When talking about the normal distribution, it's useful to think of the standard deviation as being steps away from the mean. One step to the right or one step to the left is considered one standard deviation away from the mean. Two steps to the left or two steps to the right are considered two standard deviations away from the mean. Likewise, three steps to the left or three steps to the right are considered three standard deviations from the mean. The standard deviation of a dataset is simply the number or distance that constitutes a complete step away from the mean.
Adding or subtracting the standard deviation from the mean tells us the scores that constitute a complete step. Below I've put together a distribution with a mean of 58 and a standard deviation of 5.
In stats terminology, we would say that a score of 63 falls exactly "one standard deviation above the mean. Let me explain. Take a look at the picture below. The shaded area represents the total area that falls between one standard deviation above and one standard deviation below the mean. Those Greek letters are just statistical notation for the mean and the standard deviation of a population. Regardless of what a normal distribution looks like or how big or small the standard deviation is, approximately 68 percent of the observations or 68 percent of the area under the curve will always fall within two standard deviations one above and one below of the mean.
Can you guess what proportion falls between the mean and just one standard deviation above it? If you guessed 34, you must be familiar with division. Now take a look at the next picture. It's basically the same as the first instance, only this time we're looking at two standard deviations above and below the mean. For any normal distribution, approximately 95 percent of the observations will fall within this area.
Within this range are all of the data values located within one standard deviation above or below of the mean. Finally, These percentages remain the same for all normally distributed data. I've illustrated this principle on the graph below. We can use the fact that our sample birth weight data appear Normally distributed to calculate a reference range. So a reference range for our sample of babies, using the values given in the histogram above, is:. A baby's weight at birth is strongly associated with mortality risk during the first year and, to a lesser degree, with developmental problems in childhood and the risk of various diseases in adulthood.
If the data are not Normally distributed then we can base the normal reference range on the observed percentiles of the sample, i. In this example, the percentile-based reference range for our sample was calculated as 2. Most reference ranges are based on samples larger than people.
Over many years, and millions of births, the WHO has come up with a normal birth weight range for new born babies. Low birth weight babies are usually defined by the WHO as weighing less than g the 10th centile regardless of gestational age, and large birth weight babies are defined as weighing above kg the 90th centile.
Hence the normal birth weight range is around 2. For our sample data, the 10th to 90th centile range was similar, 2.
If a group of patients is given a new drug for the relief of a particular condition, then the proportion p being successively treated can be regarded as estimating the population treatment success rate.
Thus p also represents a mean. Data which can take only a binary 0 or 1 response, such as treatment failure or treatment success, follow the binomial distribution provided the underlying population response rate does not change. The binomial probabilities are calculated from:. In the above, n! Both 0! This area totals 0. So the probability of eight or more responses out of 20 is 0. For a fixed sample size n the shape of the binomial distribution depends only on.
The number of responses actually observed can only take integer values between 0 no responses and 20 all respond. The binomial distribution for this case is illustrated in Figure 2. The distribution is not symmetric, it has a maximum at five responses and the height of the blocks corresponds to the probability of obtaining the particular number of responses from the 20 patients yet to be treated. It should be noted that the expected value for r , the number of successes yet to be observed if we treated n patients, is nx.
Uniform dispersion is observed in plant species that inhibit the growth of nearby individuals. For example, the sage plant, Salvia leucophylla , secretes toxins, a phenomenon called negative allelopathy. The chemicals kill off surrounding plants in a circle around the individual sage plants, leading to a uniform distance between each plant. Animals that maintain defined territories, such as nesting penguins, also exhibit uniform dispersion.
Random dispersion occurs with dandelion and other plants that have wind-dispersed seeds that germinate wherever they happen to fall in a favorable environment. Clumped dispersion is seen in plants that drop their seeds straight to the ground, such as oak trees, or animals that live in groups, such as schools of fish or herds of elephants. You can also have R automatically do the calculation of the Z score and look up the probability by using the pnorm function with the parameters the value, the mean, and the standard deviation , e.
The table of probabilities for the standard normal distribution gives the area i. This distribution is narrower, so values less than 30 should represent a slightly greater proportion of the population.
Test Yourself. What is the probability that a 60 year old male selected at random from this population will have a BMI less than 40? What is the probability that a male age 60 has BMI greater than 40? What is the probability that a 60 year old male selected at random from this population will have a BMI between 30 and 40? I compute the Z score as follows:. Here the value of interest is below the mean, so the Z score is negative.
The full table of Z scores takes this into account as shown below.
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