. The mean is displayed by Shift and

and the standard deviation by Shift and

. Avoid pressing Shift and AC betwixt these operations as this clears the statistical memory. There is another push on many calculators. This uses the divisor n rather than n – ane in the calculation of the standard deviation. On a Abrupt figurer

is denoted

, whereas

is denoted s. These are the "population" values, and are derived bold that an entire population is available or that interest focuses solely on the data in hand, and the results are not going to exist generalised (see Chapter
3 for details of samples and populations). Every bit this state of affairs very rarely arises,

should be used and ignored, although even for moderate sample sizes the difference is going to be modest. Recollect to render to normal fashion earlier resuming calculations because many of the usual functions are not bachelor in "Stat" mode. On a modernistic Casio this is Shift 0. On earlier Casios and on Sharps one repeats the sequence that call up the "Stat" mode. Some calculators stay in "Stat"
mode even when switched off.Mullee (1) provides advice on choosing and using a computer. The calculator formulas use the relationship

The right hand expression can exist hands memorised past the expression mean of the squares minus the mean square". The sample variance

is obtained from

The above equation can exist seen to be true in Table two.i, where the sum of the square of the observations,

, is given as 43.7l.

We thus obtain

the same value given for the total in column (3). Care should be taken because this formula involves subtracting ii large numbers to get a small one, and can atomic number 82 to incorrect results if the numbers are very large. For example, endeavor finding the standard deviation of 100001, 100002, 100003 on a computer. The correct answer is 1, but many calculators volition give 0 because of rounding error. The solution is to subtract a large number from each of the observations (say 100000) and calculate the standard deviation on the remainders, namely ane, 2 and 3.

Standard deviation from grouped data

We can also calculate a standard departure for discrete quantitative variables. For example, in improver to studying the lead concentration in the urine of 140 children, the paediatrician asked how ofttimes each of them had been examined by a doctor during the year. After collecting the information he tabulated the data shown in Table ii.2 columns (1) and (2). The hateful is calculated past multiplying column (1) past column (ii), adding the products, and dividing by the total number of observations. Tabular array ii.2

Every bit we did for continuous data, to calculate the standard difference we square each of the observations in turn. In this instance the observation is the number of visits, but considering nosotros have several children in each class, shown in column (ii), each squared number (column (4)), must be multiplied by the number of children. The sum of squares is given at the human foot of column (five), namely 1697. We and so utilise the calculator formula to discover the variance:

and

.Note that although the number of visits is not Normally distributed, the distribution is reasonably symmetrical about the mean. The approximate 95% range is given by

This excludes two children with no visits and
six children with six or more visits. Thus there are eight of 140 = 5.seven% outside the theoretical 95% range.Note that it is common for discrete quantitative variables to have what is known as skeweddistributions, that is they are not symmetrical. One clue to lack of symmetry from derived statistics is when the mean and the median differ considerably. Another is when the standard deviation is of the same order of magnitude equally the mean, but the observations must be non-negative. Sometimes a transformation will
convert a skewed distribution into a symmetrical one. When the data are counts, such as number of visits to a doctor, often the square root transformation will assist, and if at that place are no zero or negative values a logarithmic transformation will render the distribution more symmetrical.

Data transformation

An anaesthetist measures the hurting of a procedure using a 100 mm visual analogue calibration on vii patients. The results are given in Table 2.3, together with the log etransformation (the ln push button on a estimator). Table 2.3

The information are plotted in Figure two.two, which shows that the outlier does non appear so extreme in the logged data. The hateful and median are 10.29 and 2, respectively, for the original data, with a standard departure of 20.22. Where the mean is bigger than the median, the distribution is positively skewed. For the logged data the mean and median are 1.24 and one.10 respectively, indicating that the logged information have a more symmetrical distribution. Thus it would exist better to analyse the logged transformed data
in statistical tests than using the original scale.Figure two.two

In reporting these results, the median of the raw data would exist given, only it should be explained that the statistical exam wascarried out on the transformed data. Note that the median of the logged data is the same every bit the log of the median of the raw data – all the same, this is not truthful for the mean. The mean of the logged data is not necessarily equal to the log of the mean of the raw data.
The antilog (exp or

on a estimator) of the mean of the logged data is known as the geometric hateful,and is oftentimes a
better summary statistic than the mean for information from positively skewed distributions. For these information the geometric hateful in three.45 mm.

Between subjects and within subjects standard divergence

If repeated measurements are made of, say, blood pressure on an individual, these measurements are likely to vary. This is within subject field, or intrasubject, variability and we tin calculate a standard divergence of these observations. If the observations are shut together in time, this standard deviation is often described every bit the measurement mistake.Measurements fabricated on different subjects vary according to betwixt discipline, or intersubject, variability. If many observations were made on each private, and the average taken, then we can assume that the intrasubject variability has been averaged out and the variation in the average values is due solely to the intersubject variability. Single observations on individuals clearly contain a mixture of intersubject and intrasubject variation. The coefficient of variation(CV%) is the intrasubject standard deviation divided by the hateful, expressed equally a percentage. It is often quoted equally a measure of repeatability for biochemical assays, when an analysis is carried out on several occasions on the same sample. Information technology has the reward of existence independent of the units of measurement, but also numerous theoretical disadvantages. It is usually nonsensical to employ the coefficient of variation as a mensurate of between discipline variability.

Common questions

When should I use the hateful and when should I use the median to describe my
data?

Information technology is a commonly held misapprehension that for Normally distributed information 1 uses the mean, and for not-Normally distributed data one uses the median. Alas this is not so: if the data are Ordinarily distributed the mean and the median will be close; if the data are not Unremarkably distributed then both the mean and the median may requite useful data. Consider a variable that takes the value 1 for males and 0 for females. This is conspicuously not Normally distributed. However, the mean gives the proportion of males in the group, whereas the median simply tells us which group independent more than l% of the people. Similarly, the mean from ordered categorical variables can exist more useful than the median, if the ordered categories can be given meaningful scores. For example, a lecture might be rated as 1 (poor) to v (excellent). The usual statistic for summarising the result would be the mean. In the situation where at that place is a small group at ane extreme of a distribution (for instance, annual income) and so the median will exist more "representative" of the distribution. My information must have values greater than nix and yet the mean and standard deviation are about the same size. How does this happen? If data have a very skewed distribution, and then the standard deviation volition be grossly inflated, and is not a good measure out of variability to use. Equally we have shown, occasionally a transformation of the data, such every bit a log transform, will render the distribution more symmetrical. Alternatively, quote the interquartile range.

References

one. Mullee K A. How to choose and utilize a calculator. In: How to exercise it 2.BMJ
Publishing Group, 1995:58-62.

Exercises

Exercise two.one

In the campaign confronting smallpox a doctor inquired into the number of times 150 people aged 16 and over in an Ethiopian village had been vaccinated. He obtained the following figures: never, 12 people; in one case, 24; twice, 42; three times, 38; four times, 30; five times, 4. What is the mean number of times those people had been vaccinated and what is the standard deviation?Answer

Exercise ii.2

Obtain the mean and standard deviation of the data in and an approximate
95% range.Answer

Do 2.3

Which points are excluded from the range mean – 2SD to mean + 2SD? What
proportion of the data is excluded? Answers
Affiliate 2 Q3.pdfAnswer

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Source: https://www.bmj.com/about-bmj/resources-readers/publications/statistics-square-one/2-mean-and-standard-deviation

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