Standard deviation population, calculation

The sample standard deviation, s, provides an estimate of the population standard deviation, a. The (n — 1) term in equations (6.4) and (6.6) is often described as the number of degrees of freedom (frequently represented in statistical tables by the parameter v (Greek letter, pronounced nu ). It is important for judging the reliability of estimates of statistics, such as the standard deviation. In general, the number of degrees of freedom is the number of data points (n) less the number of parameters already estimated from the data. In the case of the sample standard deviation, for example, v = n — 1 since the mean (which is used in the calculation of s) has already been estimated from the same data. 

Most of the observations are likely to be within the range x 2 s, and practically all within the range x 3. . In common with the sample mean, the value of the standard deviation calculated from a set of observations is only an estimate of the true or population value of the standard deviation s becomes a better estimate of population value) as N increases and... 

This example shows that the standard deviation of the sampling distribution is less than that of the population. In fact, this reduction in the variability is related to the sample size used to calculate the sample means. For example, if we repeat the sampling experiment, but this time based on 15 rather than 10 random samples, the resulting standard deviation of the sampling is 0.159, and on 25 random samples it is 0.081. The precise relationship between the population standard deviation a and the standard error of the mean is ... 

NOTE In calculating confidence intervals, o may be substituted for s in Equation 4-6 if you have a great deal of experience with a particular method and have therefore determined its true" population standard deviation. If a is used instead of s, the value of t to use in Equation 4-6 comes from the bottom row of Table 4-2. 

Figure 4-5 illustrates the meaning of confidence intervals. A computer chose numbers at random from a Gaussian population with a population mean (p.) of 10 000 and a population standard deviation (o) of 1 000 in Equation 4-3. In trial 1, four numbers were chosen, and their mean and standard deviation were calculated with Equations 4-1 and 4-2. The 50% confidence interval was then calculated with Equation 4-6, using t = 0.765 from Table 4-2 (50% confidence, 3 degrees of freedom). This trial is plotted as the first point at the left in Figure 4-5a the square is centered at the mean value of 9 526, and the error bar extends from the lower limit to the upper limit of the 50% confidence interval ( 290). The experiment was repeated 100 times to produce the points in Figure 4-5a. 

For Rayleigh s data in Figure 4-6, we suspect that the population standard deviation from air is smaller than that from chemical sources. Using liquations 4-8a and 4-9a, we find that calculated = 21.7 and degrees of freedom = 7.22 7. This value of /cil cllUe(J still far exceeds values in Table 4-2 for 7 degrees of freedom at 95% or 99.9% confidence. 

We estimate crv, the population standard deviation of all y values, by calculating, sv, the standard deviation, for the four measured values of y. The deviation of each value of y,- from... 

The standard deviation of a set of data, usually given the symbol 5, is the square root of the variance. The difference between standard deviation and variance is that the standard deviation has the same units as the data, whereas the variance is in units squared. For example, if the measured unit for a collection of data is in meters (m) then the units for the standard deviation is m and the unit for the variance is m2. For large values of n, the population standard deviation is calculated using the formula ... 

The mean function can be used in various ways. By default this function produces die mean of each column in a matrix, so that mean (W) results in a 1 x 3 row vector containing die means. It is possible to specify which dimension one wishes to take die mean over, the default being die first one. The overall mean of an entire matrix is obtained using the mean function twice, i.e. mean (mean (W) ). Note that the mean of a vector is always a single number whether the vector is a column or row vector. This function is illustrated in Figure A.39. Similar syntax apphes to functions such as min, max and std, but note that the last function calculates the sample rather dian population standard deviation and if employed for scaling in chemometrics, you must convert back to the sample standard deviation, in the current case by typing std(W) /sqrt ( (s (1) ) / (s (1) -1) ), where sqrt is a function that calculates the square root and s contains the number of rows in die matrix. Similar remarks apply to the var function, but it is not necessary use a square root in the calculation. 

Most practical exercises are based on a limited number of individual data values (a sample) which are used to make inferences about the population from which they were drawn. For example, the lead content might be measured in blood samples from 100 adult females and used as an estimate of the adult female lead content, with the sample mean (T) and sample standard deviation (j) providing estimates of the true values of the underlying population mean (/r) and the population standard deviation (c). The reliability of the sample mean as an estimate of the true (population) mean can be assessed by calculating the standard error of the sample mean (often abbreviated to standard error or SE), from ... 

A difficulty is that the population standard deviation is not usually known and can only be approximated for a finite number of measurements by the sample standard deviation s, calculated from (26-4). This difficulty is overcome for gaussian distributions by use of the quantity t (sometimes known as Student s t), defined by... 

The calculated standard deviation s is a good estimator of the population standard deviation cr if the number of measurements is high enough. An equivalent alternative form of Eq. 16.2 is presented in Eq. 16.3 this form is more useful when a nonprogrammable calculator is used. 

When you make statistical calculations, remember that because of the uncertainty in X, a sample standard deviation may differ significantly from the population standard deviation. As N becomes larger, x and s become better estimators of and cr. 

The standard deviation represents an average (of sorts) deviation of each observation from the sample mean. Again, the only reason why we do not call this quantity the average deviation without qualification is that there really are n deviations from the sample mean, but the standard deviation is calculated using the denominator of (r - 1) instead of n. The sample standard deviation is an unbiased estimator of the population standard deviation. For our previous example of 10 age values, the value of the sample standard deviation is 16.8 years (we leave confirmation of this to you). 

Using a calculator for statistics - make sure you understand howto enter individual data values and which buttons will give the sample mean (usually shown as X or x) and sample standard deviation (often shown as population standard deviation (usually shown as [Pg.267]

Rather than calculate from first principles it would be more common to use a calculator or Excel. With a calculator there are often keys for population standard deviations (written as a or xcrn) and sample standard deviations (written as s or xcrn 1). Naturally xa , is the correct key to use for this example and gives a value of 525705. If xcrn was used you would get a standard deviation of 479900. As appealing as using a smaller value is, it is not correct to quote the standard deviation calculated in this way. 

To illustrate the difference between these two calculations, look at figure 2.7, which shows the difference between the 95% point on the normal distribution (z = 1.96) and the corresponding Student /-value for different degrees of freedom. The /-value depends on the degrees of freedom, and asymptotically approaches the value of z as the degrees of freedom tends to infinity. The important point is to note how quickly this happens. For a sample of three measurements there are two degrees of freedom and /o.o5,2 is 4.3, more than twice z0.025-Therefore a confidence interval based on a standard deviation of three results will be twice that if the population standard deviation, a is... 

Figure 2.8 shows the confidence intervals calculated for the means of the random data used earlier. Figure 2.8(a) shows 95% confidence intervals based on the population standard deviation (which we know a= 1) and z-value (1.96), which is of course the same for each value. Figure 2.8(b) is the 95% confidence interval calculated using equation 2.11. For small values of n the Student t interval is much greater than the one based on a knowledge of a, because, as discussed above,... 

Other Ways to Calculate Standard Deviations, Scientific calculators usually have the standard deviation function built in. Many can find the population standard deviation a as well as the sample standard deviation s. For any small set of data, the sample standard deviation should be used. 

The numbers in parentheses are the estimated standard deviations, calculated on the assumption that all averaged values were drawn from the same population. 

To find numerical values for the hmits of this interval we need the value of the population standard deviation. Let us assume one more time that the estimate calculated for the 140 beans is approximately correct. We then have = 0.0363/VIO = 0.0115 g. Finally, recalling that x = 0.1887 g for our sample, we can write... 

Exercise 2.14. Consider the 140 values in Table 2.2 as a single random sample from a normal population. Assume that the standard deviation calculated from these values is identical to the population value. Now, how many beans are there in a 1-kg package ... 

When comparing mean scores of one group of students to specific values (one-sample case) or to mean scores of another group of students (two-sample case), chemical education researchers can calculate z-scores only if the standard deviation of the population as a whole is known. Since population standard deviations are almost never known, researchers must estimate these values using sample standard deviations. When this is done, researchers calculate /-scores instead of z-scores. While the z-distribution (normal distribution) is independent of the number of subjects in the study, /-distributions change based on the... 

In Figure 2.5, two values are reported for the standard deviation that of the population (o) and that of the sample (s). The population standard deviation is calculated as... 

The positive square root of the variance is the population standard deviation, dx-These calculations are illustrated in Example F.3. 

These are called the sample mean and sample standard deviation, as they are estimates of the true population mean and true population standard deviation. It is assumed that the measurement set is not a complete set of possible measurements. The true mean and standard deviation will be denoted by // and c respectively. Although these are printed with many digits from the computer calculation, the real accuracy is expected to be considerably less. The question to be addressed is what can be said about the accuracy of these computed parameters relative to the true mean and true standard deviation One measure of the uncertainty of the mean is the so called standard error given by... 

The calculated bounds on a the population standard deviation at various confidence levels are shown in Table 8.3. Again the increased range that must be specified as the confidence level is increased can be readily seen in the table results. In this case a comparison of the 68.27% bounds from the Chi-Sq analysis shows a reasonable agreement with the standard error bounds. However the 95.45% confidence bounds show a larger difference when compared with the 2 standard error bounds. As the sample size is reduce the standard error bounds provides a less useful estimate of the bounds. For reference, the standard error for the standard deviation is evaluated as ... 

The mean (m) and the standard deviation (a) of the population of localized AE sources over each cell of the (ATi,AT2) histogram are calculated. 

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