One way to find the standard error of the mean is to have lots of samples. First, the average for each sample is found. Then the average and standard deviation of those sample averages is found. The standard deviation for all the sample averages is the standard error of the mean. This can be a lot of work. Sometimes it is too difficult or costly to have lots of samples.
Another way to find the standard error of the mean is to use an equation that needs only one sample. Standard error of the mean is usually estimated by the standard deviation for a sample from the whole group (sample standard deviation), divided by the square root of the sample size:
where
In general, the larger the sample, the closer the estimated standard error of the mean is to the actual standard error of the mean. As a rule of thumb, there should be at least six measurements in a sample. Then the standard error of the mean for the sample will be within 5% of the actual standard error of the mean (that is, if the whole group were measured).
There is another equation to use if the number of measurements is for 5% or more of the whole group:
There are special equations to use if a sample has less than 20 measurements.
Sometimes, a sample may come from one place even though the whole group may be spread out, while other times, a sample may be made in a short time period when the whole group covers a longer time. In which case, the numbers in the sample are not independent, and special equations are used to try to correct for this.
"A practical result:" One can become more sure of an average value by having more measurements in a sample. Then the standard error of the mean will be smaller because the standard deviation is divided by a bigger number. However, to make the uncertainty (standard error of the mean) in an average value half as big, the sample size (n) needs to be four times bigger. This is because the standard deviation is divided by the square root of the sample size. To make the uncertainty one-tenth as big, the sample size (n) needs to be one hundred times bigger.
Standard errors are easy to calculate and commonly used because:
The relative standard error (RSE) is the standard error divided by the average. This number is smaller than one. Multiplying it by 100% gives it as a percentage of the average. This helps to show whether the uncertainty is important or not.
For example, consider two surveys of household income that both result in a sample average of $50,000. If one survey has a standard error of $10,000 and the other has a standard error of $5,000, then the relative standard errors are 20% and 10%, respectively. The survey with the lower relative standard error is better, because it has a more precise measurement (the uncertainty is smaller).
In fact, people who need to know average values often decide how small the uncertainty should be—before they decide to use the information. For example, the U.S. National Center for Health Statistics does not report an average if the relative standard error exceeds 30%. NCHS also requires at least 30 observations for an estimate to be reported.
For example, there are many redfish in the water of the Gulf of Mexico. To find out how much a 42 cm long redfish weighs on average, it is not possible to measure all of the redfish that are 42 cm long. Instead, it is possible to measure some of them. The fish that are actually measured are called a sample. The table shows weights for two samples of redfish, all 42 cm long. The average (mean) weight of the first sample is 0.741 kg. The average (mean) weight of the second sample is 0.735 kg—a little bit different from the first sample. Each of these averages is a little bit different from the average that would come from measuring every 42 cm long redfish (which is not possible anyway).
The uncertainty in the mean can be used to know how close the average of the samples are to the average that would come from measuring the whole group. The uncertainty in the mean is estimated as the standard deviation for the sample, divided by the square root of the number of samples minus one. The table shows that the uncertainties in the means for the two samples are very close to each other. Also, the relative uncertainty is the uncertainty in the mean divided by the mean, times 100%. The relative uncertainty in this example is 2.38% and 2.50% for the two samples.
Knowing the uncertainty in the mean, one can know how close the sample average is to the average that would come from measuring the whole group. The average for the whole group is between a) the average for the sample plus the uncertainty in the mean, and b) the average for the sample minus the uncertainty in the mean. In this example, the average weight for all of the 42 cm long redfish in the Gulf of Mexico is expected to be 0.723–0.759 kg based on the first sample, and 0.717–0.753 based on the second sample.
Really Wild Animals
Really Wild Animals was a National Geographic television series. It had Dudley Moore as the voice of "Spin".
Adaptive radiation
Adaptive radiation is rapid evolutionary radiation. It is an increase in the number and diversity of species in each lineage. It produces more new species, and those species live in a wider range of habitats.
Some definitions phrase it in terms of a single clade: "Adaptive radiation is the rapid proliferation of new taxa from a single ancestral group". However, in the most striking cases, such as occurred in the Triassic after the greatest extinction event in Earth history, many lines underwent rapid radiation simultaneously. This must have something to do with the availability of ecological niches and relative little competition.
The Ediacaran biota were the result of an early metazoan radiation. The greatest radiation of all took place early in the Cambrian period, when most of our animal phyla evolved: see List of animal phyla.
With less competition, groups diversify to fill available habitats and niches. This is an evolutionary process driven by natural selection.
The term was introduced and discussed by George Gaylord Simpson, the palaeontologist who contributed to the modern evolutionary synthesis. Others prefer not to use the term. Robert L Carroll prefers to use the term "major evolutionary transitions", though it turns out that all or most of these could also be described as adaptive radiations. Others use terms like macroevolution, or even megaevolution, as if the processes are different from those which occur below species level. It is part of evolutionary theory that all processes take place at the level of populations. All agree, though, that the speed of evolution does change, however it is measured.
Records of timing are bedevilled by gaps in the fossil record, often at those crucial early stages when numbers are low and geographical distribution is severely restricted. "In reality, there are long periods within nearly every lineage for which the fossil record remains unknown". These gaps affect our knowledge of timing, and of changes in body shape and function.
Nevertheless, when several distinctly new lines appear within a short period, it seems reasonable to say that the rate of change has been surprisingly fast. An example would be the appearance of new reptilian groups in the Upper Triassic. Using the term 'reptile' broadly, the groups include dinosaurs, pterosaurs, Chelonia (turtles), crocodylomorphs (early Crocodilia), phytosaurs, and ichthyosaurs a bit earlier (Middle Triassic).
These radiations occurred after the great Permian–Triassic extinction event that ended the Palaeozoic era. The Triassic itself had several lesser (but still significant) extinctions. Unfortunately, the Triassic has the poorest fossil record of the whole of the Mesozoic era.